Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Pruss, Alexander Robert
2009-02-18T22:39:11Z
1996
Doctor of Philosophy - PhD
University of British Columbia
Symmetrization methods transform functions or sets into functions or sets having desirable
features such as symmetry or some kind of convexity. We consider whether the values of
various functionals do or do not increase under the transformation.
First, we answer in the negative a question of W. K. Hayman (1967) on the precise way that
Green's functions increase under circular rearrangement.
Next we study symmetrization techniques in discrete cases, obtaining convolution-rearrangement
inequalities of the form ...........
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SYMMETRIZATION, GREEN'S FUNCTIONS, HARMONIC MEASURES AND DIFFERENCE EQUATIONS by ALEXANDER ROBERT PRUSS B.Sc. (Honours Scholar's Electives) University of Western Ontario, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming /to,the rsfluir^ cL t^andard THE UNIVERSITY OF BRITISH COLUMBIA April 1996 © Alexander Robert Pruss, 1996 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer-ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Abs t rac t Symmetrization methods transform functions or sets into functions or sets having desirable features such as symmetry or some kind of convexity. We consider whether the values of various functionals do or do not increase under the transformation. First, we answer in the negative a question of W. K. Hayman (1967) on the precise way that Green's functions increase under circular rearrangement. Next we study symmetrization techniques in discrete cases, obtaining convolution-rearrangement inequalities of the form £ f(x)K(d(x,y))g(y)< £ f*(x)K(d(x,y))g*(y), x,yeG x,yeM for several graphs G, where /# is a carefully chosen rearrangement of a real function / on G. The graphs we work with are primarily the circular graphs Z„ and the p-regular trees (p > 3). These inequalities allow us to obtain a full analogue of the classical Faber-Krahn inequality in the case of subsets of a p-regular tree. We show that the convolution-rearrangement inequalities imply results on the effects of discrete Steiner-type symmetrization on harmonic measures and Green's functions, and we obtain dis-crete analogues of some of Albert Baernstein IPs (1994) results for partial differential equations. Then, we consider the collection 53 of holomorphic functions / on the unit disc D with /(0) = 0 and ^ JJn\f'(x + iy)\2 dx dy < 1. We study qualitative properties (e.g., symmetry or various kinds of convexity) of the extremals of functionals A$ on 03 defined by A$(/) = h Jo ®(f(e%l>)) d e , for / € » where $ is a fixed real function on C satisfying various properties. S.-Y. A. Chang and D. E. Marshall (1985) proved that A$ is bounded on 53 if $(z) = e^2. While it is not known whether this A$ attains a maximum over 03, we do show that there is a perturbation of <h for which the corresponding functional does not attain its supremum. Finally, a study motivated by several conjectures concerning least harmonic majorant function-als and radial rearrangement leads us to obtain, among other things, an improved version of Ame Beurling's (1933) shove theorem for harmonic measures on slit discs. ii Table of Contents Abstract ii Table of Contents iii List of Figures vii Acknowledgements ix Dedication x Foreword xi Introduction xii Chapter I. Definitions, background material and introductory results 1 1. Some conventions and notations 2 2. General definitions for rearrangement of functions and some basic results 4 3. Hardy spaces, Poisson integrals and further background material 17 3.1. Definitions of Hardy spaces 17 3.2. Nontangential limits 18 3.3. The conjugate function and the M. Riesz theorem 20 3.4. Disc algebra and BMO 21 3.5. The Nevanlinna class N 22 4. Subharmonic functions 22 5. Least harmonic majorants, harmonic measures and uniformizers 26 5.1. Dirichlet problem and harmonic measure 27 5.2. Regularity for the Dirichlet problem 29 5.3. Least harmonic majorants 31 5.4. Brownian motion and harmonic measure 32 5.5. The uniformizer and harmonic measure 34 5.6. Green's functions 37 5.7. Riesz' theorem and representation of least harmonic majorants 39 6. Some known results in symmetrization theory 41 6.1. Circular symmetrization 41 6.2. Symmetric decreasing rearrangement 46 6.3. Steiner symmetrization 51 7. Counterexamples to a question of Hayman 53 7.1. Hayman's problem 53 7.2. The three counterexamples 54 7.3. Proofs that the counterexamples truly contradict Hayman's conjecture 56 8. Radial monotonicity of Green's functions 60 iii Table of Contents Chapter II. Discrete symmetrization 66 1. Definitions and basic results 70 2. A general framework for proving discrete master inequalities 71 3. The general case of graphs 81 4. The octahedron edge graph 84 5. The circle graphs Z n 86 6. Regular trees 88 6.1. The master inequality on regular trees 91 6.2. The Faber-Krahn inequality for subsets of regular trees 93 6.2.1. Statement of the Faber-Krahn inequality 93 6.2.2. Some useful well-known results 95 6.2.3. Proof of not necessarily strict inequality in Theorem 6.2 101 6.2.4. Proof of condition for strict inequality in Theorem 6.2 102 7. Some open problems and two counterexamples 118 7.1. How the computer proved Theorem 7.1 122 8. Discrete Schwarz and Steiner type rearrangements 123 8.1. Basic definitions and results 123 8.2. Rearrangement on a product set 134 8.3. Symmetrization and preservation of symmetry 135 9. Haliste's method for exit times, harmonic measures and Green's functions 138 9.1. Definitions and statement of results for generalized harmonic measures and Green's functions 138 9.1.1. The kernel and the assumptions on it 138 9.1.2. The kernel in our main examples 139 9.1.3. The random walk on V 142 9.1.4. Generalized harmonic measure 143 9.1.5. Generalized Green's functions 148 9.2. Reducing to the case A = 0 in Assumption 9.3 149 9.3. Exit times and proofs 151 10. A discrete Beurling shove theorem 158 11. A general rearrangement method for difference equations 164 11.1. Our assumptions 167 11.2. A discrete rearrangement theorem for difference inequalities 168 11.3. Applications 179 11.3.1. Monotonicity of the system 179 11.3.2. Generalized harmonic measures 180 11.3.3. Exit times 184 Chapter III. Chang-Marshall inequality, harmonic majorant functionals, and some nonlinear functionals on Dirichlet spaces 186 1. The A$ and T$ functionals and Dirichlet spaces 188 1.1. The A$ functionals on a finite measure space 188 1.2. Dirichlet spaces • 189 1.3. The T$ functionals acting on domains and the A$ acting on holomorphic and harmonic functions 192 2. The Chang-Marshall, Essen and Moser-Trudinger inequalities 194 iv Table of Contents 3. General results on the A$ functionals on measure spaces 198 3.1. Existence of extremals 199 3.2. The A$ functionals on balls of Hilbert spaces 205 3.3. Critically sharp inequalities and nonexistence of extremals 210 3.3.1. The general results 211 3.3.2. Application to the Moser-Trudinger inequality 214 3.3.3. Application to the Chang-Marshall inequality 215 3.3.4. Proofs of the results on critically sharp inequalities 216 4. Properties of extremals of the A$ on Dirichlet spaces 225 4.1. A variational equation 225 4.2. Regularity of extremals 231 4.3. The strict analytic radial increase property (SARIP) 237 4.4. Some extensions 240 5. Symmetric decreasing rearrangement and Dirichlet norms 243 6. Baernstein's sub-Steiner rearrangement 251 Chapter IV. Radial rearrangement 257 1. Conjectures and counterexamples 260 1.1. The primary conjectures 260 1.2. Consequences of a positive answer to Conjecture 1.2 260 1.3. Radial rearrangement 262 2. Some positive results 269 3. Transferring the problems to the cylinder and the question of two-sided lengthwise Steiner symmetrization 275 4. Formulation in terms of Green's functions 281 5. The case where 1i-» ) is concave 282 6. Haliste's one-sided lengthwise Steiner rearrangement 284 7. Brownian motion, simple discrete analogues and exit times 285 7.1. Uniform motion to the right: a counterexample 287 7.2. Exit times of Brownian motion 291 8. The Beurling shove theorem and extensions 294 9. A discrete one-dimensional analogue 306 9.1. Statement of results 306 9.2. Various useful identities, formulae and some proofs 312 9.3. Proof of the formula for the probability of safe traversal 322 9.4. The one-dimensional continuous case 327 10. Horizontal convexity of extremals for some least harmonic majorant functionals . 328 10.1. Step III of the proof of Theorem 10.1 330 10.2. Step II of the proof of Theorem 10.1 334 10.3. Step I and the rest of the proof of Theorem 10.1 337 List of notations and symbols 338 1. Rearrangement-type operators 338 2. Some other operators and relations for sets and functions 338 3. Numerical operators 339 4. Miscellaneous 339 Table of Contents 5. Greek alphabet ical index 339 6. La t in alphabet ical index 340 Bibliography 343 Appendix A . Source code for cubetern.c 349 Appendix B . Source code for cm.f 354 Index 360 v i List of Figures Introduction 0.1. An example of circular symmetrization xiii 0.2. Boundary values for the Dirichlet problem associated with the wr functional on D . xiv Chapter I. 2.1. Steiner symmetrization about the real axis 10 7.1. The circularly symmetric domain Ua 54 7.2. The unsymmetrized domain Uabcd a n d the symmetrized domain U®bcd 55 7.3. The unsymmetrized domain Uab and the symmetrized domain Ufb, together with the cone C-b,s C Ufb used in the proof 56 8.1. The construction of the set Hz for z in the complement of D 61 Chapter II. 4.1. The edge graph H$ of the octahedron 85 5.1. The graph Z u 87 5.2. Symmetrization of subsets of Z n 87 6.1. The ordering on the tree T 3 89 6.2. The extremal subtrees G# of T 3 with cardinalities from 1 to 12 96 6.3. An extremal subtree G# C T 3 with cardinality 21 and the eigenfunction / corresponding to the first non-zero eigenvalue of —A 104 6.4. An extremal subtree G# C T 3 with cardinality 14 and once again with the eigenfunction / corresponding to the first non-zero eigenvalue of —A 105 6.5. Definition of A, B, A' and B' in the case where h(w) — h(v) 109 6.6. Definition of A, B, A1 and B' in the case where h(w) = 1 + h(v) 110 7.1. The cube 1i\ and the ternary plane Z 3 121 9.1. Symmetrization on Z x Zn 140 9.2. Steiner symmetrization on Z 2 147 10.1. An example of the sets H and Hi 162 Chapter III. 5.1. The functions fu f2, f and f® 245 Chapter IV. 1.1. A multiply connected domain for Example 1.1, with its radial rearrangement 263 1.2. A simply connected domain for Example 1.1. The radial rearrangement of this domain will be contained in some disc D(r') for 0 < r' < r 264 7.1. The domain WTfi 288 7.2. The domain w i 289 8.1. The decomposition of I into 7X and I2 in a case where n = 3 302 9.1. An example of the original A -^tuple and j — 18 321 vii List of Figures 10.1. The graph of f(ei8) 332 viii Acknowledgements I would like to thank Professor A lbe r t Baernstein II for his patience, encouragement, sugges-t ions, advice and for much lengthy electronic mai l correspondence wi th me. I am grateful to Professors M a t t s Essen and D . E . Marsha l l for their warm hospital i ty in Uppsa la and Seattle, respectively, and for various discussions by e-mail and in person. I am thankful to Professor A lec Matheson for a number of interesting discussions by e-mail . I wish to thank Professor Domin ik Szynal for his hospital i ty in Lub l in . I wish to express my sincere grat i tude to my supervisor, Professor John J . F . Fournier for pat ient ly bearing wi th my many questions, for proffering useful advice at many points in my research, and for his general and generous care for my mathemat ica l development. I would also like to express grat i tude to Professors Joel Fr iedman, Gregory F . Lawler and Phi l ip D . Loewen as well as M r . R a v i Vak i l for various discussions. The research was part ia l ly supported by Professor J . J . F . Fournier 's N S E R C Gran t #4822. ix Ad maiorem Dei gloriam X Foreword Most of Section 1.7 is taken and/or adapted from the author's paper [89] which will appear in the Proceedings of the American Mathematical Society; text and diagrams taken from this paper are copyright © 1994 The American Mathematical Society. The American Mathematical Society copyright agreement permits such use of the text by the author. Most of Section II.3.3 is taken and/or adapted from the author's paper [88] which will appear in the Canadian Mathematical Bulletin; text taken from this paper is copyright © 1994 The Canadian Mathematical Society. Permission to use the text in the thesis has been secured. Scattered major portions of the text in Sections III.l—III.4.4 (with a noteworthy and particular exception of the above-discussed Section III.3.3) are taken and/or adapted from the author's joint paper with Alec Matheson [75] which will appear in the Transactions of the American Mathematical Society; text taken from this paper is copyright © 1995 The American Mathe-matical Society. The author thanks Professor Alec Matheson for granting permission for the author to use his own judgement in describing the author's contribution to the joint work and for granting permission to use the text in the thesis. The American Mathematical Society copyright agreement permits such use of the text by the authors. Most of Section IV.9 is taken from the author's paper [87] which will appear in the Annales de I'Institut Henri Poincare - Probabilites & Statistiques. As of the date of submission of this thesis, the copyright in this paper remains vested in the author. Permission of the journal editor to use the text in this thesis has nonetheless been secured. xi Introduction We study symmetr izat ion and certain nonlinear functionals on collections of sets or functions. Symmetr izat ion theory strives to replace a given object (function or set) by one which is some-how related to it so that (a) the replacement object has some desirable features, such as sym-metry or some kind of convexity, and (b) the replacement " improves" the values of various functionals associated wi th the object. Note that given a symmetr izat ion method which re-places sets wi th sets, we can often automat ical ly get a method which replaces functions wi th funct ions by apply ing the method to the level sets of the functions and reassembling the sym-metr ized level sets to form a new funct ion. A typical k ind of desirable feature that fits under (a) is circular symmetry: a subset of the complex plane is said to be circular ly symmetr ic if the intersection of this subset wi th every circle centred about the origin is a connected arc centred about the positive real axis. Thus , c i rcular symmetry includes reflection symmetry about the real axis and a kind of angular convexity condi t ion. G iven a measurable set S in the complex plane, we can define the circular symmetrization S® by requir ing that S® be circular ly symmetr ic and satisfy the fol lowing three condit ions for every r > 0: (i) the circle of radius r about the origin is contained in S® if and only if it is contained in S (ii) the angular measure of the intersection of the circle of radius r about the origin wi th S® equals the angular measure of the intersection of the same circle wi th S (iii) the intersection of 5 ® wi th the circle of radius r about the origin is open in that circle. See F igure 0.1. Cond i t ion (ii) implies that the area of S® equals that of S. We now describe the kinds of funct ionals that interest us in connection wi th circular sym-metr izat ion. Let D be a domain contain ing the origin (i.e., a connected open subset of C with 0 € D). Let <f>: [0,co) —> [—co,oo) be such that t (->• <£(e*) is convex and increasing (in partic-ular if <j> is convex and increasing then this condit ion wi l l hold). Let h be the inf imum of al l real functions g which are harmonic on D and satisfy g{z) > (f)(\z\) for al l z in D. Then put T<t>(D) = h(0). A n alternate probabi l ist ic descript ion of T(f,(D) is as follows. Let Bt be a Brownian motion in the complex plane star t ing at the or igin. Let r = in f { i > 0 : Bt (f. D} be the first t ime that the Brownian mot ion leaves D. Then T^(D) is the expected value of (f>[\BT\). It is thus a kind of weighted average of <^(|z|) as z ranges over the boundary of D , wi th the weight at a point being given by the probabi l i ty density that a Brownian motion first impacts the boundary at that point. A n impor tant result on the T$ functionals is a consequence of a theorem of Baernstein's ground-breaking paper [7] and says that T^(D) < r<p(D®). Hence, circular symmetr izat ion increases the above-defined T$ funct ionals. C i rcu la r symmetr izat ion also increases convex circular means of Green's funct ions. The Green's function of a sufficiently regular domain D wi th pole at w 6 D is defined to be the (unique) funct ion g(-,w;D) vanishing on the boundary of Z), harmonic on D\{w} and such that z H-> g(z, w; D) — log . j[ . remains harmonic in a neighbourhood of w. xi i Introduction y y D X Figure 0.1: An example of circular symmetrization Baernstein's theorem [7] then states that for every r > 0 we have / ^(g{reie,w;D))M< / ^(g(reie,\w\;D®)) d0, Jo Jo for every convex increasing function tp. This is of particular interest in the case of w = 0. The case of w = 0 and ip(t) = t implies the above-mentioned result on the increase of the Tj, under circular symmetrization. An interesting question is whether we may take to be any increasing function, not necessarily convex. This was asked by Hayman [59]. The answer turns out to be negative, even in the case of w — 0, as we shall prove in §1.7. A second set of interesting functionals are the wr harmonic measure functionals. Fix r > 0. Let D be a domain containing the origin. Let Dr be the intersection of D with the disc of radius r centred at 0. Then, let h be the harmonic function on Dr which equals 1 on the parts of the boundary of Dr which lie on the circle of radius r about the origin (i.e., h equals 1 on (dDr) fl {z G C : \z\ = r}) and which equals 0 on all the other parts of the boundary of Dr. (See Figure 0 . 2 . ) Put wr(D) = h(0). A probabilistic interpretation of this functional is that it is the probability that a Brownian motion starting at the origin reaches the circle of radius r centred at 0 before impacting on any other part of the boundary of D. Hence, intuitively it measures how easy it is for a Brownian particle to reach the circle of radius r while staying in D. Once again, Baernstein [7] has shown that wr(D) < wr(D®) and hence circular symmetrization increases the wr functionals. The intuitive reason for this in terms of the probabilistic interpre-tation is that the circular symmetrization straightens out and consolidates the roads leading from 0 to the circle of radius r; see Figure 0.1. xiii Introduction k.y Figure 0.2: Boundary values for the Dirichlet problem associated with the wr functional on D In Chapter I we shall set up definitions and give some theorems on general and specific sym-metrization methods, as well as cite and summarize some material on the notions needed to define and study our functionals. We shall also give the most important results of Baernstein's famous paper [7], as well as our answers to the question of Hayman [59] mentioned above. Finally, we shall conclude Chapter I by obtaining a lower bound on the size of the set on which the function g(-,0;D) is radially decreasing; this will be useful to us in Chapter IV. In Chapter II, we shall consider symmetrization theory in discrete settings. For instance, we shall prove a generalization of a full analogue of Baernstein's above-cited results on the increase in Green's functions and tor-functionals under circular symmetrization in the setting of subsets of the discrete cylinder Z X Z T O . (Note that special cases of our results can be proved by the methods of Quine [90].) Of course the discrete cylinder is a discrete version of the continuous cylinder E X {z € C : \z\ = 1}, which is conformally equivalent to the punctured plane C\{0} under a natural exponential conformal equivalence, and Baernstein's results can be lifted to the continuous cylinder, which is why we can say that our results on the discrete cylinder are analogous to his theorems. The method of Chapter II proceeds by proving convolution-symmetrization inequalities of the form £ f^)K(d(x,y))g(y)< £ f*(x)K(d(x,y))g*(y), x,y£G x,yeG where G is a specific connected graph such as Z m , d is the shortest-distance metric on G, K is a decreasing function and / * and indicate symmetrizations of arbitrary functions / and g on xiv Introduction G. Symmetrizations inducing such inequalities do not exist for all graphs; indeed, we shall use a computer-based proof to show that they do not exist for the cube G = 7i\ or for the ternary plane G — Z 3 . However, we shall prove the inequalities in the special cases of the circular graph G = Z m , the p-regular tree G = Tp and the octahedron edge graph G = Hs- And, of course, the case of the linear graph G = Z goes back to Hardy and Littlewood (see [58, Thm. 371]). Our method for proving these kinds of convolution-symmetrization inequalities will be a discrete version of a method of Beckner [18, 19, 20, 21], and Baernstein and Taylor [15], as presented by Baernstein [11]. This method in the discrete setting is in fact completely elementary. Given appropriate convolution-symmetrization inequalities, we then get several consequences. First, given the case of G = Tp we obtain a full Faber-Krahn inequality for subsets of the p-regular tree. Recall that the classical Faber-Krahn [47, 68] inequality stated that out of all domains D in K n of a fixed area, the first non-zero Dirichlet1 eigenvalue of the negated Laplacian —A = — 57J™=i ~§xl 1 S minimized precisely for a ball. Our Faber-Krahn inequality will provide a characterization of the subsets D of Tp of a given size which minimize the first non-zero eigenvalue of the negated combinatorial Laplacian with the Dirichlet boundary condition that our functions vanish outside our subset D. Not surprisingly, these subsets look like a discrete tree version of the balls which were optimal in the classical Faber-Krahn case (see Figure II.6.3 on p. 104). Our proofs, are completely elementary, although the proof of the uniqueness of the optimal subsets is quite involved. Secondly, given a convolution-symmetrization inequality on a constant degree graph G, we can, as already mentioned, obtain analogues of Baernstein's results. Indeed, we shall obtain results on symmetrization for domains which are subsets of Z x G (where G is Z, Z m , Tp or Hs); these will concern the increase in certain generalized harmonic measures and generalized Green's functions. In fact, we shall proceed by two methods. Firstly, we shall prove these results via a probabilistic approach going back to Haliste [56]. Secondly, we shall show even more general results for difference equations onZxG (even for some non-linear ones) by a modified version of a method of Baernstein [11] and Weitsman [99] who proved such results for partial differential equations. In Chapter III we shall engage in an analysis of various functionals on some collections of holomorphic functions. For instance, let 2$ be the collection of all functions / holomorphic on the unit disc {z G C : \z\ < 1} which satisfy /(0) = 0 and have \f'(x + iy)\2dxdy < 1. Note that \f'(x + iy)\2 is the Jacobian of the mapping / , and hence the displayed condition can be interpreted as saying that the area of the image of / counting multiplicities does not exceed 7r. If / is one-to-one then this condition simply says that the area of the image of / does not exceed w. Let $ be a Borel-measurable function on C. Put 1 f2ir A * ( / ) = ^y o The Dirichlet eigenvalues are obtained by having —A act on those functions on our domain D which vanish on the boundary of D. xv Introduction for / G 03. (This makes sense since functions in 03 have radial limits almost everywhere on the unit circle.) If <j> is a Borel-measurable function on [0,oo), then we abuse notation and write for A$ where &(z) = 0(|z|). The A^ functionals are closely related to the T^ functionals. Indeed, we have the inequality A*(/) < r*(D) (0.1) whenever <f>: [0, 00) —> [—00,00) is such that t 1-7 <£(ef) is convex and increasing while D is the image of / (assuming / is non-constant). In the displayed inequality, we will actually have equality if / is one-to-one. Moreover, if / G 03 then AreaZ3 < ir. Hence, there are interesting connections between the A^ functionals on 03 and the functionals on B, where B is the collection of all domains containing the origin and having area at most IT. The result that had sparked our interest in the A^ functionals was the Chang-Marshall inequal-ity [32] (another proof was later given by Marshall [72]) which says that sup A^(/) < 00, (0.2) if d){t) = e*2 for t € [0, 00). This inequality is rather difficult to prove and very sharp. Indeed, we shall show that if 4> is a function on [0, 00) such that supye<g A^(/) < co then there exists a C < 00 such that for all t € [0, co) we have <p(t) < Ce*2. The Chang-Marshall inequality was improved by Essen [44] to assert that supr^(D) < 00, (0.3) where <p(t) again equals e*2. (That this is actually an improvement follows from (0.1).) We spend some time in Chapter III examining the A$ functionals for their own sake. We shall in particular be interested in the extremals of these functionals, namely functions / £ 03 such that A$(/) > sup5G(g A$(</). For instance, as a generalization of a result of Matheson [73], we have a theorem that if $ is upper semicontinuous on C and <P(Z) = o(e'zl2) as \z\ —>• 00 (uniformly in arg 2), then A$ has an extremal on 03. We also obtain some results on the A^ functionals in a few other settings such as analogous functionals acting on balls of Hilbert spaces of measurable functions. The question of existence of an extremal function is a quite interesting one in the case of the Chang-Marshall inequality. Indeed it is still not known whether A^ has an extremal if <f>[t) = e* . We shall prove, however, that even if it does, still there exists an infinitely differentiable convex increasing function <f>i{t) satisfying 4>i(t) < e* for all t € [0, 00) such that </>i(i)/e* -> 1 as t —> 00 but A^t has no extremal. On the other hand, we shall also show that there exists an infinitely differentiable convex increasing function <fo(t) satisfying <bi{t) > e*2 for all t € [0, 00) and again such that (foW/e* —> 1 as t —>• 00, but this time with A ,^2 achieving a maximum. These results will show that the existence or nonexistence of an extremal function in 03 for A^ if <p(t) = e'2 depends on the precise nature of the function e*2 and not just on its asymptotics, and that this existence or nonexistence is not stable under perturbations of <j>. Given an extremal for A$, we will under some hypotheses on $ be able to obtain a variational equation that this extremal will have to satisfy. The variational equation is not a differential xvi Introduction equat ion, but involves a pseudo-differential operator, which complicates the analysis. However, the equation is good enough to yield qual i tat ive results. For example, jo int ly wi th A lec M a t h -eson [75] it has been shown that if $ is inf initely d i f ferent ia te on C then the extremals, if they exist, must extend to be infinitely differentiable functions on the unit circle (recall that they were assumed to be holomorphic inside the unit disc; this did not by itself say much about their regulari ty on the boundary of the unit disc). We wi l l see that some of the results mentioned in this paragraph continue to work in more general settings such as a-weighted Dir ichlet spaces. F ina l ly , we wrap up Chapter III by connecting the results wi th symmetr izat ion theory. We shal l prove a result on the relation of a-weighted Dir ichlet norms (0 < a < 2) and symmetr ic decreasing rearrangement, and use this result to prove that if / is extremal for A $ where $(z) is of the form <f)(Re z) for <f> a convex funct ion on R str ict ly convex at 0, then / is one-to-one and has a Steiner symmetr ic image. We also present an alternative rearrangement method due to Baernstein replacing Steiner symmetr izat ion but keeping some of its desirable properties. We shal l examine the relation between this alternative method and Steiner symmetr izat ion. In Chapter IV we come back to symmetr izat ion theory. Th is t ime, our interest is in replacing a domain D by a domain D of not larger area which is s imply connected and hopefully star-shaped 2 as well , satisfying T^D) > T^D) for al l <f> such that t i-» <t>(et) is convex and increasing. We are unable to determine if such a replacement exists. If it does exist, then this shows that Essen's inequal i ty (0.3) is a consequence of the Chang-Marsha l l inequali ty (0.2). Th is is so because of the relation between the A ^ and functionals for one-to-one functions and simply connected domains. Moreover , if such a D exists then we can obtain results on the existence of extremal domains in B for the funct ionals. In fact, we make an explicit conjecture on what we think D should be, namely a combinat ion of circular symmetr izat ion (which by itself does not always produce simply connected domains) and M a r c u s ' radial rearrangement [70] (which may sometimes decrease the Tj, if applied by itself, as we shal l show). If our conjecture is correct, then moreover we have wr(D) < wr(D). Whi le this weaker inequali ty is also st i l l open, we shall use another method of Hal iste [56] to prove that wr(D) < wr(D) for our conjectured choice of D if D is s imply connected. O f course, the main point in the construct ion of D was to obtain a s imply connected domain so that if D is a priori s imply connected, we do not gain much. A l though we do gain something: our D is star-shaped, whereas D need not be. We shal l also discuss some other part ia l results giv ing support to our conjectures. For instance we shall note that it essentially follows f rom known results that T^(D) < T^(D) if t H - » D ^ ^ is concave (which is of course a major restr ict ion). We also consider some generalizations of Beur l ing 's shove theorem [23]. These wi l l allow us to, for instance, prove our ful l conjecture on the increase of the for domains D of the form D \ 7 , where D is the unit disc and I is a finite collection of closed intervals on the negative real axis. We shall also describe what our conjectures look like when transferred to the continuous cyl in-der. In part icular, our conjectures can be transformed so as to concern Brownian motion on the cyl inder. We shall show that the transformed version of the weaker wr(D) < wr(D) conjecture does not hold if the lengthwise component of the Brownian mot ion on the cyl inder is replaced by a uni form mot ion to the right. Th is would seem to provide evidence against our conjectures, 2 A set D C C is star-shaped if for all z G D the line segment joining z with 0 lies in D. xv i i Introduction but the counterexample in fact satisfies the original wr(D) < wr(D) conjecture as it comes from a simply connected domain D. Then, we shall discretize and collapse the cylinder to the set Z. We formulate a simple analogue of the wr(D) < wr(D) conjecture as a conjecture on a random walk on {1, 2,..., AT + 1} with a reflecting boundary condition at | and with dangers distributed on the points 1,2, ...,AT. The latter conjecture then concerns the optimal distribution of the dangers so that the random walk has greatest chance of surviving to reach N + 1 when started at 1. We shall prove this conjecture (which incidentally is a variation on a problem considered by Essen [43]) via an involved but completely elementary method. Now, our main conjectures were formulated for functionals defined as infima of the values at 0 of the harmonic majorants of <£(|z|). We can likewise consider similar functionals which are infima of the values at 0 of the harmonic majorants of cf>(Re z) for convex <j>. In the final section of our thesis we prove a result in this case, analogous to a weak form of our conjectures. The proof of this result connects in a crucial way with the work of Chapter III by using the variational expression for the extremals of the A$. Moreover, the proof uses either Steiner symmetrization discussed in Chapter I or Baernstein's alternative to it discussed in Chapter III. Note on numbering and organization Theorems are numbered in the form X.Y, where X is the section number and Y the theorem number. A reference to "Theorem X.Yn refers to theorem Y of section X of the current chapter. A reference to "Theorem A.X.Y" refers to theorem Y of section X of chapter A. Note that A e {I, II, III, IV}, while X and Y are Arabic numerals. What has just been said about theorems also applies to lemmas, definitions, remarks, etc. Sections are sometimes subdivided into subsections and even sub-subsections. For instance, a reference to §2.1.4 would refer to sub-subsection 4 of subsection 1 of section 2 of the current chapter. A reference to §111.2.3 would refer to subsection 3 of section 2 of chapter III. Note that theorems, lemmas, etc., are not renumbered by subsections or sub-subsections, but their numbering is reset precisely each time one comes to a new section. Each chapter begins with an overview. Some possible reading tracks A reader whose only interest is symmetrization theory can read Chapters I, II and IV as well as sections 1.1.2, III.5 (omitting Corollary III.5.2) and III.6 (omitting Corollary III.6.2). The only adverse effect on symmetrization theory that this will have is that some of the motivations in §IV.1.2 for various conjectures will be obscure and some of the proofs in §IV.10.will refer to unread material. A reader whose only interest lies in discrete symmetrization theory can read §1.1, §1.2, all of Chapter II as well as §IV.9. In doing so, no necessary background material will be omitted, though it may be desirable to read sections 1.5 and 1.6 to see continuous versions of some of the discrete theorems; these continuous versions are from time to time parenthetically mentioned in the discussion of the discrete results. xviii Introduction A reader whose only interest is in the A $ and T$ functionals as well as in the Chang-Marsha l l and Essen inequalit ies can read §1.3-1.5 together wi th §111.1-111.4.4. The omission of background mater ial on rearrangements wi l l necessitate the dropping of §111.5, §111.6 and §111.10 which do contain some informat ion on the funct ionals. If a reader is not interested in discrete symmetr izat ion theory, Chapter II can be omit ted with no loss of continuity. Likewise, so can §IV.9, al though if the latter is dropped then the proof of Theorem IV.7.1 wi l l have to remain a mystery. There are a few results throughout the thesis which are of some interest in and of themselves and which are independent of other work: (i) A reader interested in an answer to Hayman 's problem on circular symmetr izat ion and Green's functions and already famil iar wi th the basic notions of symmetr izat ion need only read §1.7. (ii) A reader already fami l iar wi th basic notions about Green's functions and the Riesz repre-sentation theorem for subharmonic funct ions, and whose interest lies in the radial mono-tonici ty properties of the Green's function need only read §1.8. (iii) A reader interested only in the Faber -K rahn inequali ty on regular trees need only read sections 1-3 and 6 in Chapter II. (iv) A reader interested in the results on the existence/nonexistence of extremal functions for perturbat ions of the Chang-Marsha l l or Moser-Trudinger inequali ty need only read sections 2, 3.1, 3.2 and 3.3 of Chapter III together wi th the basic definitions in §111.1. A reader already fami l iar wi th the Chang-Marsha l l or Moser-Trudinger inequality and interested only in the existence/nonexistence result for perturbat ions wi l l only really need a quick glance at the sections other than §111.3.3. (v) A reader whose only interest lies in the elementary combinator ia l rearrangement result for random walks in a dangerous bl ind alley need only read §IV.9. x ix Chapter I Definitions, background material and introductory results Overview Af ter having given some basic conventions (§1), we begin the thesis proper by defining the notion of a rearrangement method and stat ing a few useful and simple results on rearrangements (§2). O f part icular importance wil l be the fundamental Hardy-L i t t lewood rearrangement inequalities (Theorems 2.3 and 2.4) which, fol lowing Kawoh l [65], we shall give for al l measure-preserving rearrangement methods. A standard construct ion central to §2 wi l l be the rearrangement of a funct ion based upon the rearrangement of its level sets (equation (2.1)). Then , we shall review basic results on Hardy spaces on the unit disc and on Poisson integrals there (§3). Then , we discuss basic notions concerning subharmonic functions (§4). Then , in §5 we shall review such notions as harmonic measure, least harmonic majorants, the connection between Brownian mot ion and harmonic measure, the uniformizer, Green's functions and finally the Riesz decomposit ion theorem for subharmonic functions (Theorem 5.9). These notions and standard results wi l l be used in the subsequent parts of Chapter I and in Chapters III and IV . Hav ing reviewed these notions, we then consider in §6 the circular and Steiner rearrangements as well as the symmetr ic decreasing rearrangement, give a few very basic facts, and then discuss various results of Baernstein [7], [9], [14], Beckner [20] and Hal iste [56]. Af ter these basic results and reviews, we proceed in §7 to give a negative answer to a question of Hayman [59] concerning circular rearrangements and Green's functions. Note that §7 is 1 Chapter I. Definitions, background material and introductory results essentially taken from the author's paper [89]. Then, we discuss the question of how large we can take the set on which the Green's function is radially decreasing away from its pole (§8). In particular, we shall show that this happens on the largest disc centred on the pole and contained in the domain in question (Theorem 8.1). In fact this will usually happen on a larger set, and we obtain another lower bound on this set (Theorem 8.2). While our result here may be of some interest in and of itself, the reason we give it is because we will use it in §IV.8 where we shall obtain an improved version of Beurling's shove theorem [23]. The material of §8 is adapted from the author's paper [84]. 1. Some conventions and notations We have Z+ = {1, 2,...}, Z " = { - 1 , -2,...}, Z+ = {0}uZ+, Z Q = {0}DZ~ and Z = Z ^ U Z " . The terms "positive" and "negative" shall be taken to mean "non-negative" and "non-positive", respectively. Likewise, the terms "increasing", "decreasing", "decrease", "increase", "smaller" and "greater" shall be understood in the weaker sense. When we wish to make a stronger statement, we will explicitly add an auxiliary term such as "strictly". The term "countable" means "at most countable". We shall use the term domain to mean any non-empty connected open subset of C. We call a set in the plane convex if whenever z and w are two points in it, then the line segment joining z and w lies in this set. We call a set star-shaped if whenever z is a point in the set, then the line segment joining z and 0 lies in this set. We call a set horizontally convex if whenever z and w are two points in this set lying on the same horizontal line (i.e., having Im 2 = Imw), then the line segment joining z and w lies in the same set. Measurable functions and semicontinuous functions are assumed to be almost everywhere finite unless otherwise provided for. 2 Chapter I. Definitions, background material and introductory results We shall use | • | for absolute values of numbers, for cardinal i t ies of discrete sets, and for (sometimes normalized) Lebesgue measures in non-discrete settings. We write s u p p / = {x : f(x) # 0} for the support of a funct ion / defined on a topological space. If / is defined on a discrete set, then supp / = {x : f(x) ^ 0}. Let D(w; r) = {z G C : \w - z\ < r} be the disc of radius r centred on w. P u t D( r ) = D(0; r) and let D = D ( l ) be the unit disc. P u t T(w; r) = {z € C : \z - w\ = r} for r > 0. Set T ( r ) = T(0 ; r) and let T = T ( l ) be the unit circle. Note that T(0) = {0}. We write Ac for the complement of a set A and 2A for its power set, namely 2A = {B : B C A}. P u t A\B = {a £ A : a £ B}. If A is a collection of sets, wri te \JA = \J A = {x:3AeA.xeA}. If A is a set then write 1A for the funct ion which is 1 on A and 0 outside A. If P is a proposit ion, then put l{p] = 1 if P is true and l{py = 0 if P is false. Wr i te 0*3;^ = ^-{x-y} f ° r the Kronecker S. P u t = max ( i , 0) and t~ = (-t)+ so that t = t+ - t~ for al l t £ R. For a real function / write for the funct ion x (flx))^. Wr i te sgnx = x/\x\ if x ^ 0. The choice of sgnO wi l l never be relevant. Wr i te arg z for any choice of the argument of a complex number z so that z = \z\eiarzz. For a subset S of C and a complex number A we write XS = {\z : z £ S}. Likewise, \ + S = {X + z:zeS}. 3 Chapter I. Definitions, background material and introductory results The terms "ana ly t ic " and "holomorphic" are synonymous for us. The term u n i v a l e n t means "one-to-one and ho lomorphic" . A real funct ion / on an open interval (a, b) is said to be c o n v e x if whenever a < a < 3 < b and t E [0,1] then f(ta+(l-t)3)<tf(a) + (l-t)f(3). It is said to be s t r i c t l y c o n v e x a t x € (a, b) if whenever a < a < 3 < b and t G (0,1) are such that x = ta + (1 — t)3 then / » < t / ( a ) + ( l - * ) / ( / ? ) . Given a funct ion $ on W1, we wri te = J^r for the part ia l derivative of $ wi th respect to Xj (j = l,...,n). Given a measure space (I,p), we say that the measurable functions fn on I c o n v e r g e i n m e a s u r e to a measurable funct ion / on I providing l im p{x € I : \f(x) - fn(x)\> e} = 0 n for every fixed e > 0. 2. General definitions for rearrangement of functions and some basic results We are interested in ways of rearranging a set so as to make it more symmetr ic , or at least so as to give it some special property such as star-shapedness. We wish to formulate the definitions in the greatest generality we can. D e f i n i t i o n 2 . 1 . Let T be a collection of subsets of a set X. We cal l T a a - p s e u d o t o p o l o g y providing: (i) 4 Chapter I. Definitions, background material and introductory results (ii) X e T, and (iii) whenever A\ C A2 C A 3 C • • • are members of T, then U^Li An € T. A cr-pseudotopology is essentially a one-sided monotone class. Clearly, any topology is a cr-pseudotopology. Likewise, a u-algebra is a cr-pseudotopology. Now let T and Q be <r-pseudotopologies on X and Y, respectively. Definition 2.2. A map T —> Q is a rearrangement providing: (i) we have A* C whenever A and are two sets in T such that A C B, (ii) we have whenever Ai C A 2 C A 3 C • • • are sets in J", and (iii) 0* = 0 and A"# = Y\ Remark 2.1. Note that condition (i) actually follows from condition (ii), but we retain the definition as above for clarity. Given an extended real-valued function / on X, define the level set at height A to be fx = {x € X : f{x) > A}. Likewise, for an extended real-valued g on Y, put <7A = {y G Y : g{y) > A}. Remark 2.2. Let / be any extended real function. Then as can be easily seen. Likewise 5 Chapter I. Definitions, background material and introductory results Thus , if / is any extended real funct ion then /•oo rO f(x) = lh(x)d\+ (lh(x)-l)d\. JO J-oo Note that this last expression always makes sense, because if one of the two terms on the right side is ±00 then the other vanishes. Remark 2.3. Let / and g be any extended real functions on a set X. Then / = g if and only if fx = g\ for every A 6 R . Th is follows immediately f rom the preceding remark. D e f i n i t i o n 2 . 3 . Say that / is ^ " - l o w e r m e a s u r a b l e providing f\ £ T for every A G K . If T is a topology, then .F-lower measurabi l i ty agrees wi th J--\ower semicontinuity. If is a cr-algebra, then / is .F-lower measurable if and only if it is ^ "-measurable. Now, for an ^ "-lower measurable extended real-valued funct ion / on X let /# ( * / )= sup {A : j / e ( / A ) # } , (2.1) where # : T —> Q is a rearrangement. Th is is a standard and well-known technique for taking a rearrangement of sets and making it into a rearrangement of functions by rearranging the level sets of the funct ion and then reassembling the rearranged level sets into a new funct ion. T h e o r e m 2 . 1 . Let f be an extended real J7-lower measurable function on X and let #: T —» Q be a rearrangement. Then for every A £ [—00,00] we have (f*)x = (fx)*- (2.2) In particular, /* is (J-lower measurable. Proof. F i x A £ [—00,00) f irst. Then f rom the definit ion of / * ( y ) we see that f*(y) > A if and only if there exists a p > A such that y £ (fp)*- Let pn = A + If there is a p > A such that V S (fp)* t r i e n choose n £ Z+ so that pn £ (X,p\. Then we have (/p)# C (fPn)# by property (i) of rearrangements since evidently fp C fPn as p > pn. Hence, y £ ( / p „ ) * - Since al l the pn are 6 Chapter I. Definitions, background material and introductory results strictly larger than A, it follows that /*(y) > A if and only if there exists n £ Z+ such that V G ( / „ „ ) * • But f*(y) > A if and only if y e (f*)x- Thus, U ( / p J # n=l / oo \ ^ (/#)A = {y : Brz G Z + . y € (/Pn)#} (2.3) where we have used property (ii) of rearrangements since fPl C fP2 C fP3 C • • • as pi > p2 > /?3 > • • •. But, it is easy to see from the definition of f\ that fx = [J fpn) n=l so that by (2.3) we see that ( / * ) A = (fx)* for all A € [—00,00) as desired. The remaining case is if A — 00. But then both sides of (2.2) are empty sets and we are done. The proof is now complete since the lower measurability of / * follows from the fact that ( / # ) A = ( / A ) # G G for all real A. • Remark 2.4- If Y is a topological space and # is a rearrangement such that A* is open whenever A G T, then / * is lower semicontinuous whenever / is "^-lower measurable. To see this, it suffices to note that by Theorem 2.1 we have ( / * ) A open for every real A which precisely says that is lower semicontinuous. We say that a function <f> on [-00,00] is left lower semicontinuous providing lim inf <f>(u) > <f>(t) for all t G (—00,00]. Theorem 2.2. Let f be an extended real T-lower measurable function on X, and suppose that <f>: [—00,00] —>• [—00,00] is a monotone increasing and right lower semicontinuous function. Then ((f) o /)* — <j)o ( /*) if T —» Q is a rearrangement. Chapter I. Definitions, background material and introductory results Proof. In light of Remark 2.3, it suffices to prove that (<f>o f)f = (<f>o ( / * ) ) A whenever A e R . So f ix A e R . Let t = mf{u : 4>(u) > A}. Then if u < t, we have (j>(u) < A. B y left lower semicontinuity we have <f>(t) < A. O f course, for u > t we have 4>(u) > A. Then , (<f>of)x-={x:f(x) >t} = ft (2.4) and (<p o f*)x = {y : f*(x) >t} = (f*)t = (ft)*, by Theorem 2.1. B u t by Theorem 2.1 and (2.4) we have ( ^ o / ) * = ( ( 0 o / ) A ) * = ( / t ) * so that ((4> o / ) # ) A = (4> o (f*))\ as desired. • D e f i n i t i o n 2 .4 . Suppose that we have a measure p on X such that al l elements of T are p-measurable and a measure v on Y such that al l elements of Q are immeasurable. We say that a rearrangement # : T —> Q is m e a s u r e - p r e s e r v i n g providing p(A) = v(A#) for al l A £ T. Example 2.1 (Decreasing rearrangement to B.Q). Let Y be the interval [0, oo) equipped wi th Lebesgue measure. Let (X, T, p) be any measure space. G iven A € T, let A* = [0,p(A)). Then * is easily seen to be a rearrangement, and it is also clearly measure-preserving. Since A* is always open, by Remark 2.4 it follows that / * is lower semicontinuous on [0, oo) whenever / is ^"-measurable. Example 2.2 (Decreasing rearrangement on Z Q j . Consider the discrete space X = Y = Z+. P u t T = Q = 2 z o + . Define 5* = { i e Z + : i < |5|}, where \S\ is the cardinal i ty of a subset S of X. Then , clearly \S*\ — \S\ and it is easy to see that * is a measure preserving rearrangement. 8 Chapter I. Definitions, background material and introductory results Given an extended real / on Z Q , we may describe the / * in a very intuit ive way. Indeed, for any n £ Z + , the numbers / * ( 0 ) , . . . , f*(n - 1) are a list of the n largest values of / . We cal l / * the d e c r e a s i n g r e a r r a n g e m e n t of / . The reader is advised to often keep in mind the previous two examples which are very typical and are really the most basic types of rearrangement. Example 2.3 (Schwarz symmetrization in Kn). Let U be an arbi t rary Lebesgue measur-able subset of R™. Let U® be an open ball in R r a centred on the origin and having the same volume as U. (If U has infinite volume, then put U® = R n . ) It is easy to see that ® is a measure preserving rearrangement on the <r-algebra of all Lebesgue measurable subsets of R n , where "measure preserving" is asserted wi th respect to Lebesgue volume measure. Example 2.4 (Steiner symmetrization on <C). Let U be a subset of the plane. Define Y(X]U) = \{y:x + iyeU}\, where | • | is Lebesgue measure on R. Set UB = {x + iy : x £ R , \y\ < Y{x; U)/2}. C a l l UB the S t e i n e r s y m m e t r i z a t i o n o f U a b o u t t h e r e a l a x i s (frequently the words "about the real ax is" wi l l be omit ted) . It is easy to verify that B is a measure preserving rearrange-ment on the a-algebra of al l Lebesgue measurable subsets of C , where "measure preserving" is predicated wi th respect to Lebesgue area measure. Steiner symmetr izat ion was invented by J . Steiner who proved that it decreased circumferences of sets and used it in 1838 to prove that if there is a domain in the plane which minimizes circumference for fixed area, then this domain is a circular disc [98]. Note that in Brascamp, L ieb and Lut t inger [26, Appendix] one may f ind a modern proof of the fact that given a bounded measurable set in the plane, one can find an infinite sequence of Steiner symmetr izat ions about different axes which transforms the set into a disc. See F igure 2.1 for a simple example of Steiner symmetr izat ion at work. 9 Chapter I. Definitions, background material and introductory results \ 1 1 nJ [ J J ^ X Figure 2.1: Steiner symmetr izat ion about the real axis. 10 Chapter I. Definitions, background material and introductory results Example 2.5 (Circular symmetrization on C). Let U be a measurable subset of C . F i x r £ [0, oo). If {\z\ = r} CU then let 0(r; U) = co, otherwise let 6>(r; U) = \{6 6 [0, 2TT) : reiB € U}\, where | • | indicates Lebesgue measure on IR. Define to be the c i r c u l a r s y m m e t r i z a t i o n of U. Note that Area(C/®) = Area( f / ) as can be easily seen. A n example of circular symmetr izat ion is given in F igure 0.1 on p. xi i i of our Introduct ion. Note that © is a measure preserving rearrangement on the collection of al l open subsets of C (see Remark 6.1 in §6.1). A g a i n , "measure preserving" is meant wi th respect to Lebesgue area measure. C i rcu la r and Steiner symmetr izat ions are two of the main symmetr izat ions in which we are interested. Some further properties of them can be found in sections 6.1 and 6.3, respectively. The fol lowing proposit ion is useful. It is not intended to have opt imal condit ions, but it suffices for our purposes. P r o p o s i t i o n 2 . 1 . Let <fi be any Borel measurable function on K, f any T-lowtr measurable function, and # a measure-preserving rearrangement. Then, providing either side makes sense and at least one of the following conditions is satisfied: (ii) / > 0 and 4> is monotone increasing and left lower semicontinuous with <f>(0) — 0. Proof. Suppose first that (i) holds. Then , if we let F(X) = p(f\), we wi l l have U® = {rew : |0| < 0(r;U)/2} (2.5) (i) p(X) < oo while 11 Chapter I. Definitions, background material and introductory results where G(x) = v((f*)\). But by Theorem 2.1 and the assumption that # is measure-preserving, we have F = G and so (2.5) follows. Suppose now that (ii) holds. Then, using Theorem 2.2 we may replace / by <f> o f and /# by d>o /# = (ch q /)# and thus we may assume that <f> is the identity function. But, by Remark 2.2 and Fubini's theorem we have, r poo / fdp= / p(fx)d\ Jx Jo and p poo f*du = v{{f*)x)d\. JY JO Hence, / fdp= f f*dv, Jx JY since p(f\) — ^ ( ( / * ) A ) by Theorem 2.1 and by the measure-preserving property of our rear-rangement. • The following result is very well known. It is essentially due to Hardy and Littlewood (cf. [58, Thm. 368]) in the case of decreasing rearrangement on Z "^, and has been more generally expounded by Kawohl [65, Lem. 2.1] whose proof we adopt. Assume that X and Y are equipped with the measures p and v respectively. Theorem 2.3 (Hardy-Littlewood inequality). Let T —> Q be a measure-preserving re-arrangement. Let f and g be -lower measurable extended real functions on X. Then, if f and g are both positive, then we will have f f9dp< f f*g*dv. (2.6) Jx JY Moreover, if f is any T-lower measurable function in Lp(p) and g is any Q-lower measurable function in Lq(p), where p'1 + q~x = 1 and either 1 < p < oo or p is a-finite and 1 < p < oo, then (2.6) continues to hold providing at least one of the following conditions holds: (i) one of f and g is positive 12 Chapter I. Definitions, background material and introductory results (ii) p(X) < oo. It is not intended that the disjunct ion of condit ions (i) and (ii) should be opt imal , but only that it should cover the cases which we need. Before we prove Theorem 2.3 we give a result which is of independent interest although it is doubtless also well known. Theorem 2.4. Let T —>• Q be a measure-preserving rearrangement. Let fx, /2,..., fn be any positive J- -lower measurable functions on X. Then, I hf2.-.fndp< I f*f*...f*du. (2.7) Jx JY Proof. We first prove (2.7) for fi = 1 ^ where Ai 6 T for i = l , . . . , n . It is clear that ( 1 ^ . ) * = 1A#. Thus , the desired inequali ty in this case is equivalent to the relation p{Ain• • • n An) < v{A*n-nAf). Let A = A i n • • • n An and B = Af n • • • D A*. Then , A C A,- for all i so that A* C Af for all i since rearrangements preserve inclusions. Thus , A * C B. Hence, p{A) = v{A*) <p(B), where we have used the assumption that our rearrangement was measure preserving. Now, consider the case of general functions /,-. Let = In light of Remark 2.2 and Theorem 2.1 we have poo fi= / fid\ Jo and POO f * = / ( / ? ) # ^ A . B y Fubin i 's theorem, we then have p poo poo p h - - - f m d p = • • • / / fi1 • • • fn dp*d\i- • - d\n, (2.8a) Jx Jo Jo Jx 13 Bu t Chapter I. Definitions, background material and introductory results and r roo roo r ff •'••/* dp = • • • / / ( / 1 A l ) # - - - ( / „ A ) # ^ d A 1 - - - d A n . (2.8b) Jx Jo Jo Jx f tfl-fidp< f (f^)*---(f*)*dp, Jx Jx since we have already proved our inequali ty for characterist ic funct ions. Then (2.7) follows from (2.8a) and (2.8b). • The fol lowing result wi l l be of some use in the proof of Theorem 2.3. Lemma 2.1. Let fn be a sequence of extended-real T-lower measurable functions increasing pointwise to a function f. Let T —> Q be a rearrangement. Then, /* is a sequence of extended-real functions increasing pointwise to /*. Proof of Lemma. It is clear f rom the definit ion of ff and property (i) of rearrangements that the sequence increases pointwise. Let g be the pointwise l imit of the fn-F i x A € R . Note that ( / I ) A Q (/2)A Q • • • since we have fx < fa < • • • pointwise. Moreover, note that oo / A = U (/»)*• (2-9) 71=1 For , if x G / A then f(x) > A and hence for sufficiently large n we must have fn(x) > A so that x £ (/n)A) while, conversely, if x e (fn)\ then f(x) > fn(x) > A and so x 6 f\. In the same way, oo 9X = (J [f*)x. 71=1 Bu t , by Theorem 2.1 we have ( / * ) A = ( ( / T I ) A ) * - Hence, oo 9X=\J{{fn)x)*. n=l 14 Chapter I. Definitions, background material and introductory results But ( / I ) A C ( / 2 )A C • • • so that by property (ii) of rearrangements we have ( CO U(f^x n=l Thus, gx = (fx)* by (2.9). By Theorem 2.1 it follows that gx = (f*)x and, since A was arbitrary it follows that g = f#, which completes the proof. • Proof of Theorem 2.3. We have already proved the result in the positive case in Theorem 2.4. Suppose now that / £ Lp(p) and g 6 Lq(p) for 1 < p < oo, and that if p 6 {l,oo} then p, is c-finite. Let 4>(t) = t+ and ip(t) = — (t~). Then, (p and ip are continuous monotone functions, and hence commute with rearrangements by Theorem 2.2. It follows that f* = cp o (/#) + ip o (/#) = (</> o /)# + (iP o /)# , while of course f = <f>o f + ipo f. Suppose we could prove (2.6) in the special case where / and g have constant (but not necessarily the same) sign. Then, the general case would follow by linearity and the above two displayed equations (together with their analogues for g) since (p and ip have constant sign. Hence we need only prove our result in the case of functions / and g with constant sign. We now reduce the problem further. Let <pn(x) = x if \x\ < n and let <pn(x) = nsgnz otherwise. This is a continuous monotone function so that (4>N o f)* = <Pnof* and (4>n°g)* = <Pn°g*-Thus, <pn ° f* f* and (pn o </# -> g# as n —> oo. In the constant sign case, if we could prove (2.6) for cpn o / and <pn o g, then we could take a limit as n —> oo (using the monotone 15 Chapter I. Definitions, background material and introductory results convergence theorem) and thus obtain the result for / and g. But the functions <j>n o / and 4>n° 9 are bounded. Hence, we may assume that / and g are bounded and of constant (but not necessarily the same) sign. Suppose first that / < 0 and g > 0. If g G L9(fJ-) for some q < oo then g is integrable, since it is bounded and L1 C Lq D L°°; let <?„ = gr in that case for every n G Z + . If g = oo then we are in the cr-finite case, so let An be an increasing sequence of sets with finite //-measure whose union is X, and put gn = g • l ^ n , so that gn is integrable because of the boundedness of 9-Choose M so that / + M > 0 (we can do this as / is bounded). By (2.6) for positive functions (which is already proved) we have / (f + M)gndp< [ {f+M)*g*dv. Jx JY But (/ + M)# = /# + M, and Jxgndp = Jy gt dv < oo (the last equality follows from Proposition 2.1). Hence, / fgndp< / f*(gn)*dv. Jx JY Taking the limit as n —> oo and applying the monotone convergence theorem as well as Lemma 2.1 we obtain the inequality f f9dp< j f*g*dv, Jx JY as desired. The case where / > 0 and g < 0 is analogous. Consider now the case where / < 0 and g < 0. Under our assumed conditions, this case can only occur if p(X) < co. Let M be again such that / + M > 0. We then have f (f + M)gdp< f (f + M)*g*dp, Jx JY since we have already proved (2.6) in the case of functions of which one is positive and the other is negative. But (/ + M)# = / * + M and jx gdp = jy g* dp < co since g is bounded and p is finite, and so (2.6) follows. • 16 Chapter I. Definitions, background material and introductory results Finally we give the following definition. Definition 2.5. A rearrangement # mapping a u-pseudotopology T into itself is said to be a symmetrization if (A*)* = A* for all A £ T. Example 2.6. Schwarz, Steiner and circular symmetrizations (Examples 2 . 3 , 2 . 4 and 2 . 5 , re-spectively) are easily seen to be symmetrizations in the sense of the above definition. Likewise, the decreasing rearrangement on Z Q (Example 2 . 2 ) is a symmetrization. Useful results on and applications of various concrete symmetrizations can be found in [ 1 6 ] , [ 4 0 ] 3. Hardy spaces, Poisson integrals and further background ma-terial 3.1. Definitions of Hardy spaces Useful references on Hardy spaces are Duren [ 4 1 ] , Garnett [ 5 5 ] , Hoffman [ 6 3 ] and Koosis [ 6 7 ] . A real or complex valued function / is said to be harmonic on a domain D if / £ C2(D) and Af = 0 everywhere on D, where A = + ^ 2 is the Laplacian. The following mean value property is very basic. It follows from rescaling the z = 0 case of [ 9 4 , Thm. 1 1 . 9 ] . Theorem 3.1. Let f be harmonic on D. Let z £ D and choose any r > 0 such that D(z; r) C Remark 3.1. Let / be harmonic on D and let g: U —>• D be holomorphic. Then / o g is harmonic on U, i.e., harmonicity is conformally invariant. To see this, note that by linearity it suffices to show it for real / . But any real harmonic function is locally the real part of a holomorphic function (see, e.g., [ 9 4 , Thm. 1 1 . 9 ] ) so that f = Re F for some holomorphic F. Then, fog = Re(Fog). Now, Fog is holomorphic. Moreover, the Cauchy-Riemann equations easily show that the real part of a holomorphic function is harmonic, and so / o g = Re(F o g) is harmonic as desired. and [ 8 2 ] . D. Then. 1 7 Chapter I. Definitions, background material and introductory results Let / be a holomorphic or harmonic function on D. Define I / P m. for 0 < p < oo and 0 < r < 1. Set rnoo{r,f)= sup \f(ret9)\. 0£[O,27r] Then, the holomorphic Hardy space Hp = HP(D) is defined to be the collection of all functions / holomorphic on D with Likewise, the harmonic Hardy space hp = hp(D) is defined as the collection of all real harmonic functions / on D with Definition 3.1. Let D be a domain in the plane. Fix p 6 (0,oo). Then, D is said to be an Hp domain providing that every holomorphic function / on D whose image lies in D satisfies / G Hp. In such a case, we write D € Hp. Theorem 3.2 (Sakai [95, Thm. 8.1]). If AreaD < oo then D e Hp for every finite p. Moreover, if f is holomorphic on D with /(0) = 0 then for 1 < p < oo we have \\f\\HP= sup mp(r,f) < oo. 0<r<l ||/||w= sup mp(r,f) < oo. 0<r<l \\f\\HP<c(p)y/Al^M, for some constant c(p) depending only on p. 3.2. Nontangential limits Define the cone Ca = {l- re,e : r > 0, |0| < a} with opening angle 2a at the point 1. 18 Chapter I. Definitions, background material and introductory results Definition 3.2. Let z £ T. We say that a sequence zn £ O tends to z nontangentially providing \zn — z\ —> 0 and there exists a £ (0, ft/2) such that zn £ C a for all sufficiently large n. Definition 3.3. Let / be a function on D. Say that / has a nontangential limit L at z £ D if f{zn) —> £ whenever zn tends nontangentially to z. Write n. t. lim f(z) = L. Given / £ L J(T), define f(n) = ±- f(e*e)e-*nSde 2 n Jo to be the nth Fourier coefficient. Let 1 - r 2 PJe%a) = — =• 1 — 2r cos a + r 2 be the Poisson kernel. Given a function / £ L1(T), define I / - 2 7 T (P* /)(re«'«) = — / F(e'"*)Pr(e^-*)) d<f>, (3.1) for re10 £ D. Then, P * / is necessarily harmonic [55, p. 11]. Moreover, it is well known that oo —1 P * / = £ / ( n ) z " + £ f(n)zn. n=0 n= — oo The crucial result on non-tangential limits is as follows. The Hp case for p = 1 is essentially due to F. and M. Riesz and is given in Koosis [67, p. 49]. The hp case is due to Fatou (cf. Garnett [55, Thm. 1.5.3]). The last "Moreover" can be found in Garnett [55, p. 59]. Theorem 3.3. Let f be in HP(B>) for some p £ [1, oo] or in hp(B>) for some p £ (1, oo]. Then f has a nontangential limit at almost every point ofT. Moreover, the function n. t. lim / £ LP(T) and / = P * n. t. lim / . For 1 < p < oo, the map f t-t n.t. lim / is an isometry of hp(U>) onto ReLp(T), so that if F £ 3?LP(T) and f = P * F then F = n.t. lim / . Moreover, for 1 < p < oo, the map n. t. lim is an isometry of HP(B) onto Hp(T) = {f £ LP(T) : f{n) = 0,Vn £ Z~}. 19 Chapter I. Definitions, background material and introductory results Frequently, we shall identify n. t. l im / wi th / . In (3 .1) , we say that / is the P o i s s o n e x t e n s i o n o f F i n t o D. 3.3. The conjugate function and the M . Riesz theorem Let u be a real harmonic funct ion on D. It is well known that there exists a unique real harmonic funct ion u on D such that u(0) = 0 and u + iu is holomorphic on D. We cal l u the c o n j u g a t e f u n c t i o n of u. We may likewise define u for a complex valued harmonic function u by u = Ui + iu2, where u\ = Re u and u2 = Im u. Then , u + iu = (u\ + iui) + i(u2 + iu2) is also holomorphic. Wr i te Vu = u + iu for u harmonic on D. Note that Wu = Vu for all harmonic u o n f l wi th u(0) = 0 since Vf = f if / holomorphic on D wi th / (0 ) = 0 by the uniqueness of the conjugate funct ion. We cal l V the S z e g o p r o j e c t i o n . Let u be a funct ion in LP(T) for some p > 1. Then , P * u is harmonic on D (cf. Theorem 3.3) and satisfies n. t. l im P*u = u. (Of course Theorem 3.3 only says this for real valued functions, but this extends to the complex case by linearity.) Then , write u = n. t. l im P * u. Likewise, put Vu — n. t. l im V(P * u). The fol lowing theorem is very impor tant . Theorem 3.4 ( M . Riesz; see, e.g., [55, Thm. III.2.3]). The map u >->• u is a bounded op-erator from LP(T) to itself for every p > 1, and the map V is a bounded operator from LP(T) toHp(T). O f course, as mentioned in §3.2, there is an identif ication between HP(T) and Hp(Bi) v ia / i—^ P*f. The fol lowing remark is somewhat useful. Remark 3.2. Let u be a real valued harmonic funct ion on D satisfying u(z) = u(z) for al l z £ D . Then , u(z) = — u(z). The easiest way to see this is to note that , by replacing u w i th 20 Chapter I. Definitions, background material and introductory results ur(z) = u(rz) for 0 < r < 1, it suffices to prove the remark for u bounded on D. Bu t then the desired result follows f rom the expression (see [55, p. 102]) u(reie) = ^-Qr(6 - <]>)u(e1*) dcf>, where —2rsin9 • Qr(0) = 1 — 2r cos 9 + r 2 is ant isymmetr ic . F ina l l y note that L 2 ( T ) is identified natural ly wi th the space £2(Z) v ia the map F : / /. Under this ident i f icat ion, the operator V (or, more precisely, FVF-1) acts on a function / 6 £2(7i) by sending / to l z + • / . In other words, V is the orthogonal project ion operator from L2(T) to H2(T). 3.4. Disc algebra and B M O We define the d i s c a l g e b r a to be the collection of al l functions / continuous on D and holo-morphic on D. Now, let / be a local ly integrable funct ion on T . Let I be an arc of T . Let be the mean of / on 7, where | • | is normal ized Lebesgue measure on T . Define 11/11* = sup l - j l f - f j ] . Wri te B M O = { / : l l /H * < o o } , for the space of functions of b o u n d e d m e a n o s c i l l a t i o n . If we identify functions which differ almost everywhere by a constant, B M O wil l become a Banach space. It is known that LP(T) D B M O D L°°(T) for al l p < oo, w i th all inclusions proper. Define B M O A = { / € tf^D) : n. t. l im / e B M O } . 21 Chapter I. Definitions, background material and introductory results 3.5. The Nevanlinna class iV Let / be a holomorphic function on D. Then, we say that / is in the N e v a n l i n n a c lass N providing where log+ t = max(0,log£). We say that a domain D is a N e v a n l i n n a d o m a i n if for every holomorphic / on D whose image lies in D we have / 6 N. By Theorem 3.2, any domain of finite area is a Nevanlinna domain since clearly Hp C ./V for any strictly positive p. T h e o r e m 3.5 ( [55, T h m . I I .5 .3 ] ) . Let f 6 N. Then f has a nontangential limit almost everywhere on T, and log+ | n.t. lim/| £ i x (T). 4. Subharmonic functions D e f i n i t i o n 4 . 1 . A real function /:£>—» [—co, co) where D C C is a domain is said to be s u b h a r m o n i c providing: (i) / is upper semicontinuous (ii) / for every z € D there is an ro > 0 such that D(z; ro) C D and for all r 6 (0, ro], where A is Lebesgue area measure on C. It is clear that subharmonicity is a local criterion. Note that any harmonic function is subhar-monic by Theorem 3.1. A positive multiple of a subharmonic function is obviously subharmonic, and the sum of two subharmonic functions is subharmonic. D e f i n i t i o n 4 . 2 . A real function f:D—t (—00,00] where D C C is a domain is said to be s u p e r h a r m o n i c if —/ is subharmonic. sup / 0<r<l70 (4.1) 22 Chapter I. Definitions, background material and introductory results Theorem 4.1 (Maximum principle). Let /: D —> [—00,00) be a subharmonic function. Assume that f attains a maximum on D. Then f is constant. _ Proof. Suppose that z € D is such that f(z) = supD f. Then , let U = {w S D : f(w) = f(z)}. This is a relatively closed set in D since {w e D : f(w) = f(z)} — {w 6 D : f(w) > f(z)} and / is upper semicontinuous. We shall show that U is likewise open. To see this, f ix w £ U. Let r > 0 be such that D(w; r) C D and B u t f(v) < f(w) for all v 6 D ( io ; r ) by choice of w. Hence, since A(D(w;r)) = irr2 it follows that f(v) = f(w) for almost every v in D(w;r). B y upper semicontinuity then it follows that f(v) > f(w) for every v 6 B>(w;r). B u t f(v) < f(w) and so f(v) — f(w) for al l v € D ( w ; r ) . Hence D(w; r) C [/. Hence U is both open and closed. Since it is nonempty and D is connected, The fol lowing well-known theorem explains the reason for the term "subharmonic" . Theorem 4.2 ([55, Thm. 1.6.3]). Let f: D —> [—00,00) be upper semicontinuous. Then f is subharmonic if and only if whenever u is a harmonic function on a bounded open subset U of D with everywhere on U. The above condit ion basically says that if / is smaller than a harmonic funct ion on the boundary of a bounded domain U, then it is smaller than that harmonic funct ion inside U, too. it follows that U = D and we are done. • l imsup(/(2:) — u(z)) < 0 for every £ G dU, where the l im sup is taken as z tends to £ from within U, then f<u 23 Chapter I. Definitions, background material and introductory results Remark 4-1- Let / : D —y U be univalent and let g be subharmonic on U. Then / o g is subharmonic on D. This follows from Theorem 4.2, the conformal invariance of harmonic functions (Remark 3.1) and the fact that the definition of subharmonicity was a local one. The following well-known result will often be implicitly used. Theorem 4.3. Let &(z) — <j>(Rez) for some function (j) on K. Then <& is subharmonic on {z : Im z £ (a, b)} for (a, b) a non-empty open interval in K if and only if <j> is convex on K. Proof. Suppose first that <f> is convex and put §(z) = <f>(Rez). Let D — {z : Im z £ (a, b)}. Choose any z £ D and any r sufficiently small that 0(2; r) is contained in D. Then, W\ $(z) dA(z) = ff <f>(Rew) dA(w) <4>\-^ff RewdA(w)) \irr2 JJ D ( z ; r ) J = 4>(Rez), where we have used Jensen's inequality (see, e.g., [55, Thm. 1.6.1]) and the easy fact that the mean of w H-> Re w over a disc centred at z is Re z. Hence, $ is subharmonic as desired. Conversely, suppose that &(z) — (j>(Re z) is subharmonic on D. In fact, by translation invariance of subharmonicity and since subharmonicity is a local condition, it follows that $ is subharmonic on all of C. Then, to obtain a contradiction, assume that d> fails to be convex. We can then find real numbers x < y in (a, b) and t £ (0,1) such that <l>(tx + {l-t)y)>td>(x) + {l-t)<f>(y). • (4.2) Let a = ^ vlzt(x)- Let v = tx + (1 - t)y. Put i>(u) = 4>(u) + a(y - u) - <j>(y). Then ij)(x) = tp(y) = 0. Moreover, it is easy to see that ifi(v) > 0 in light of (4.2). Let $1 (2) = ^(Rez). Since \Pi differs from $ only by a linear function and linear functions 24 Chapter I. Definitions, background material and introductory results are harmonic, it follows that $ 1 is subharmonic. Let $ 2 ( 2 ) = ip(lmz). This too must be subharmonic, since subharmonicity is invariant under rotation. Thus, is subharmonic. Let Q be the square {z : x < Re z < y, x < Im z < y}. I claim that $ attains a maximum on Q and is moreover non-constant on Q; by the maximum principle we will then have a contradiction. To prove the claim, first let w £ (x, y) be such that ib(w) is maximal. Since v £ (x,y) has ih(v) > 0 it follows that ib(w) > 0. Now, because \P is continuous on Q, it suffices for us to verify that sup ^(z) < ty(w + iw) zEdQ since w + iw £ Q. Fix z £ dQ. Then either Re z £ {x,y} or Im z € {x,y} or both. Consider the case where Rez £ {x, y}, as the other case is analogous. Then, = ib(Rez) + ^ (Imz) = tp(lmz) since ip vanishes on {x,y}. But ty(w + iw) = 2tp(w) and ih(w) > ^(Imz) by the choice of w since Im z £ [x, y]. Since tp(w) > 0 it follows that ^(w + iw) > ^ (z) as desired. Hence indeed $ attains a strictly positive maximum in Q and thus must be constant on Q. But is not constant since ty(x + ix) = ih(x) + VK^ ) = 0 a n o ^ ^ ' s continuous. Hence we have a contradiction as desired. • Theorem 4.4. Let &(z) = </>(|.z|) for some function <p: [0,oo) —> [ — 0 0 , 0 0 ) . Then $ is subhar-monic if and only if t ^ 4>(et) is convex and increasing on R and cj) is continuous at 0 (though possibly equal to — 0 0 at 0). Proof. We shall first show that $ is subharmonic on C\(—00,0] if and only if t >^(e4) is convex on R. To see this, let ip(t) = <f>(et) for t £ R. Consider the exponential map exp sending DA={z € C : |Imz| < 7r} into C\(-oo,0]. Then, by conformal invariance of subharmonicity (Remark 4.1), we have <& subharmonic on C\(—00, 0] if and only if $oexp is subharmonic on D. 25 Chapter I. Definitions, background material and introductory results But (<fr o exp) ( 2 ) = (^le^ D = ip(Rez). Hence, by Theorem 4.3, <& is subharmonic on C\(—00,0] if and only if ip is convex. In fact, by rotation invariance, if <^ (|.z|) is subharmonic on C\(—00, 0] then it is subharmonic on all of C\{0}. Hence, $ is subharmonic on C\{0} if and only if t ^ 0(e*) is convex. We now complete our proof. Suppose first that $ is subharmonic on C. We have only to prove that <p is increasing and has cp(0) = limr^ .0+ <Kr)- Suppose first that <p is not increasing on (0, 00). Then, since t H-> (p(et) is convex, it follows that lim -^oo (p(ef) = 00. But this is impossible, since $ is upper semicontinuous at 0 and has 3>(0) < 00. Thus, cp indeed is increasing on (0,00). Moreover, for sufficiently small r we have Since $ ( 2 ) = and <p(r) is continuous increasing function on (0,oo), it follows that <p(0) < lim inf r_».o+ <p(r) so that cp is lower semicontinuous at 0. By upper semicontinuity of $ at 0 we see that we must have <p continuous at 0 as desired. Conversely, suppose that t \-t 0(e*) is convex and increasing and cp is continuous at 0. Then $ is subharmonic on C\{0}. Clearly it is also continuous at 0. Moreover, the continuity of <p at 0 and the increasing character of cp imply that Remark 4-2. In particular, if <p is convex and increasing on [0,00) then z H-> (p(\z\) is subhar-monic on C. To see this, note that in such a case t t-» (p(et) must also be convex since t »-> et is convex. 5. Least harmonic majorants, harmonic measures and uniformiz-for every r 6 (0,oo), and so we see that indeed $ must be subharmonic everywhere. • ers This section still contains no really new results, but is intended to give a precise meaning to our terminology and to collect some known facts which we will later use. 26 Chapter I. Definitions, background material and introductory results We work all the time on domains D in the complex plane C. For harmonic measures, Dirichlet problems, Brownian motions, etc., our basic reference is the book of Doob [39]. 5.1. Dirichlet problem and harmonic measure Definition 5.1. Let D be a domain and / a function on the Euclidean boundary 3D. Then we say that the Dirichlet problem on D with boundary value / is solvable if / is resolutive, i.e., if there exists a PWB solution [39, §1.VIII.2] F on D. We then say that F is the solution to the Dirichlet problem on D. The above definition is rather technical, but in practice this shall not concern us. All that is necessary for intuition is to note that F is a harmonic function on D which in some sense (which sense is made precise by the invocation of the PWB method) has the limit / at the boundary of D. Definition 5.2. A domain D is Greenian if there exists a positive nonconstant superharmonic function on D. Remark 5.1. As Doob [39, §1.11.13] notes, any domain D which is not dense in C is Greenian. For, if w G C\D, then the function f(z) = c + log \z — w\ is harmonic (hence superharmonic) and nonconstant on D, while for large enough c it will be positive. Remark 5.2. By Doob [39, §1.V.6], the plane C is not Greenian. The domains with which this thesis will be concerned are primarily the domains of finite area. The following proposition implies that all domains of finite area are Greenian. Proposition 5.1. Any domain D which is simply connected or whose complement has positive Lebesgue area measure must necessarily be Greenian. The proof of the case of the complement having positive Lebesgue area measure will be delayed until §5.4. 27 Chapter I. Definitions, background material and introductory results Proof in simply connected case. Suppose first that D is simply connected. Let / be a Riemann map from D onto O, i.e., a univalent map sending D onto D whose existence is guaranteed by the Riemann mapping theorem [94, Thm. 14.8]. Then z 4 - log is superharmonic on D Definition 5.3. Let A be a subset of dD where D is domain. Then, the harmonic measure of A in D is defined to be the function z \—>• u>(z, A; D) which is the solution of the Dirichlet problem on D with boundary value 1 on A and 0 on dD\A, if this solution exists. Harmonic measure exists for any Borel set [39, §§1.VIII.4 and 1.VIII.6]. Remark 5.3. Harmonic measure is monotone with respect to A and D. More precisely, if A C A' and D C D' with A C dD and A' C dD' then u(z, A; D) < u(z,A';D'). This follows from the fact that o>(-, A; D) is easily seen to lie in the lower PWB class for the Dirichlet problem of which u(-t A';D') is the solution. (See [39, §1.VIII.2] for definitions.) For convenience, we will sometimes write = OJ(Z, •; D). The following result is very impor-tant, although rather technical. Theorem 5.1 (cf. Doob [39, §1 .VIII .8 ] ) . Harmonic measure exists for every Borel subset A of dD where D is a Greenian domain. The set of subsets A for which harmonic measure exists is a a-algebra %D> and for each fixed z, the function A i-» u(z, A; D) is a measure on Tip. Technical remark 5.1. This is given by Doob [39, §1.VIII.8] in the case of dD being given by a metric compactification, and not the Euclidean boundary. However, the case of the Euclidean boundary follows from the fact that for an unbounded Greenian domain, the singleton {oo} has null harmonic measure [39, Example 1.VIII.5(a)]. Definition 5.4. Let D be a Greenian domain. We say that / € L1 (u>D) if / is a %£> measurable function on dD such that (see [55, p. 34]) and clearly positive and nonconstant. • (5.1) 28 Chapter I. Definitions, background material and introductory results for every z £ D. Remark 5-4- In fact, it suffices to verify (5.1) for any single point z £ D; see [39, §1.VIII.8]. The following result is quite important. Theorem 5.2 ([39, §1 .VIII .8 ] ) . Let f £ Ll(uD) for a Greenian domain D. Then, the Dirichlet problem with boundary value f on dD has a solution F given by F(z) = f fdu,?. JdD Remark 5.5. The harmonic measure UQ coincides with normalized Lebesgue measure on T for Borel sets. The easiest way to see this is to simply note that both measures are rotation invariant finite measures, and hence must coincide on the Borel sets by the uniqueness of Haar measure (see, e.g., [93, Thm. 14.19]). 5.2. Regularity for the Dirichlet problem Definition 5.5. Let D be a Greenian domain. Call a point z of dD regular if for any / £ Ll(uD) on dD which is continuous at z £ dD we have F(C)-•/(*), as C —>• z from within D, where F is the solution of the Dirichlet problem on D with boundary values / . A domain D is said to be regular if every point of its boundary is regular. The results given in Doob's book [39, §1.VIII] show that these definitions are equivalent to the standard definitions in the case of Greenian domains. Note that if / £ Ll[uD) is continuous on dD for a regular domain D, then the solution to the Dirichlet problem on D extends to a continuous function on D which agrees with / on dD. We shall have occasion to use the following simple criterion for regularity and for the Greenian character of a domain. 29 Chapter I. Definitions, background material and introductory results Theorem 5.3. Let D be any domain. Let z £ dD. Suppose that there is an curve w(r) = z + reie(r\ 0 < r < e, lying outside D, for some e > 0 and a continuous function 9. Then D is Greenian and z is regular. Only the Greenian character needs to be proved, since, given this Greenian character, the regularity is a consequence of [60, Thm. 2.11]. We shall prove the Greenian character in §5.4. Definition 5.6. We say that a domain D is a C 1 domain if for every z £ dD there exists a function <f>: (—1,1) —¥ dD which is continuously differentiable with <f>(0) = z and <f>'(0) / 0. Corollary 5.1. Let D ^ C be a C 1 domain. Then D is Greenian and regular. Proof of Corollary. Fix z £ dD. (Of course dD ^ 0 since D ^ C and since domains are by definition non-empty.) Let <p be as in the definition of a C1 boundary. Then (f)(0) = z. Choose Si £ (0,1) such that <f>(t) ^ z for all t £ (0, <5i). (Such a 8i exists since 4>'(0) ^ 0.) Define p(t) = \<f>(t) - z\ for t £ [0, o"i). Using the fact that (f)'(0) ^ 0 it is easy to see that p is continuously differentiable on [0,$i), with p'(0) > 0. Thus, there is an e > 0 such that there is a continuous function p~x: [0,s] —> [0,5) with p(p~l(r)) = r for all r £ [0',e], and S £ (0,<$i). We have p(t) > 0 for t £ [0, 6]. For t £ (0, 8} let 4>(t) ~ z m = \<p(t) - z\ Then, tb is a continuous map of (0,6] into T. The (one-sided) differentiability of p at 0 and the strict positivity of this derivative implies that limt_>.o+ ip(t) exists by L'Hopital's rule, so that we may extend ip to a continuous function from [0, 8] to T. We may then choose a continuous logarithmic function L: [0,8] —> C such that ip(t) = eLM for all t £ [0,8]. In fact L will map [0,8] into iR. Define 9(r) = L(p 1(r))/i. Then, for 0 < r < e we have z + re 30 Chapter I. Definitions, background material and introductory results • 5.3. Least harmonic majorants Let $ be a subharmonic function on a domain D. Write" LHM(z, <&; D) = inf{/i(z) : h is a harmonic function on D with h > <&}, for the least harmonic majorant of $ at z £ D. Note that LHM(-,$;/J) is either identically +00 on D or else it is harmonic there [39, §1.111.1]. If it is harmonic, then we say that $ has a harmonic majorant on D. The following result is quite well-known. Theorem 5.4 (cf. [46, equation (1.4)] and [39, Example l.VIII.3(a)]). Assume that® is a subharmonic function on D with an extension to D such that the extended function is continuous at every point of D while $\QD € Ll(uD). Assume that $ (as a function on D) is continuous at every point of dD. Then, Proof. It is clear that $ is in the lower PWB class for cj>=$\dD (see [39, §1.VIII.2] for defini-tions). Let $ be the solution of the Dirichlet problem on D with boundary value 4> on dD (this exists by Theorem 5.2). Since $ is in the lower PWB class, it follows that f < $ on D. I now claim that LHM(-, <1>; D) is in the upper PWB class for <f>. If this is true then on D. But since LHM(•,<&; D) is the least harmonic majorant while ^ is harmonic, it then follows from the above inequalities that $ = LHM(-, $; D). The conclusion of the theorem then follows from Theorem 5.2. $ < * < LHM(-,$;D) 31 Chapter I. Definitions, background material and introductory results Thus we must prove that LHM(-, D) is in the upper PWB class for cj). To do this, let zn —> w where zn £ D and w £ dD. We must prove that liminf LHM(zn,$;D) > Mw). n-¥oo But LHM (2 n , D) > <&(zn) —$(w) = <f>(w), where we have used the continuity of $ on 3D. • 5.4. Brownian motion and harmonic measure Standard Brownian motion on R" is a Markov process {5i}ie[o,oo) with almost surely continuous paths, values in E" and Gaussian increments; see [39, §2.VII.2] for a rigorous definition (in the case n = 2, see also [38]). We often drop the word "standard" from the term "standard Brownian motion". We use Pz(-) and Ez[-] to indicate probabilities and expectations when the Brownian motion is conditioned to start from the point z £ K" at time 0. Given a domain D C K 2 , let TD = inf {t >0:Bt£D} be the first exit time of Brownian motion from the domain D. We then have the follow-ing very useful connection between Brownian motion, harmonic measure and PWB solutions of Dirichlet problems. Theorem 5.5 (see, e.g., [39, §2 . IX .10 and §2 . IX.13] ) . LetD bea Greenian domain. Then P(rr) < oo) = 1. Let A be an rio-measurable subset of dD. Then, u(z,A;D) = Pz(BTDeA). (5.2) Moreover if f £ LM^OJ0) and F is the solution of the Dirichlet problem on D with boundary value f on dD, then F(z) = E*[f(BTD)}. (5.3) 32 Chapter I. Definitions, background material and introductory results Note that (5.2) is equivalent to (5.3) by Theorem 5.2. The above result shows that harmonic measure of A at z in D is the probabi l i ty that when a Brownian mot ion started at z hits dD, it hits i t wi th in the set A. Thus , it measures how large A is compared to the rest of dD as seen f rom the point of view of z. A very impor tant result in two dimensions is the theorem of Levy that if / is a non-constant analyt ic funct ion and Bt a Brownian mot ion in the plane, then f(Bt) is a Brownian motion moving perhaps at a variable speed. Th is is known as the conformal invariance of Brownian mot ion. M o r e precisely, we have the fol lowing result. Theorem 5.6. If f is a non-constant analytic function on a domain D and z a fixed starting point in D then there exists a strictly increasing continuous function a (depending on z and f) such that the process {f(Ba^)}0<t<a-i^TD^ is a standard Brownian motion started from f(z) if B0 = z. A proof can be found in Doob [39, §2.VII I .14] or in M c K e a n ' s book [76, p. 108]. Tak ing this together wi th the connection between harmonic measure and Brownian motion we can, if we l ike, obtain a result on the conformal invariance of harmonic measure. Remark 5.6. We may extend the process f(Ba(t)) to be a Brownian motion for all t ime if we wish; see Dav is [38, T h m . 2.4]. We may now give two proofs which we have hitherto delayed. Proof of Proposition 5.1. Suppose that C\D has posit ive Lebesgue area measure. F i x any z £ D. Then , there exists an r > 0 such that T(z;r)\D has posit ive one dimensional Lebesgue measure. Then , Pz(Tj^z.r^ £ D) equals the harmonic measure of T(z;r)\D at z in B ( z ; r ) . Bu t this harmonic measure is equal to precisely the angular measure of T(z;r)\D in light of Remark 5.5, and this angular measure is str ict ly posit ive. Hence, Pz(r^z;r) £ D) > 0. Hence, Pz(3t. Bt£D)> 0. B y [39, T h m . 2.IX.10], we see that D is Greenian. • 33 Chapter I. Definitions, background material and introductory results Proof of Greenian character in Theorem 5.3. The condition we have assumed implies that there exist two points z ^ w in Dc and a continuous path 7 lying in Dc joining z to w. We shall use a very visual proof. Fix a large positive integer iV to be specified later. Let wi, w2, • • •, W J V be the vertices of a regular polygon with N sides of length \z — w\ centred on 0. Put WQ = W J V - For n = 1,..., N, let YJ- be a translated and rotated copy of 7 so that 7,- starts at u;„_i and ends at wn. As N tends to infinity, the distance of the Wi from the origin also tends to 0 0 . Moreover, 7 is a bounded set. Hence, we may choose N sufficiently large that the unit disc D is contained in the complement of the union of the convex hulls of the sets 7,-. Let T = 71 U • • • U JN- Then, T is a bounded curve winding its way precisely once around the origin. Hence, any continuous path from the origin to infinity must intersect T. Since, with probability 1, Brownian motion is an unbounded process, we conclude that with probability 1 a Brownian motion starting at the origin must intersect T and thus it must intersect one of the j{. By rotation invariance of Brownian motion, the probability of intersecting any particular 7; is the same, it follows that with probability at least 1/N, the Brownian motion must intersect 71. By [39, Thm. 2.IX.10] it follows that any domain contained in the complement of 71 is Greenian. Rotating and trans-lating this statement, we see that any domain contained in the complement of 7 is Greenian, and hence D is Greenian. • 5.5. The uniformizer and harmonic measure Let D be a domain in the plane. Let D be the universal covering surface of D and let 7r: D —>• D be the universal covering map. See Beardon [17] for details. The Riemann surface D will be conformally equivalent to one of C U {00}, C and D [17, §9.1]. The following result is very standard. Proposition 5.2. Let D be a Greenian domain. Then D is conformally equivalent to D. Proof. If D is conformally equivalent to C U {00} then it is compact, and hence ir[D] = D is compact, a contradiction. Suppose now that D is conformally equivalent to C. By the Greenian 34 Chapter I. Definitions, background material and introductory results property, there exists a posit ive nonconstant superharmonic funct ion / on D. Then , / o n is a posit ive nonconstant superharmonic funct ion on D. Bu t C is not Greenian (Remark 5.2), and hence we have a contradict ion. Thus the only remaining case is that of D conformally equivalent to C . • If D is a Greenian domain , then let p: D —>• D be a conformal isomorphism, and put / = 7r o p: D —>• D. Then / is a surjective map. We cal l / a u n i f o r m i z e r o f D. Note that if D is s imply connected then / is just a Riemann mapping. The fol lowing result gives us another way to compute harmonic measures, and was kindly pointed out to the author by Professor A lec Matheson . Recal l that the notion of a Nevanl inna domain was defined in §3.5. T h e o r e m 5.7 (cf . F i s h e r [50, §2 .4 ] ) . Let D be a Greenian domain, and suppose that A is a Borel subset of 3D. Let f: D -» D be a uniformizer. Assume that f lies in the Nevanlinna class N. Then, u>(f(0),A;D) = ±j\A(f(et6))dd. (5.4) If (j> is a Borel measurable function on 3D with f £ Lx(uD) and $ is the PWB solution to the Dirichlet problem on D with boundary value <p, then * ( / ( 0 ) ) = ^ - C Hf(el$))de. (5.5) Remark 5.7. The assumption that / 6 iV wi l l be necessarily satisfied if D is a Nevanl inna domain , and in part icular if D is an Hp domain . We give a Brown ian mot ion proof of Theorem 5.7. Proof. Equat ion (5.5) follows f rom (5.4) and Theorem 5.2. Thus we need only prove (5.4). W i thou t loss of generality, / ( 0 ) = 0 £ D. Identifying D and D, we may moreover assume that D = D and that f = n. Let Bt be a Brownian mot ion in the plane star t ing at 0 = Bo. Let Xt — f(Ba^t)) be a standard Brownian mot ion, where a is as in Theorem 5.6. Let T = a~l(rn). Note that P(T < oo) = 1. 35 Chapter I. Definitions, background material and introductory results For , Xt is a Brownian motion in the Greenian domain D unt i l t ime T, while wi th probabi l i ty 1 it exits D at some finite t ime by Theorem 5.5. Moreover, since Xt extends to a Brownian mot ion for al l t ime (Remark 5.6), the expression XT makes sense and satisfies XT = lim t_>x_ Xt with probabi l i ty 1. I now cla im that under this extension T = mf{t > 0 : Xt i D} w i th probabi l i ty 1. (5.6) It is clear that T > in f {£ > 0 : Xt £ D} since Xt G D whenever t < T. Thus , to prove our c la im it wi l l suffice to show that l im XT 4: D w i th probabi l i ty 1. To do this, let 5 be the set of points of our underly ing probabi l i ty space such that on S we have: (i) T < oo and ru < oo (ii) t i—>• Xt is continuous on [0, T] (iii) t H-> BT is continuous on [0, TO] (iv) XT = f(B^). We have already seen that (i)-( i i i) happen wi th probabi l i ty 1. We now remark that so does (iv). To see this, note that BT G T since BT is continuous on [0,Tp], and remark that the " / " in " / (BT )" is short for the non-tangential l imit n.t. l i m / . (Of course, the non-tangential l imi t of / exists almost everywhere by Theorem 3.5.) Bu t the standard connection between non-tangential l imi ts of harmonic functions and their l imits along Brownian paths (see Brelot and Doob [27], Constant inescu and Cornea [37], as well as Burkholder and Gundy [29]) then shows that /(B^) = l i m ^ . ^ - f(BT) almost surely, f rom which the desired result follows upon replacing t w i th cx(f). We shall prove that everywhere on S we have l im t _, .x- XT £ D. For suppose that we are work ing at a point u> G 5 and that all our random variables are sampled at precisely CJ, and 36 Chapter I. Definitions, background material and introductory results finally assume that lim^x- XT € D. We shall obtain a contradiction. To do this, let T(t) = BT and j(t) = f(BT) for t £ [0, T p ) . Set 7(715) = XT- Since lim XT = lim y(t), it follows that 7 is a continuous function on [0, TJJ]. Because w. D = D —>• D is a universal covering map, it follows that there is a continuous lifted path 7 : [0, ru] —> D such that ^ 0 7 = 7 on [0,7Tu] and 7(0) = 0 (see, e.g., [17, Chapter 7]). By uniqueness of lifts [17, Thm. 7.4.3], since 7(0) = T(O) and 7r o 7 = 7 = n o T on [0, T U ) , it follows that 7 = T on [0, TJU). But lim j(t) = 7(7D) € D, *—*-n>— while lim r(t) = B ^ ^ D , a contradiction. Hence, (5.6) is valid. Applying Theorem 5.5 to the standard Brownian motion X T , we see that LU(0,A;D) = P(Xt £ A). But, given the set S defined before, which event has probability 1, we have XT = f(BTD). Thus, u(0,A;D) = P(f(BTD)eA). But B T D has uniform distribution on T when BQ = 0. (This can be seen from rotation invariance of Brownian motion and uniqueness of Haar measure on T [93, Thm. 14.19].) Hence u(0,A;D) = P(f(BTD)£A) = ±- lA{f^d)) d6. It Jo • 5.6. Green's functions There is more than one equivalent definition of a Green's function that could be given. The one that we shall give will be a two step definition. First suppose that D has regular boundary and 37 Chapter I. Definitions, background material and introductory results that C\D is non-empty. Then, a Green's function g(-, w; D) for D is defined to be any function such that: (i) g(z, w;D) — 0 whenever z fi D or w fi D (ii) g(-,w;D) is continuous on C\{u;} for each fixed w (iii) g(-,w;D) is harmonic on for each fixed w (iv) z \-t g(z,w;D) — log is a harmonic function in a neighbourhood of w if to G J D is fixed. For uniqueness and existence see [60, Thms. 1.14 and 3.13]. Now suppose that D is any Greenian domain. It is easy to see that we can find a sequence Dn of regular domains with compact closures such that Di C D2 C • • • and D = U^Li Dn. (It is easy to construct such domains, using Theorem 5.3 for this purpose. As Hayman and Kennedy [60, p. 253] remark, it is easy to even make sure that each Dn is a union of finitely many discs.) We then define the Green's function of D as g(z,w;D)= lim g(z,w;Dn). n—>oo This limit exists and is independent of the choice of the Dn [60, Lem. 5.6]. It is worth noting that g(z, w; D) = g{w, z; D) for all z and w [60, Thm. 5.26]. Remark 5.8. Note that «jf(z,0;D) = log T^T , (5.7) \z\ for z G D. This is easily verified since log ^ is locally the real part of an analytic function away from the origin, and hence harmonic there, since condition (iv) is trivial and since log ||r vanishes for z G #B. More generally, for z and w in D we have (5.8) g(z,w;ty = log 1 — zw w 38 Chapter I. Definitions, background material and introductory results To see this, note that the function vanishes as desired for z and w in dD and that it is locally the real part of an analytic function of z on B\{u;}. The only thing left to verify is condition (iv). But, g(z, w; D) — log j—-—j- = log |1 — zw\ — Relog(l - zw), and hence is locally the real part of an analytic function for z near w. The following very well known result is worth noting. Theorem 5.8. Let U ^ C be a simply connected domain. Let w G U and suppose that / : D —> U is a Riemann map from D onto U with /(0) = w. Then U is Greenian and g(z,w;U) = log I for all z G U. 5.7. Riesz' theorem and representation of least harmonic majorants Theorem 5.9 (Riesz; see [60, Thm. 3.9]). Let $ be subharmonic in a domain D, with $ ^ —oo. Then there exists a unique positive Borel measure p in D such that for any compact subset E of D the function z H-> $(z) — / log\z — w\dp(w) JE is harmonic on the interior of E. We shall write p = p§ and call it the Riesz measure of Remark 5.9. The uniqueness of /x$ implies that //$ depends only locally on $. More precisely, given $ on D and given a subdomain U C D we have p$\v = p$ on U. If $ G C2{D) then = ^ A $ , where A = £ ^ + (See [39, §1.8].) This together with the local dependence implies that if U C D is a subdomain in which $ is harmonic, then p has support in D\U since A$ = 0 on U. 39 Chapter I. Definitions, background material and introductory results The following result, although very trivial, will in fact be useful. Theorem 5 . 1 0 . Let 3>(z) = <j>(\z\) be subharmonic on a disc U>(R). Then there is a positive measure v§ on [0, R) such that for every Borel set A C D(i?) we have = fR r lA(reie) dOdvtir). (5.9) JO Jo Proof. The uniqueness of p, = p§ implies that it is rotation invariant (since $ is rotation invariant and so are the conditions of Theorem 5.9). Now, define the measure v on [0,R) as follows: v(S) = (2TT)-V(W :reS,9e [0,2TT)}). We must prove the validity of (5.9). In fact, it suffices to show that (5.9) holds for all sets A of the form {reie : r € S, 9 € T} for 5 a Borel subset of [0, R) with 5 C [0, R) and for T a Borel subset of [0, 27r ) . The general case then follows from the fact that the collection of these kinds of sets generates all Borel sets in D(R). Now then, suppose we have a set such as we mentioned. First note that i/(S) < oo since A C B(R) and if i/(S) = oo then {reiB : r G 5, 0 € [0, 27r)} has infinite /i-measure, and since this set is a compact subset of D(i?), this contradicts Riesz's theorem. Consider the measure a on T defined by a(U) = p({reie : r e S,el6 £ U}). This is then a finite rotation invariant measure on the Borel sets. Hence, it must be a multiple of the Haar measure on T (see, e.g., [93, Thm. 14.19]) so that But a(T) = 2KV(S). Hence, r2ir p({rete :reS, eid 6 U}) = v{S) / lu(eie) d9. Jo Putting U — {e10 : 9 e T}, we easily obtain (5.9) in the case of our set A = {reld : r 6 S,9 G T}. • 40 Chapter I. Definitions, background material and introductory results Remark 5.10. We would like to say something about the Riesz measure of the function 3>(z) = <f>t(Re z) for z e C where <pt{x) — max(0, x — t). I claim that in this case where c £ (0, oo) is some constant independent of t, St is the point mass measure in E con-centrated at t and m is Lebesgue measure on E. This claim will be of some use to us in §111.6. Without loss of generality t = 0 (the general case follows by translation). Note that $ is infinitely differentiable away from imaginary axis z'E with A$ = 0 there, and hence /i$ has support on z'E. Thus, for some measure A on E. Now, A is a translation invariant measure on E as $ is invariant with respect to translation in the direction of the imaginary axis so that p$ is likewise invariant with respect to such translation. Moreover, by Riesz's theorem, A must be a finite measure on the compact subsets of E. Hence, A is a translation invariant measure on the Borel subsets of E and finite on compacta. Thus, A = c • m for some c G [0, oo) (see, e.g., [93, Thm. 14.21] Of course, c ^ 0 since otherwise p$ vanishes which by Riesz's theorem implies that $ is harmonic, and this is evidently not true on a neighbourhood of z'E. Finally, the following result known as the "Riesz decomposition of subharmonic functions" is quite important. Theorem 5.11 ([60, Thm. 5.25]). Suppose that $ is subharmonic in a Greenian domain D and has a harmonic majorant. Then, 6. Some known results in symmetrization theory 6.1. Circular symmetrization Let U be a measurable subset of C and define U® as in Example 2.5. Call a set U circularly symmetric if U = U®. Note that for any set U we have (U®)® = 17®, which shows that © is 41 p.§ = cSt x m, p® = SQ x A Chapter I. Definitions, background material and introductory results indeed a symmetrization on the collection of all open sets in the plane. Remark 6.1. It is not quite true that © is a rearrangement from the a-algebra of all Lebesgue measurable sets in C to that same cr-algebra in the sense of Definition 2.2. The difficulty is that Definition 2.2(ii) fails to be satisfied.1 However, it is a rearrangement on the collection (topology) of all open subsets of C . In fact, to see this we need only prove that for all r £ [0, oo) we have where A = U^Li An. Now, if T(r) n A ^ T(r) then this result easily follows from the fact that Lebesgue measures behave nicely with respect to increasing unions. Suppose now that A Pi T(r) = T(r). Let Fn — T(r)\An. This is a decreasing sequence of compact sets. Since A n T(r) = T(r) and A is the union of the An it follows that f|^ =i Fn = 0- Hence, by compactness it follows that for sufficiently large n we have Fn = 0 so that T(r) C An for large n and (6.1) follows. Hence, when we apply Theorem 2.1, we must be careful to ensure we only apply it to functions / which are lower semicontinuous since lower measurability in our case coincides with lower semicontinuity. But the careful reader will notice that this will always be the case. The following proposition will henceforth be often implicitly used. Proposition 6.1. Let D be an open subset of C . Then, the function r i-» 6(r;D) is lower semicontinuous on {r G [0, oo) : 6{r; D) < oo} and D® is open. Proof. Write 0(r) = 6(r; D) for short. First we prove the lower semicontinuity of 6 on R — {r £ [0,oo) : 0(r) < oo}. Define 9(r) = \{8 G [0,2JT) : reie £ D}\. Then, 9 = 9 on R. I claim that § is lower semicontinuous on [0, oo). Let r „ - ) r £ [0, oo) be a sequence of positive numbers. We must prove that 9(r) < liminf 9(rn). Fix e > 0. Assume e < 9(r). Let S = DO T(r). Then, !Let An = {eie : \0\ < n - 1} U {-1}. Then A® = {eie : \0\ and U~ = " A M } while [J"=1 An = T so that (U~=1 An)® = T # T\{-1} = U"=i(^l) a n d Definition 2.2(h) indeed fails. OO (6.1) 71=1 42 Chapter I. Definitions, background material and introductory results there exists a compact set K C S such that \{9 : rel6 G K}\ > 9{r) — e. Since D is open and K C D is compact , we have 5 = f inf \z — tul > 0. z€C\D,wEK Then , if n is sufficiently large that \rn — r\ < S, we must then have (rn/r)K C Z). Thus , for such rt, * ( r „ ) > |{0 : r n e i e G ( r „ / r ) A ' } | > fli(r) - e. Hence l im inf 0(rn) > 6(r) — s. Since e > 0 was arbi trary, we see that 6(r) < l im inf 6{rn). We must now prove that D® is open. Let D i = {re8'* : |0| < 6(r)}. Because 9 is lower semicontinuous, it follows that Di\{0} is open. Pu t D 2 = (J T ( r ) . rG[0,oo)\R It is easy to see that [0, oo)\R is open in [0, oo), since [0, oo) \ i? = {r G [0, oo) : T ( r ) C £>} and D is open. Hence, it follows that D2 is open. Bu t it is clear that D® = Di U D2. (6.2) Moreover, D® = ( L > i \ { 0 } ) U / J 2 . To see this, note that by (6.2) it suffices to show that if 0 G D® then 0 G D2. Bu t this is clear since T(0) = {0}. Hence D® is open since D i \ { 0 } and D2 are both open. • Given a measurable funct ion / on C , we may define /® v ia (2.1), even if / is not lower semi-continuous. C a l l a funct ion / on C circularly symmetric if / = f®. C a l l it symmetric de-creasing if for every r G (0, oo) we have f(retd) — f(re~ie) for each 9 G [0, ir] while 0 H-> f{rel6) is decreasing on [0, TT]. 43 Chapter I. Definitions, background material and introductory results Remark 6.2. Clearly a circularly symmetric function is symmetric decreasing. Conversely, a lower semicontinuous symmetric decreasing function is circularly symmetric. For, to see this it is only necessary to note that if / is lower semicontinuous and symmetric decreasing, then each level set f\ is necessarily a circularly symmetric set since it is open and of the form {reie : |0| < 0(r)/2}. We now define the Baernstein *-function of a measurable function / on C. First put Jg(rel6)= / j ( r e ' > , J-\e\ for a measurable g on C and 0 G [—TT, 7r]. Then, the (circular) Baernstein *-function of a function / is defined to be / ° = J ( / @ ) . Remark 6.3. We always have Jf < everywhere. To see this, fix r € [0,oo). Consider ® restricted to measurable subsets of T(r)\{—1). This is a rearrangement, and it is measure-preserving with regard to the one-dimensional Lebesgue measure A on T(r). Let M(re¥>) = l(-0,0)(9) o n T(r). Then, u® — u so that by the Hardy-Littlewood inequality (Theorem 2.3) we have [ u-f<[u- f®. Jt(r) JT But it is clear that J T ^ u • g — Jg(ret6) for any g, and so we are done. Remark 6.4- For 6 € [— IT, TT] and r > 0 we have f>{rj°) = sup f f{re^)d<f>, A JA where the supremum is taken over all measurable subsets A of [—TT,7T] with measure 2\9\. Baernstein uses the above identity as a definition of /^(re^) and proves that this agrees with our definition of (see [7, Prop. 2]). We now give Baernstein's result on circular symmetrization and Green's functions. Theorem 6.1 (Baernstein [7]). Let D be a Greenian domain in the plane, and let u(z) = g(w,z;D) for some fixed w € C. Let v(z) = g(\w\,z;D®). Then, v is symmetric decreasing and < Jv everywhere on C. 44 Chapter I. Definitions, background material and introductory results The fol lowing Coro l la ry is well known. Corollary 6.1. Let D be a Greenian domain in the plane, and let $(z) = <K|z|) be a subhar-monic function in the plane. Then, L H M ( w , $ ; D ) < LHM(|H,$;£>®). Th is corol lary is a direct consequence of the fol lowing result together wi th Theorem 6.1 and the fact that if w is fixed and u(z) = g(w, z; D) then u^(relv) — fQ2v g(w, rexB; D) d6. Proposition 6.2. Let D and D\ be Greenian domains in the plane. Fix w £ D and wi £ Di with \w\ — \w\\. Assume that / g(w,rei9;D)d6 < g(wl,reie;Dl) d6 Jo Jo for every r £ [0, 00). Then, for a subharmonic $ of the form $(z) = 0 ( | z | ) we have L H M ( w , D) < L H M ( t » i , £ > i ) . The above proposit ion is quite wel l -known. The author would like to thank Professor A lber t Baernstein II for having pointed it out to h im. Proof of Proposition. Let v$ be the measure on [0, 00) given by Theorem 5.10. F i x w £ C. Then , by Theorem 5.11 we have rco r2ir L H M ( w , $ ; L > ) = $ ( « ; ) + / / g(w,reie;D) dO dv^r) Jo Jo and /•oo />27r L H M ( t D i , $ ; Di) = + / / g{wu relB; Dx) dd du^r). Jo Jo (Actual ly , the integrals should really be restricted a l i t t le so as to be taken over the sets where relB £ D and retB £ D\, respectively, but this wi l l make no difference since the Green's functions vanish outside their respective domains.) The desired result follows immediately f rom Fubin i 's theorem and the posi t iv i ty of v^. • 45 Chapter I. Definitions, background material and introductory results Final ly , Baernstein [7] gives the fol lowing symmetr izat ion theorem for harmonic measures. In the language of Brownian mot ion, it says that the probabi l i ty that a Brownian motion start ing at w £ D hits the circle of radius r about the origin before hi t t ing any other part of dD is increased if we replace w by \w\ and D by D®. Intuitively, the obstacles for the Brownian mot ion to move outward become less prominent. (See F igure 0.1 on p. x i i i of the Introduction.) T h e o r e m 6.2 ( B a e r n s t e i n [7]). Let D be a domain containing the point w and contained in the disc D ( r ) . Then, co(w,D(r)ndD;D) < u(\w\, D(r) n dD®; D®). Remark 6.5. Ac tua l ly , Baernstein [7] proves that if u(w) = u(w,H(r) HdD;D) and v(w) — co(w, B(r)ndD®; D®) (with u and v set identical ly to zero in D(r)\D and D(r)\D®, respectively, and both set identical ly to one in C \ B ( r ) ) then vP < Jv. However, the inequali ty u(w) < v(\w\) fol lows. To see this, fix e > 0. If w ^ D then u(w) — 0 and the inequal i ty is t r iv ia l as v > 0. Hence assume that w G D. Then \w\ G D® as is easily seen. Wr i te w = re%e. There exists S > 0 such that if.|0' — 0\ < 8 then u(rel6') > u(w) —e since u is continuous at w. It follows that u®(rel6') > u(w) — e for 6' G (—8/2,8/2) by definit ion of u®. Hence, for 0 < 9' < 8/2 we have u°(relB') >29' -(u(w)-e). O n the other hand, v is continuous at f rom which it follows that «(|«;|) = X\m(29')-lv°(reie') > l im sup(29')-^°(re10') > u(w)-e. 0'lo e>\a Since e > 0 was arb i t rary we are done. 6.2. Symmetric decreasing rearrangement Let S be a subset of T . Let to be Haar measure on T , i.e., one-dimensional Lebesgue measure normal ized so that u(T) = 1. If S = T then let S® = T . Otherwise, let • S® = {ei$ : |0| < TTU;(S)}. 46 Chapter I. Definitions, background material and introductory results Then ® is almost a rearrangement on the a-algebra of all Lebesgue measurable subsets of T (the same difficulty as in Remark 6.1 occurs), and in fact is a rearrangement on the collection of all open subsets of T. We shall, however, feel free to continue to define f® by (2.1) even if / is not lower semicontinuous; however, in such a case, Theorem 2.1 will be unavailable. As in the previous section, call a function / on T symmetric decreasing if f[eld) = f(e~te) for all 9 and if f(ez6) is decreasing in 9 for 9 G [0, IT]. For a function / on T, we shall call f® its symmetric decreasing rearrangement. Remark 6.6. Let / be symmetric decreasing on T. Then / = f® at all but countably many points. To see this, let E be the set of all points of T at which / is discontinuous. This is a countable set by the monotonicity properties of / . Let 0(*) = liminf/(O for z G T. Then, g is lower semicontinuous and agrees with / outside E. Since E has measure zero, it easily follows from the definition of our symmetric decreasing rearrangement that g® — f everywhere on T except possibly at —1. But g® — g by Remark 6.2 specialized from circular rearrangement on all of C to circular rearrangement on T. Hence f — f® outside EU {—1}. Remark 6.7. Say that functions / and g are equimeasurable if for all A G K the measures of f\ and gx are equal. I claim that for any measurable function / on T we have / and /® equimeasurable. If / is lower semicontinuous then / is lower measurable with respect to the standard topology on T, and the desired result follows from Theorem 1.2.1 and the measure-preserving character of ®. To handle the general case, we proceed by defining a new rearrangement © on the a-algebra of all Lebesgue measurable subsets of T. Simply let S® = {ei9:\0\<nu(S)}, where 9(r;S) is as in §6.1. It is clear that © is a rearrangement, and that S® and 5® can differ by at most the point —1. Hence, /® and / are equimeasurable by Theorem 2.1 and the measure-preserving character of ©, while /® and f® are equal almost everywhere (in fact, equal everywhere except possibly at —1). 47 Chapter I. Definitions, background material and introductory results Theorem 6.3 (Baernstein [9, 14]). Let f, g and h be measurable functions on T, with f 6 LP(T), g € Li(T) and h <E Lr(T) for 1 < p, q, r < oo and 1 = p'1 + q~l + r~K Then: r2ir P2TT p2n p2w j l f{eiB)g{e^°-^)h{e^)'ded<i>< / / f® (ei9)g®{e^9-^)h®(j*) d0d(j). Jo Jo Jo Jo This is a circular version of the well known Riesz-Sobolev rearrangement inequality (see [92]; in [26] an improved version can be found). Given two functions / and g on T, write (/*</)(^) = ^ / Q /(e^))p(e^)#. Baernstein's inequality then says that (/ * g * h)(l) < (/® * g® * h®)(l). This is of particular interest when g = g®, and it is in that case that we shall use it. Definition 6.1. A function / on T is said to be strictly symmetric decreasing if / is symmetric decreasing and 8 i - > f(et9) is one-to-one on [0,7r]. In the case where g is strictly symmetric decreasing, the following result is an improvement on Theorem 6.3. Theorem 6.4 (Beckner [20, Lemma on p. 225]). Let f and h be positive functions on T, and let g be symmetric decreasing. Assume moreover that f, g and h satisfy the conditions listed in Theorem 6.3. Then, (f*9*h)(l)<(f®'*g*h®)(l). (6.3) Suppose moreover that g is strictly symmetric decreasing and neither f nor h is almost every-where equal to a constant function on T. Then equality holds in (6.3) if and only if there exists a w € T such that for almost all z £ T we have f®(z) — f(zw) and g®(z) = g(zw). Technical remark 6.1. The statement of the result in Beckner [20, Lemma on p. 225] erroneously omits the hypothesis that neither / nor h is almost everywhere constant. If, say, h is almost everywhere constant then g * h is constant and coincides everywhere with g * h®, so that 48 Chapter I. Definitions, background material and introductory results ( / *<7*/ i ) ( l ) = (/® *g*h®) since / and / ® are equimeasurable. The error in Beckner's proof is on the top of his p. 227 where he asserts under some condit ions the existence of a certain pair of sets A and B, which existence cannot be guaranteed if either of / and h is constant. C o r o l l a r y 6 . 2 . Let g and h be symmetric decreasing functions on T with g £ LP(T) and h £ L 9 ( T ) where 1 < p, q < oo and p - 1 + q-1 = 1. Then g * h is symmetric decreasing and continuous. Moreover, if g is strictly symmetric decreasing and h fails to be almost everywhere constant, then g * h is strictly symmetric decreasing. Proof. P u t F = g*h. The cont inui ty assertion is a standard fact (see, e.g., Hewi t t and Ross [61, T h m . 20.16]) We shal l prove that for every set A C {—ix, ix] of normalized measure 2a, we have Suppose that this is so. B y Remark 6.4 (specialized to the case of circular symmetr izat ion on T and not on al l of C) we have J-a A JA where the supremum is taken over al l sets A of measure 2|or|. Since [—a, a] is such a set, it follows f rom the above identi ty and (6.4) that Hence F = F® almost everywhere on T , and since F is continuous it follows that F is symmetr ic decreasing on T . g® = g and h® — h almost everywhere (Remark 6.6). F ina l ly , suppose that g is str ict ly symmetr ica l ly decreasing and h is not almost everywhere constant. We have F symmetr ic decreasing. Suppose that F is not str ict ly symmetr ic de-creasing. Then , there exist 0 < 6i < 82 < ix such that F(etB) is constant for 0 £ {0i,92]. (6.4) We now need to prove (6.4). B u t (6.4) follows f rom Theorem 6.3 wi th / = l{ e;e :0eA} since 49 Chapter I. Definitions, background material and introductory results Let 8 = 0 2 - 0 i . P u t A = (-01,01) U (0i + 5 / 2 , 0 i + 6). Let / = l { e . e . e e A } . Clear ly /® = l^ff1_s/4,e1+8/4) a n d s o / is n o t a rotat ion of / ® . Thus , by Theorem 6.4 we have (f*F)(l) = ( / * 5 * / i ) ( l ) < ( /® *g*h®)(l) = (/® * F ) ( 1 ) . Bu t , the choice of 0i and 0 2 implies that ( / * ^ ( 1 ) = ( /® * F)(l) and we have a contradict ion, as desired. • C o r o l l a r y 6 . 3 . Let u be a function in / i 1 (D ) . Suppose that the boundary values u\j are sym-metric decreasing. Then, ur\T is symmetric decreasing for every r £ [0,1], where ur(z) = u(rz). Also, u(x) is monotone increasing for x £ (—1,1). Moreover, if u is non-constant then u r | T is strictly symmetric decreasing. Proof. W i t h o u t loss of generality r £ (0,1) . Let 1 - r 2 Pr(elU) 1 - 2 r c o s 0 + r 2 ' Then , the Poisson extension formula (Theorem 3.3) shows that ur\T = Pr * (u\T), and since Pr is evidently symmetr ic decreasing, we thus have ur\T symmetr ic decreasing by Coro l lary 6.2. We prove that u{x) is monotone increasing for x £ [0,1). The result on (—1,0] follows by apply ing the result on [0,1) to the funct ion z 1-4- —u(—z) which is evidently also symmetr ic decreasing on T . Now, f ix 0 < xx < x<i < 1. B y the max imum principle we have u(xi) < supu(x2e%e). 6 Bu t UX2\Y is symmetr ic decreasing so that sup# u(x2el9) = u(x2) and the proof of the monotone character on [0,1) is complete. C lear ly Pr is str ict ly symmetr ic decreasing, and so the "moreover" also follows f rom Coro l -lary 6.2. • F ina l l y we give the fol lowing result which we wi l l have an occasion to use in Chapter IV . T h e o r e m 6 . 5 . Let $ ( 2 ) = (^M) be upper semicontinuous on C . Let D be a circularly sym-metric Greenian domain containing the origin. Assume that (—L,0] C D. Let h(z) = L H M ( 2 , $ ; D ) . 50 Chapter I. Definitions, background material and introductory results Then h is monotone increasing on (—L,0]. , Proof. By circular symmetry we have D(L) C D. Replacing 4>{t) by max(<^ (i), 4>(L)), which does not change h (this can be seen from Theorem 5.4 and the fact that the change does not affect <f> on dD) and which preserves subharmonicity as can be easily seen, we may assume that $ is constant on D(L). By Theorems 5.10 and 5.11 we have roo r-2it h(z) = <f>(z) + / g(z,reie;D)ded^(r) Jo Jo for a positive measure u^. Since $ is constant on D(£), the support of lies in [L, oo). It thus suffices to show that for any fixed r £ [I, oo) we have gr monotone increasing on (—L, 0], where 9r{z) = ^J2J g{z,re*6;D). I claim that gr is circularly symmetric on O(L) if r > L. To see this note that for z G B(£) we have i r2n / 1 \ 1 f2v 1 since g(z,-;D) — log j-^ -q is harmonic on D(L). Now, on the right hand side of the above displayed equation, the only term which has any dependence on argz is g(z,0;D). But z i-4 g(z,0;D) is circularly symmetric (Theorem 6.1) so that indeed gr is circularly symmetric on D(L). Moreover, it is harmonic on D(L). The desired monotonicity of gr follows by scaling and an application of Corollary 6.3. • 6.3. Steiner symmetrization Recall the definition of Steiner symmetrization as given in Example 2.4. We call a set U with U = UB Steiner symmetric about the real axis. We shall sometimes omit the words "about the real axis". Note that UB is always Steiner symmetric. Note also that Steiner symmetrization is a rearrangement in the sense of Definition 2.2 on the cr-algebra of all Lebesgue measurable subsets of C. 51 Chapter I. Definitions, background material and introductory results Note that a set is Steiner symmetric about the real axis if and only if every vertical line meets this set in an open interval symmetric about the real axis. As in Proposition 6.1, we easily see that if D is open then Y(-;D) is lower semicontinuous on E and DB is open. (Actually, the proofs are now easier because we do not need to distinguish any exceptional set such as the previously exceptional set of those r where 6(r; D) = oo.) Analogues of Theorems 6.1 and Theorem 6.2 can also be proved in this setting by the methods of Baernstein [7]. We shall not give the proofs, but we do state the following two results. T h e o r e m 6 .6 . Let Q(z) = (f>(Rez) be subharmonic and let D be Greenian. Then, LHM(z, D) < LHM(Re z, DB), where DB denotes Steiner symmetrization. The proof of this result follows via the Steiner analogue of Theorem 6.1 analogously to Corol-lary 6.1. We only give this result because in §IV.10.3 we shall parenthetically mention that it could be used in a slightly modified proof of Theorem IV. 10.1. The following result is an analogue of Theorem 6.2. It could also be proved by the same methods of Baernstein [7], although it was first proved by Haliste [56] by Brownian motion methods similar to the methods we shall employ in §11.9. Haliste actually only gave the result for certain sufficiently regular domains. However, an approximation such as the one in the proof of [7, Thm. 7] easily yields the general case. T h e o r e m 6.7 ( H a l i s t e [56]). Let D be a Greenian domain. Assume that D C {z G C : Re z < M}. Then, for any z £ D we have co(z, {Re z = M} C\D;D) < u(Re z, {Re z — M} n LP; DB). Remark 6.8. A Steiner symmetric domain is necessarily simply connected. (This does not hold for a circularly symmetric domain. For instance, D\D(—\; \) is circularly symmetric but 52 Chapter I. Definitions, background material and introductory results evidently not simply connected.) This observation will be of great importance in §IV.10. To see the validity of this observation, let 7 be an arbitrary curve in a Steiner symmetric domain D. Define Ft(z) = Rez + i(l — t) Imz for t £ [0,1] and z £ C. Then, Ft is a homotopy with Ft o 7 £ D for all t £ [0,1], Fo o 7 = 7, and F\ o 7 C R. Hence, 7 is homotopic in D with a curve lying on the real axis. But clearly any such curve is homotopic with the trivial curve as can be seen by applying the homotopy Gt(z) = (1 — i) Rez + iQz for t £ [0,1] and z £ C . 7. Counterexamples to a question of Hayman 7.1. Hayman's problem The material of the present section is basically taken from the author's paper [89]. Let U be a Greenian domain with w € U. UWT\ = {z ell : g(z, w; U) > A}. Then, UW<X is circularly symmetric for every A > 0 if and only if U is circularly symmetric and w > 0 (one implication follows from the fact that UQ — U; the other is due to Baernstein [7, Corollary on p. 154].) Hayman [59, Question 5.17] had asked whether we necessarily have (Uw,x)® C (U®)\W^\. As Baernstein [7] notes, this is the same as asking whether we always have g(ret6,\w\;U®)>g(re'e, w;U), (7.1) where g(relB', w; U) is u®(ret6) for u(retd) = g(retd, w; U). The equivalence of the two questions then follows from the fact that {z £ U : g(z, w; U) > A} = (UWJ\)® (see Theorem 2.1 as g(-, w; U) is lower semicontinuous and © is a rearrangement when confined to open sets). Recall that Baernstein [7] had proved the weaker inequality that g(rel9,\w\;U®)d6> / g(rel6\w;U) d0, Jo for 0 < 61 < IT (Theorem 6.1, above). However, we will show that in general the stronger inequality (7.1) is not valid, and the answer to Hayman's question is negative, even when 53 Chapter I. Definitions, background material and introductory results Figure 7.1: The circularly symmetric domain Ua. The pole will be at a or at —a. restricted to U being simply connected and w = 0. In one of our examples, (7.1) will be false even though U is circularly symmetric (but of course w cannot lie on the non-negative real axis then). 7.2. The three counterexamples We give three counterexamples. The first is the easiest, and this is the one with U circularly symmetric. Fix any 0 < a < | . Let Ua be a disc of unit radius centred on the point a. Clearly, Ua is circularly symmetric and U® = Ua. (See Figure 7.1.) Theorem 7.1. There exists r\ S (a, 1 — a) such that for any r 6 (a, r j we have min g{rel6, a; Ua) = g(-r, a; Ua) < min g{rel6, -a; Ua). 0 0 The completely elementary proof will be given later. This gives a counterexample to (7.1) since Ua = U® and since min# g(rel6, -a; Ua) = g{-r, -a; Ua) by definition of g. We now restrict the pole to lie at zero. This will make things a little more difficult. Theorem 7.2. There exists a domain U in the plane and strictly positive numbers r and e such that g(reie,0;U®)<g(re*e,0;U), 54 Chapter I. Definitions, background material and introductory results Figure 7.2: The unsymmetrized domain Uabcd and the symmetrized domain Ufbcd. The poles will be at the origin. whenever 0 < \w — 0\ < e. Moreover, U may be taken to be simply connected. This is of course also a counterexample to (7.1). We shall present two such counterexamples, one simply connected and one not, because the two examples have rather interesting and different proofs. The multiply connected example is constructed as follows. Fix 0<a<b<c<d< 1. Let Uabcd = D\([-d, -c] U [a, b]). Clearly U®cd = D\([-«f, -c] U [-b, -a]). (See Figure 7.2.) Lemma 7.1. There exists r £ (a, b) U (c, d) and e > 0 such that 9(relS,0;U®J < g(reier®;Uabed), whenever 0 < \TT — 6\ < e. The simply connected example is constructed as follows. Fix 0 < a < b < 1. Let Vj = {z : \z\ < 1, Re z > —b} be a disc with a piece sliced off, and let Uab be V& slit along the positive real axis starting at a, namely Uab = Vb\[a,l). 55 Chapter I. Definitions, background material and introductory results Figure 7.3: The unsymmetrized domain Uab and the symmetrized domain U®b, together with the cone C-b,s C U®b used in the proof. The poles will be at the origin. Then clearly Uab is simply connected and Ufb = V&\(-6, -a]. (See Figure 7.3, ignoring the cone C_6,s C U® for now.) Lemma 7.2. There exists a' 6 (a, b) with the property that for every r £ [a', b) there is an e > 0 such that g(ret6,0;U®)<g(reid,0;Uab), whenever 0 < \n — 0\ < e. 7.3. Proofs that the counterexamples truly contradict Hayman's conjecture Proof of Theorem 7.1. Let g(-,w) be Green's function for the unit disc with a pole at w, so that by (5.8) we have 1 — zw g(z,w) = log z — w (7.2) Since 1 - a > \ > a, we need only work with 0 < r < 1 - a. Write U = Ua = U®. Then, g{reid, -a; U) = g(rel6 - a, -2a) and g{reie, a; U) = g(reie - a, 0). We can see directly that min g{rei6, a; U) = a(-r - a, 0) = log j-^—r. (7.3) 56 Chapter I. Definitions, background material and introductory results Note that in fact in general if V is a circular ly symmetr ic domain and v is on the positive real axis then g(rel9 ,v;V) is symmetr ica l ly decreasing wi th respect to 9 € [—7r,7r] by Theorem 6.1. O n the other hand, 1 + 2a(reie - a) g{rei0, -a; U) = g(re*» - a, -2a) = log re™ + a (7.4) Us ing the Maple computer algebra package (one could presumably also do this by hand), we find that dg(reie, -a; U) _ ( -1 + 6 a 2 + 2 r 2 - 8 a 4 - 8 a 2 r 2 ) ar sin 0 d~6 ~ A ' where A = 4 a V - 8 a 2 r 2 c 2 - a 2 - 2arc + 4 a 3 r c + 4 a 4 - 4 a r 3 c - 4 a 6 + 1 6 a V c 2 - 8 a 4 r 2 - 4 a 2 r 4 - r 2 , and c = cos#. Since g(re%e, —a; U) is certainly not constant in 6, it follows that dg(re%e, —a; U)/dd can only vanish at 0 = 0 and at 9 — n, so that ming(rel9,-a; U) = mm(g(r, -a; U),g(-r, - a ; U)). (7.5) 8 Bu t a completely elementary analysis of the expl ici t formulae (7.3) and (7.4) shows that for 0 < a < | and r sufficiently close to a we have g(—r, -a; U) > g(-r,a; U), while the inequali ty a < r < 1 - a implies that g(r,—a;U) > g(—r,a;U). B y (7.5), the proof is complete. B y a more precise but st i l l elementary analysis it should in principle be possible to determine the exact range of values of r for which the result holds. • Proof of the Lemmas. We first give the common part of the two proofs. Let U be Uabcd or Uab, depending on which of the two examples we wish to work w i th . B y symmetry in both cases we need only consider ir < 9 < ir + e. Define G(z) = g(z,0;U) and G*(z) = g(z,0;U®), and for x e ( - 1 , 1 ) let „. . .. G(x + iy) - G(x) G*(x + iy)-G*{x) H(x) = hm — 1 — — — and H (x) = hm — ^ . 2/-+0+ y y^o+ y 57 Chapter I. Definitions, background material and introductory results We shall later prove that if U = Uabcd then H(x) > H*(-\x\) for some x £ (a, 6) U (-d,c). (7.6) Assume this for now. In this case we let r = \x\. We may assume that x £ (a,b) since the reader wi l l easily see that the proof in the case of a; £ (—d, — c) would be quite analogous. On the other hand, in the case of U — Uab we shall show that H*(x) ->• 0 as x 1 -b. (7.7) Aga in , assume this for now. B u t , it is easy to see that l im infr-p6 H(r) > 0, so that we may choose a' £ (a, b) such that for every r £ [a', b) we have H*(—r) < H(r). F i x such an r, then. Thus , in either case we are work ing wi th an r such that H*(—r) < H(r). Assuming this, the proof f rom now on wi l l actual ly be the same in both of the cases. For t £ [0, n], let f(t) = G(reu) and g(t) = G* ( re * ( t + 7 r ) ) . Then , f rom the inequali ty H(r) > H*(—r) together wi th the standard fact that in both of the cases under consideration we have / ' (£ ) —> rH(r) and g'(t) —> rH*(—r) as £ —>• 0+, it follows that there must then be an e~i > 0 such that f'(t) > g'(t) whenever 0 < t < £]_. Since the Green's funct ion of a domain vanishes on the boundary, in both of our cases we easily see that / (0 ) = g(0) = 0 and it follows that f(t) > g(t) for 0 < t < e\. Let F be the increasing rearrangement 2 of / on [0, n], i.e., an increasing funct ion equimeasurable wi th / on [0,7r]. The posi t iv i ty of / ' near zero, the vanishing of / at zero, together with the easy fact that in both of our cases / is bounded away f rom zero on every interval [8, ir] where <5 > 0, al l imply that we may f ind an e<i > 0 such that f(t) < f(t') whenever 0 < t < £2 and t < t' < ir. Then , it follows that f(t) = F(t) whenever 0 < t < e2. Hence, F(t) = f(t) > g(i) for 0 < t < s, where e = m i n ( e i , £ 2 ) - B y the symmetry of U we have F{t) = G( re ' ( * + 7 r ) ) , and the desired conclusions of the lemmas fol low. It remains to prove (7.6) for U = Uabcd and (7.7) for U = Uab- F i rs t let U = Uabcd- Let a be Green's funct ion for the unit disc as in (2). B y Green's formula, much as in [23, pp. 46-48] or 2 W e can define F — fx via (2.1), where A is the symmetrization on [0, TT) defined by Sx = (ir — \S\, ir). This wil l give us F on (0, TT). To get F on [0, TT], impose continuity at the endpoints. 58 and Chapter I. Definitions, background material and introductory results [79, p. 105, eqn. (5.4)], we can see that for z 6 Z)\{0} we have: G(z) = log -!- - 2 I e(z,x)H(x)dx- 2 f g(z, -x)H(-x) dx (7.8) \ z \ Ja Jc G*{z) = log 7^7 - 2 f g{z,-x)H*{-x)dx-2 f g(z, -x)H*(-x) dx. (7.9) \ z \ Ja Jc Now in order to obtain a contradiction, suppose that we had H(x) < i7*(— \x\) for every x G (—d, —c) U (a, b). By (7.9), we would have G*(z) < log - 2 jT 0(2, -a:).ff (a:) dx-2 g(z, -x)H(-x) dx. (7.10) Now, fix 2 € (—d, —c). From the explicit formula for $(z,w) one can verify that g(z, —x) > Q(Z,X) whenever x > 0 and z < 0. Since i7 is known to be strictly positive on (a, 6), it would follow from (10) that G*(z) < log 7^7 - 2 /" g(z,x)H(x)dx-2 f Q(Z,-x)H(-x) dx. \ z \ Ja Jc But by (7.8), the right hand side is precisely G(z) so that we would have G*(z) < C(z). On the other hand, since both G and G* vanish on (—d, —c), we have G(z) = G*(z) = 0. We thus obtain a contradiction, and so (7.6) holds. Now let U = Uab- Note that G* is harmonic and bounded on the cone C-b,s = {—b + rel6 : 0 < r < 6, 0 < # < 7 r / 2 } , where S is chosen sufficiently small so that the cone fits inside U® and S < b — a so that G* vanishes on the two edges [—6, —b + fi] and [—b, —b + i<S] of the cone. (See Figure (7.3).) Thus, (7.7) will clearly follow as soon as can we prove that whenever h is a harmonic function on the translated and dilated cone C = Co,i = {rei0 : 0 < r < 1, 0 < 9 < 7r/2}, with h bounded on C and vanishing on the two edges [0,1] and [0, i], then the normal derivative of h at x € (0,1) tends to zero as x 1 0. To prove this claim, note that 59 Chapter I. Definitions, background material and introductory results is a univalent map of C onto the upper half plane, with R(0) — 1. Now, define hi — ho on the upper half plane. This will be a bounded harmonic function, vanishing on the interval ( 0 , oo) since this interval is the image under R of the edges [0,1) and [ 0 , i) of C. It is easy then to see that the normal derivative of hi will have to be bounded by some finite constant K on the interval [\, §]. Then, since R is analytic in a neighbourhood of x if x £ (0,1), it follows that the normal derivative of h = hi o R for x sufficiently close to 0 is bounded by K\R'{x)\. But as x 1 0 then we see directly that R'{x) —> 0 , and so the normal derivative tends to zero as desired. This proves ( 7 : 7 ) . • 8. Radial monotonicity of Green's functions The results of the present section are taken from the author's paper [84]. In this section we study the radial monotonicity properties of the Green's function. Let D be a Greenian domain in the plane with 0 £ D. Let Gpy be Green's function for D with pole at 0 , i.e., put GD{Z) = g(z,0;D). Because of the character of the logarithmic pole of GJJ at 0 , it is easy to see that GJJ is radially decreasing on some neighbourhood U of 0 , i.e., if zi and 22 are in U , with a r g 2 i = a r g 2 2 and 0 < \zi\ < \z2\, then GD{Z{) > G \ D ( 2 2 ) . It is natural to ask how large we may take U. In answer to this, one may well conjecture the following result which we shall prove later in this section. Theorem 8.1. Let D be a domain in the plane containing the origin. Let U be the largest open disc centred about the origin and contained in D. Then G D is strictly radially decreasing on U . In fact, we can do even better than this. Given a point z £ C \ { 0 } , let Lz be the line through z perpendicular to the ray from the origin to z. Let Hz be the component of C\LZ which contains the origin. The domain Hz is a half-plane with z lying on its boundary. Explicitly, Hz = {w £ C : (w,z) < \z\2}, where (•, •) is the Euclidean inner product in C = I K 2 . See Figure 8.1 for an example of Hz where z £ dD. 6 0 Chapter I. Definitions, background material and introductory results Figure 8.1: The construction of the set Hz for z in the complement of D. Define & = D Hz. (8.1) zeC\D Note that D' is convex, being an intersection of half-planes. Theorem 8.2. Let D be a domain in the plane containing the origin. Then Go is radially decreasing on D' n D and radially strictly decreasing in the interior of D'. Corollary 8 .1. Let D be a simply connected domain in the plane with 0 € D. Let f be the Riemann map from D onto the unit disc, with /(0) = 0. Then f is starlike on the interior of D', i.e., R e ^ r ^ > 0 there. Proof of Corollary. One may use Theorem 8.2 and the fact that G D i z ) = ]0g\fJz)\ GI Chapter I. Definitions, background material and introductory results (Theorem 5.8) to prove that | / | is radial ly increasing on In t£> ' . It follows that Re zJj^ > 0 there. A n easy appl icat ion of the max imum principle shows that , since / is non-constant and zj(^ is holomorphic near 0, we must in fact have str ict inequality, as desired. • In §IV.8 we shall apply Theorem 8.2 to produce an improvement of Beur l ing's shove theorem [23, pp. 58-62]. C lear ly Theorem 8.1 follows f rom Theorem 8.2 since if a disc centred about the origin is con-tained in D then it is also contained in D'. W i t h regard to Theorem 8.2, we may ask how good is the idea of choosing Hz to be a half-plane. To answer this, f ix z £ C and let Ds = C \ D ( z ; e ) . It is not difficult to see that if we let T£ be the set of points where the radial derivative of G D C is non-posit ive, then T e tends to Hz as e —> 0. (To prove this, one can either compute the Gryt explicit ly, or else one should be able to extract this informat ion f rom the proof of Theorem 8.2 given below.) It now remains for us to give a proof of Theorem 8.2. To do it, we need a certain auxi l iary result on harmonic measures. L e m m a 8 . 1 . Let F be a compact set in the plane with Cx boundary. Fix a disc U outside F with the property that if w G F and z £ U, then z £ D (u ; / 2 ; |t<;|/2). Assume that 0 ^ U. Let R be sufficiently large that U>(R) D Futj. Define <pR(z)=u(z, T(R);U(R)\F). Then there exists Ro < oo such that for all R> R0 and every z G U, the radial derivative of cpR is positive at z. Assume the L e m m a for now. Proof of Theorem 8.2. Consider the funct ion GD(Xz), A = l which is harmon ic 3 in 79\{0}, and str ict ly negative near the origin because of the character of the pole of G D at 0. If we could show that G D is radial ly decreasing on I n t D ' , then it would 3The easiest way to see the harmonicity of Er> on -D\{0} is to note that GD is locally the real part of a 62 Chapter I. Definitions, background material and introductory results follow that ED < 0 there, and by the maximum principle we would in fact have ED < 0 on lnt.0', so that Go would be strictly decreasing there. And, of course, it would then also follow that GD is radially decreasing on D' n D by continuity of Go on 7J\{0} . We now prove that GD is radially decreasing on IntD'. By approximation we may assume that D is bounded and has a C 1 boundary. Fix wo £ lnt£>'\{0}. Choose a disc W containing w0 such that W C Int_D'\{0}. Fix wi in W with argu>i = argwo and |u;i| > \wo\. If we can show that GD(WO) > GD{W\) then we will be done. So fix £ > 0. Because of the character of the pole of GD at 0, it follows that we may choose a small 77 = 77(e) > 0 with the properties that D(r?) C D\W and that for every S £ (0,77] and every point £ £ T(S), we have \GD(Q — GD(8)\ < e. Fix S £ (0,77]. Since GD is harmonic in D \ {0} and vanishes on 3D, it follows that for any w £ D\D(S) we have GD(w)= [ GD(C)du^(0, JT(5) where the measure u^^^ is the harmonic measure u^m(A)=u(w,A;D\m) for A C D\D(S). But because every value of Go on T(S) is within e of GD($), it follows that \GD(w) - GD(S) • u(w,T(6);D\D(6))\ < e. (8.2) This estimate holds uniformly for all w £ D\D(5). Now let $s(w)=u{w,T(6);DXD(6)). We shall prove that for S > 0 sufficiently small we have < $s(wo)- (8-3) It will then follow by (8.2) that GD(W\) —GD(WQ) < 2c". Since e > 0 was arbitrary it will follow that GD{W\) < GD{WO) as desired. holomorphic function there. But if F is holomorphic, then z H-> -jjF(Xz) is holomorphic, too, and the real part of a holomorphic function is harmonic. 63 Chapter I. Definitions, background material and introductory results Now, to prove (8.3), we use the conformal invariance of harmonic measure under the conformal map T(W) = 1/w on C\B(£) as follows. Let F = {l/w : w € C\£>}. Since D was bounded and contained the origin, it follows that F is compact. Let R = 1/5. Let U = {l/w : w £ W}. This will be a disc. Moreover, by conformal invariance we have <!>s(w) = chR{l/w), (8.4) where cpR is defined as in Lemma 8.1. Since r maps the closed half-planes Hw onto the closed sets C\B(z/2; |z|/2), where z = l/w, and since W lies in the interior of D', it is easy to see that the conditions of the Lemma are satisfied. It follows that 4>R(l/w{) < 4>r{1/WQ) for all sufficiently large R, which in light of (8.4) completes the proof. • Proof of Lemma 8.1. By dilation invariance we may assume that U and F are both contained in D. Let Q ( - , W) be Green's function for the unit disc with pole at w. Then, (8.5) Extend tj>R to all of B(R) by setting <f>R(z) = 0 for z € F. Then <pR is a subharmonic function on IS>(R). Since it is harmonic outside F, we only need to verify subharmonicity on F. But on F it equals 0 and it is positive everywhere else, so that the only part of subharmonicity to be verified is the upper semicontinuity. But, D\F is a regular domain (Corollary 5.1) so that in fact <f>R is continuous. Now <f>R is identically 1 on T(JR), and hence its least harmonic majorant on D(i?) is the function which is identically 1. From Theorem 5.11 we immediately conclude that <t>B.{z) = 1 - / &{z/R,w/R)dpR(w), JF where pR is short for the positive Riesz measure p(j)R, since Q(z/R,w/R) = g(z,w;B(R)) by scaling. I claim that z (->• g(z/R,w/R) is radially decreasing on U for any fixed w € F and for R sufficiently large, with the size of R depending on U and F but not on the particular choice 64 g(z,w) = log zw z — w Chapter I. Definitions, background material and introductory results of w £ F. The desired result follows immediately from this claim and the above representation of <J>R(Z). Given any two points z and w of C, let 0(z, w) £ [0, TT] be the size of the angle subtended by the rays from the origin to the points z and w, providing z and w are both non-zero. If at least one of z and w vanishes, then arbitrarily define 8{z, w) = 0. Now, if z £ U and w £ F, then by the condition z fi ID(w/2; \w\/2), it follows that \w\ cos6(z, w) - \z\ < 0. By a compactness argument it follows that there in fact exists e > 0 (depending only on the choice of the sets U and F) such that |u;| cos8(z,w) — \z\ < —e whenever z £ U and w £ F. We now prove our claim about the radial monotonicity of z >-» Q(Z/R,W/R) for large R. Fix any z £ U and w £ F. Write z = Zeia and w — Wel@ for real a and j3, and non-negative Z and W. For conciseness let 8 = 8(z, w). Then, . 1-2ZW cos 8 + Z2W2 ^ W ) = 2 l 0 g Z2-2ZWcose + W2' so that d_ i a _ (1 - W2) (W cos 8 + WZ2 cos 8 - ZW2 - Z) ~dZ^ 6 , W ' ~ (l-2ZWcos8 +Z2W2)(Z2 + W2-2ZW cos 8)' Providing z ^ w, the denominator is strictly positive (and, in fact, we must have z ^ w, since U and F are disjoint). Thus, for R > 1 we may write ^ , ( ( | ) A - ) . M , ^ ( ^ . _ , + ! ! - 2 f L ^ ) . ( 8 „ for some strictly positive function h(z,w,R). But Z and W do not exceed 1 since we have assumed that ( / U F c D . Moreover, since z £ U and w £ F, we must have Wcos8 — Z < —e. It is easy to see then that there exists an RQ > 1 depending only on e (and on the fact that F U U C D) such that for all R > R0 the expression (8.6) is strictly negative (indeed any i?o > max(l, y/2/e) will work here). This proves the claim. • 65 Chapter I I Discrete symmetrization Overview In this chapter we shall examine various discrete symmetrization results. Our first concern (sections 1 through 6.1) is with proving convolution-rearrangement inequalities of the form £ f(x)K(d(x,y))g(y)< £ f*(x)K(d(x,y))g*(y), i , i /6G x,y£G on some graphs G, for decreasing functions K. Our graphs will be equipped with an ordering order-isomorphic to a subset of Z + . Our approach will be the a discrete version of a method of Beckner [18, 19, 20, 21], and Baernstein and Taylor [15], as presented by Baernstein [11]. This method will in fact give us some even more general inequalities, known as the "master inequalities" (inequality (2.1)). The method is based on a partial reordering induced by an involution of G, and it is valid only in the presence of a set of involutions satisfying certain stringent requirements. In §1 we shall define the kind of rearrangement induced by our ordering that we are interested in. In §2 we shall describe how under some rather difficult to satisfy assumptions the master inequality holds on a discrete metric space, and how the master inequality implies a convolution-rearrangement inequality such as the one mentioned above. (Note that it is not thought that the assumptions are necessary for the master inequality, though a counterexample is not at present available.) In §3 we shall define some graph-theoretic notions and will give a proposition simplifying, in the case of a graph, the verification of the assumptions under which in §2 we have proved the master inequality. In §4 we shall apply this to verify that the master inequality 66 Chapter II. Discrete symmetrization holds for the edge graph H% of an octahedron. O f course, this is not a major result, since after al l the octahedron is only a 6-vertex graph, but we give it in order to warm up to proving the master inequal i ty in more interesting cases. In §5 we shall prove the master inequality for the case of the circular graphs Z „ . Th is result is of some interest. Note that the analogous convolut ion-rearrangement inequali ty for Z has been proved by Hardy and Li t t lewood (see [58, T h m . 371]). The case of the circular graphs wi l l allow us in §9 to generalize some results of Quine [90] on discrete circular rearrangement and harmonic measures. F ina l ly , after handl ing the circular graphs, we proceed to the most difficult case, namely that of the regular tree Tp (§6). In that case, after having reviewed some basic notions, we also prove that the assumptions for the master inequal i ty hold (§6.1). Nex t , we proceed to apply the master inequali ty on trees to obtain an analogue of the classical Fabe r -K rahn inequal i ty for the first nonzero Dir ichlet eigenvalue of the negated Laplac ian (§6.2). The most difficult part of our proof, and probably of the whole thesis, wi l l be the proof of the uniqueness of the extremal domains (§6.2.4). The reader may wish to skip that section on a first reading. In §7 we give some open problems connected wi th the above-mentioned mater ial . Moreover, we use a computer-based proof to show that no convolution-rearrangement inequali ty of the type we are interested in can be found on the cube Z | or the ternary plane Z | . Then , in §8 we examine two types of discrete rearrangements, which we shall cal l "Schwarz" and "Ste iner" , respectively. A l l the rearrangements considered in the previous sections of this chapter were Schwarz type symmetr izat ions. However, for some of the fol lowing mater ial it is good to generalize to what we cal l a "Steiner type" rearrangement, namely a rearrangement which is essentially a disjoint combinat ion of Schwarz rearrangements. We state and prove a number of basic results on Schwarz and Steiner rearrangements in §8.1. O f part icular usefulness wi l l be the rather t r iv ia l Propos i t ion 8.4 which characterizes those functions which arise out Steiner rearrangement. A lso very useful wi l l be Proposi t ion 8.6 which in a sense allows us to "undo" rearrangements in an appropriate way. 67 Chapter II. Discrete symmetrization In §8.2, we shall give a construct ion of a Steiner type rearrangement on the subsets of a product set Z x X f rom a Schwarz (or even Steiner) type rearrangement on the subsets of X. In §8, we shal l show how a convolut ion-symmetr izat ion inequali ty in general implies the preservation of symmetry under certain convolut ions; this fact is similar to Coro l la ry 1.6.2. The results of sections 8-8.3 wi l l be used in the remaining three sections of this chapter. The remaining three sections are concerned wi th discrete analogues of rearrangement results for harmonic measures, Green's functions and part ia l differential equations. In §9 we shall work by using a probabi l ist ic method of Hal iste [56]. We wi l l obtain discrete rearrangement results for generalized harmonic measures (Theorem 9.1), Green's functions (Theorem 9.2) and exit t imes (Theorem 9.4). We shal l be work ing wi th a random walk on the product set Z x X, where there is assumed to be an appropriate convolution-rearrangement inequali ty for a Steiner type symmetr izat ion on X. In §9.1.1 we shall state our basic assumptions on the kernel of our random walk. In §9.1.2 we shal l show that our basic assumptions are satisfied if X is one of our previously considered graphs Z , Z „ , Tv and Hs- In §9.1.3 we shall define our random walk o n Z x I given the kernel discussed in §9.1.1. In §9.1.4 we wi l l construct a generalized harmonic measure based on our random walk en-countering dangers of various probabi l i t ies at various points. More classical discrete harmonic measures are given by the special case where these probabil i t ies can only have the values 0 and 1. In the same section we shall state our main result on rearrangement and generalized har-monic measure (Theorem 9.1), which result generalizes work of Quine [90]. Then , in §9.1.5 we shall define an analogous generalized Green's funct ion as the expected number of times that our random walk visits a given point. We shall also give our main result on rearrangement and this generalized Green's funct ion (Theorem 9.2). Then , we proceed to the proofs. In §9.2 we make a prel iminary reduct ion. F ina l ly , in §9.3 we shall give our result on exit t imes (Theorem 9.4) and the probabi l ist ic proofs of our theorems. These proofs wi l l be based on an i terat ion of our assumed convolut ion-rearrangement inequali ty (see L e m m a 9.1). M u c h of the mater ial f rom §9 is adopted and extended f rom the author 's paper [83]. 68 Chapter II. Discrete symmetrization In §10 we shall prove a discrete version of Beurling's shove theorem [23, pp. 58-62] by using the results of §9. We shall return to the shove theorem and consider the continuous case in §IV.8. Much of the material of §10 is also taken from the author's paper [83]. Finally, in §11 we develop a very general method for proving results such as Theorems 9.1 and 9.2. We shall prove a quite general version of these results (Theorem 11.1) by using a modifica-tion of the methods of Baernstein [11] and Weitsman [99]. The technical difference is that while Baernstein had used a specially-constructed elliptic differential operator and a standard max-imum principle, we instead use a customized maximum principle and no specially-constructed differential operator since the construction of such an operator would likely run into problems in our discrete setting. Our methods allow us to take a convolution-symmetrization inequality and from it obtain a rearrangement theorem for discrete analogues of the nonlinear partial dif-ferential equation — Au(x) = —c{x)u{x) + <p(u(x)) + X(x), where c > 0 and <p is increasing and convex. First in §11 we define our difference operators and give the assumptions on our kernel functions, effectively assuming that we have a convolution-rearrangement inequality. In §11.1 we give a precise statement of the difference equations (more generally, difference inequalities) that we are working with. In §11.2 we give our main rearrangement theorem and prove it. Then, in §11.3.1 we prove that this rearrangement theorem even gives a meaningful result for the identity rearrangement—it yields a monotonicity result for our system of difference equa-tions. In §11.3.2 we demonstrate how Theorem 11.1 implies Theorem 9.1 and mention that it also similarly implies Theorem 9.2. Finally, in §11.3.3 we show that Theorem 9.1 implies the exit times result of Theorem 9.4. This shows that Theorem 11.1 can also be used to prove results about exit times. On first reading, the reader may wish to omit sections 6.2.2, 6.2.4, 7.1, 9.2, 9.3, 10 and 11.3.3. It is worth noting that results in a spirit somewhat resembling some of our work on discrete rearrangement can be found in the book of Marshall and Olkin [71]. 69 Chapter II. Discrete symmetrization 1. Definitions and basic results For the convenience of the reader, while developing the general theory we shall use the concrete metric space Z equipped with the standard metric d(m, n) = \m — n\ as an example. Let M be a countable set. In this chapter, we say that two real functions / and g on M are equimeasurable if they are equimeasurable with respect to counting measure on M, i.e., if I/A| = \g\\, for every positive A, where | • | indicates the cardinality of a set and f\ and g\ are defined as in §1.2. We assume that we are given a specific fixed well-ordering -< of M with the property that every element of M has at most finitely many predecessors, i.e., an ordering under which is M is order isomorphic to a subset of (ZQ ,<). (If M = Z then we will use the ordering 0-<l-<—1-<2-<—2-<3-<---.) Note that in our notation x < y implies that x ^ y. Then, the (discrete) symmetric decreasing rearrangement of a real function / on M is defined to be the unique -^ -decreasing1 function / * which is equimeasurable with / . We call a -^ -decreasing function symmetrically decreasing. Given a subset S of M, if = oo then define 5* = M, and otherwise let S* = {ei,e2,... ,e|5|}, where e\ < e2 -< e3 -< • • • is an enumeration of M. We call S symmetric if S = 5*. Note that if we let T = Q be all the subsets of M then it is easy to see that #: J7 —> Q is a symmetrization in the technical sense of §1.2. Moreover, f* as defined above is easily seen to agree with the definition in (1.2.1) since it is easy to see that in our current definition we have (/#)A = (fx)*, which agrees with Theorem 1.2.1. Definition 1.1. Two real functions / and g on a set X are said to be similarly ordered providing that for every x and y in X we have f(x) < f(y) if and only if g(x) < g(y). A function F is -^decreasing if x -< y always implies F(x) > F(y). 70 Chapter II. Discrete symmetrization The fol lowing result is very well known [58, T h m . 368] 2 and has sl ightly less restrictive hy-potheses than Theorem 1.2.3. T h e o r e m 1.1 ( H a r d y - L i t t l e w o o d ) . For a pair of real functions f and g we have E fWsw ^ E t1-1) assuming that both sides make sense. If moreover the left hand side is finite, then equality holds if and only if f and g are similarly ordered. 2. A general framework for proving discrete master inequalities Let ( M , d) be a metr ic space such that M is at most countable. G iven two real functions / and g on M, an increasing posit ive convex funct ion $ on [0, oo) and a decreasing posit ive funct ion K on [0, oo), define Q(f,g;*,K)= E *(!/(*) " S(y)\)K(d{x,y)). Then , we say that t h e m a s t e r i n e q u a l i t y h o l d s f o r M w i t h t h e o r d e r i n g -< providing that for al l real / and g, and al l $ and K as above, we have Q(f,g;$,K)>Q(f*,g#;$,K). (2.1) Such inequalit ies were considered in continuous cases by Beckner [18, 19, 20, 21] who generalized the work of Baernstein and Taylor [15]; see also [11]. We now outl ine a general approach to proving master inequalit ies in our discrete sett ing v ia an adaptat ion of the Baernstein-Taylor-Beckner approach; our proofs are based on the descript ion of the approach as given by Baernstein [11]. Because we cannot hope to get (2.1) for a general M and a general -<, we must make a number of assumptions. (Indeed, not every M has a well-ordering -< under which the master inequal-i ty holds—see Theorem 7.1 in §7, below, for two very simple and natural counterexamples.) 2 T o see that [58, Thm. 368] implies our result, note that ( M , -<) is order isomorphic to ([0, N] f l Z j , <) for some ./v e zJu {oo} . 71 Chapter II. Discrete symmetrization Basically, we shall make assumptions which will ensure that the method of proof given for the continuous case in [11] works. Then, to prove the master inequality in any concrete case, we will only need to prove the assumptions in that case. Recall that an isometry p of M onto itself is said to be an involution if p o p is the iden-tity function. Whenever we say "involution" we shall mean "isometric involution"; the term "isometric" will sometimes be explicitly given and sometimes dropped. Given an involution p of M, we set ftp = {x G M : x -< px}, and Fixp — {x G M : px = x}. Then, fip, Fixp and pf)p are three disjoint sets whose union is M. Assumption A. There is a transitive set 3 of isometric involutions of M such that whenever p is in 3, and the points x and y are both in S)p, then the inequality d(x,y)<d{x,py) (2.2) holds. The assumption of transitivity just says that for all x and y in M there exists a p G 3 such that px = y. In the case M = Z, we may take 3 to be the collection of all isometric involutions of Z. It is easy to see that any element of 3 is defined by px = xo — x for some XQ = xo(p) 6 l Z= f {0 , ± | , ± 1 , ± | , . . . . } . It is also easy to see that for XQ < 0, we have 9)p = {x G Z : x > Zo } , while for x0 > 0 we have 9yp = {x G Z : x < x0}. Finally, it is likewise easy to verify that Assumption A holds. The principal result of the present section is as follows and as mentioned before is essentially due to Baernstein, Taylor and Beckner. A proof will be given later in this section. 72 Chapter II. Discrete symmetrization Theorem 2.1 (Master Inequality). If Assumption A holds, then the master inequality for M with ordering -< must also hold. I.e., if Assumption A holds, then for any real f and g, and for any functions $ and K from [0, oo) to [0, oo) such that $ is increasing and convex and K is decreasing, we must have Q(f,g;*,K)>Q(f*,g*;*,K). (2.3) Now, for x £ M and r £ [0, oo), let V(x;r) = \{y : d(x,y) < r}\. In order to get an appropriate convolution inequality, we make the following assumption. Assumption B. For every fixed r £ [0, oo) the number V(x;r) is finite and independent of x £ M. An easy immediate consequence of Assumption B is that if K is any positive real function then the function x E K(d(x,y)) y€M is actually constant. -Remark 2.1. Note that if V(x; r) is always finite and there exists a transitive set of isometric automorphisms of M, then Assumption B holds automatically. In particular, if V(x; r) is always finite and Assumption A holds, then Assumption B holds likewise. Our convolution-rearrangement inequality is then as follows. Theorem 2.2. Assume that the master inequality holds for M with the ordering -<, and that Assumption B also holds. Let K be a decreasing positive function on [0, oo). Then, for any positive functions f and g we have E f(*)KWx,y))g(y)< E f*(x)K(d(x,y))g*(y). (2.4) x,y£M x,y£M Moreover, this also holds in the case where f and g are not necessarily positive, providing that they have finite support. 73 Chapter II. Discrete symmetrization Proof. B y approximat ion (monotone convergence theorem) it suffices to consider the case where / and g have finite support . In that case we may also assume that K has bounded support ; indeed, if we set K(t) = 0 for t sufficiently large that t > d(x, y) for all x € supp / U s u p p / * and y G supp 17 U supper* , then neither the left nor the right side of (2.4) wi l l change. Wr i te yeM B y Assumpt ion B , as noted above, ot does not depend on x and is finite (the latter fact uses the boundedness of the support of K). Let $(£) = t2. Then we have Q(f,g;*,K) = -2 £ f(x)K(d(x,y))g(y) + a j ^ f2(x) + a j ^ / ' f a ) , (2-5) x,y£M x£M y(zM and the analogous identi ty for Q(f#, g#;$>, K) also holds. Bu t , £/ 2(*)=E(/ #) 2(*)> (2-6) x£M xeM since / and are equimeasurable. The identi ty (2.6) also holds wi th g in place of / . Combin ing (2.6) and its analogue for g w i th the master inequali ty Q(f,g;$,K) > <3( /* , 3>, K), and wi th (2.5) and its analogue for / * and g*, we obtain (2.4) as desired. • Note that we have seen that if M — Z then Assumpt ion A holds. Assumpt ion B is tr iv ial in this case, so that (2.4) must also hold. In fact, (2.4) in this case precisely coincides with [58, T h m . 371]. T h e o r e m 2 .3 (cf . [58, T h m . 375 ] ) . Suppose that Assumptions A and B are both satisfied. Given a symmetrically decreasing real valued function g on M which either has finite support or is positive, and given a decreasing function K on [0, 00), the function K * g defined by {K*g)(x) = K(d(x,y))g(y) yeM is symmetric decreasing on M. 74 Chapter II. Discrete symmetrization Proof. Our proof is adapted f rom that of [58, T h m . 375]. F i x any positive function / on M wi th finite support . Then , by Theorem 2.2 we have E*</)(*)< E f*(x)K(d(x,y))g*(y)=Y/f*(x)(K*9)(x), xeM x,y£M xeM since g = g*. Therefore, by [58, T h m . 369] it follows that and K * g are simi lar ly ordered. Since this holds for any non-negative / , it follows that K * g must be symmetr ic decreasing. • We now proceed to prove Theorem 2.1. We need a lemma first, which we state in a slightly stronger form than is needed for Theorem 2.1; we wi l l later need the stronger form in the proof of the condi t ion for equali ty in Theorem 6.2. Recal l that a funct ion $ is said to be str ict ly convex if we have the strict inequali ty $(fcc + (1 - t)y) < t$(x) + (1 - t )$ (y ) , whenever x ^ y and t G (0,1) . Lemma 2.1 (cf. [11, Thm. 1]). The master inequality holds for any two point metric space M = {£, v} where £ ^ v. Moreover, if $ is strictly convex while -R'(O) > K(5) where S = d(£, v) is the distance between the two points of M, then Q(f,g;$>,K) = Q(f#,g#;G>, K) if and only if f and g are similarly ordered. The straightforward proof by consideration of the various cases is left to the reader. Now, for p £3, define m a x ( / ( z ) , f(px)), if x € f)p fP(x) = <j min( / (a; ) , f{px)), if x £ p9)p f{x) if x G F i x p. It is easy to see that / and fp are equimeasurable. Note that intui t ively the rearrangement f fp w m often br ing / closer to Indeed, our general strategy for the proof of Theorem 2.1 wi l l be to choose a sequence of isometric involut ions pn such that if / 0 = / and go = g while 75 Chapter II. Discrete symmetrization fn = (fn-i)Pn and gn = (gn-i)Pn, t h e n fn ->• f* and gn ->• Given such a sequence, we wi l l basical ly derive the master inequali ty f rom the fol lowing lemma which is a precise analogue of [11, T h m . 2]. L e m m a 2 .2 . Suppose that p £ 3 and that Assumption A holds. Then, Q(f,9;*,K)>QVP,gp;*,K), (2.7) whenever f, g, $ and K are as in the definition of the master inequality. Moreover if the right hand side of (2.7) is finite, while $ is strictly convex and there exist x and y in 9)p such that both of the following two conditions hold: (i) K(d(x,y))> K(d(x,py)) (ii) (f(x)- f(px))(g(y)-g(py))<0, then strict inequality holds in (2.7). Proof. Assume that Q(fp,gp;$,K) < oo (otherwise use an approximat ion argument). We now have Q(f,g;$,K)= £ a{x,y), where a{x,y) = mf(x)-g(y)\)K(d{x,y)), and (x,y)£M2 where b(X)y) = ${\fp{x)-gp(y)\)K{d(x,y)). Now, I c la im th&tJ2(x,y)eM> a(x,v) = E(*,j/)eJtf' b ( x > J/)» w h e r e M' = {(x, y) £ M X M : x £ F\xp or y £ F ix / ) } . 76 Chapter II. Discrete symmetrization For, write M ' = M i U M 2 U M 3 , where Mi = Fix p x Fix p, M2 = Fixp x (SjpUp?)p), and M3 = {Si)pUp^p) x Fix p. Note that the M,- are pairwise disjoint. To prove the claim it suffices to prove that (x,y)eM{ (x,y)eMi for i = 1,2,3. In fact, it suffices to show it for i = 1,2 since the case i = 3 is analogous to i = 2. Now, if (x,y) G Mi then /p(a?) = /(#) and yp(y) = g(y) so that a(x,y) = b(x,y), and so the proof in the case where i = 1 is complete. Suppose i = 2. Note that E a(x>2/)= E (a(a;'y) + a(:c'Py))' (2-8) (x,y)€M2 x£Fixp yes)p and the analogous expression holds with b in place of a. But if £ G Fix p and y G f)p then d(a;, y) = d(px,py) = d(x,py) since p is an isometry fixing x. Then, a(z, y) + a{x,py) = K{d(x, y))M\f{x) - g{y)\) + - y(py)|)], and b(x, y) + b(x, py) = K(d(x, y))[<I>(|/P(z) ~ 9P(V)\) + " 9P(py)\)]-But it is easy to see that the sets {/(z) - g(y),f(x) - g{py)} and ifp(x) - 9p(y),fP(x) - gP(py)} are equal since fp(x) = f(x) and since the sets {g(y),g(py)} 77 Chapter II. Discrete symmetrization and {9p(.y),9P(py)} are equal by definition of gp. It follows that a(x, y) + a(x, py) = b(x, y) + b(x, py) so that by (2.8) the proof of the case i = 2 is completed, and so the claim is proved. Because of the claim, we have Q(f,g;*,K)-Q(fp,gp;$,K) = £ (a(x, y) - b{x, y)). (2.9) (x,y)eM2\M' But, Y a(x,y)= Y (a(xiy) + a(xipy) + a(pxiy) + a(px'Py)) (x,y)eM2\M> {x,y)ef)l (2.10) and of course the same expression holds with b in place of a. Let A(x, y) = a(x, y) + a(x, py) + a(px, y) + a(px, py) and B(x, y) = b(x, y) + b(x, py) + b(px, y) + b(px, py), for x and y in $jp. Then by (2.9) and (2.10) we have Fix x and y in Sjp. Consider the two point metric space X = {1, 2} with a metric D defined by D(l, 2) = 1, and with the well-ordering 1 -< 2. Define F(l) = f{x), F(2) = f(px), G(l) = g(y) and G(2) = g(py). Then, F#(l) = fp(x), F#(2) = fp(px), G*(l) = gp{y) and G*(2) = 9p{py)i where the symmetric decreasing rearrangement on X is defined in terms of the ordering <. Define K'(0) = K(d(x,y)) and K'(l) = K(d(x,py)). Note that d(x,y) = d(px,py) and d(x,py) = d(px,y) since p is an isometric involution. It is easy to see then that A(x,y) = Q(F,G;®,K') and B(x,y) = Q(F*,G*;$,K'). Thus, A(x,y) > B(x,y) by Lemma 2.1. The desired inequality then follows by (2.11). Moreover, if $ is strictly convex while x and y are in S)p and satisfy conditions (i) and (ii), then F and G will fail to be similarly ordered (because of (ii)) and we will have K'(l) < K'(0) (2.11) 78 Chapter II. Discrete symmetrization (because of (i)). Thus, in that case we will have A(x, y) > B(x, y) by Lemma 2.1, and since we have already seen that A(x', y') > B(x',y') for all other pairs of x' and y' in Sjp, we conclude by (2.11) that in that case Q(f,g;®,K) > Q(fp,gp;$>, K). • It only remains to prove Theorem 2.1 itself. Proof of Theorem 2.1. For m £ Z + , let [TO] be the set of the first m elements of M (with respect to -<). Let [0] be the empty set. For conciseness, write Q(-,-) instead of Q(-, •; K). Define a real function I/J on pairs of non-negative integers by tp(m,n) = 2min(m, n) if m ^ n and •0(m, TO) = 2m — 1. Given a real function F on M, let p(F) = sup{m:F\[m] = F* }, where F\^ is the restriction of F to the set [TO]. Clearly F = F* if and only if p(F) = \M\. We shall inductively define a sequence {(/n, 5n)}^Lo °f P a i r s of functions with the following properties: (a) fo = / and g0 = g (b) fn and gn are equimeasurable with / and g respectively (c) Q{fn-l,9n-\) > Q(fn,9n) (d) 1>(n(fn),it(gn)) > min(n- 1 , 2 | M | - 1 ) . Assume this has been done. By (a) and an inductive application of (c) we find that Q(f,9)>Q(fn,9n) (2.12) for all n. Conditions (b) and (d) imply that $(\fn(x) - gn{y)\) tends to $(\f#(x) - g*(y)\) as n —>• oo for any fixed x and y in M; indeed for n sufficiently large (the size of n depending on x and y) we will have fn{x) = f*(x) and <7n(y) = g*(y). Also observe that if M is finite, then in fact fn and gn eventually stop changing with n, since if /4>{p{fn)i p{9n)) > 2 |M| — 1 then we 79 Chapter II. Discrete symmetrization must have p(fn) — p{gn) — \M\ and so /„ = /#. and gn = g#. Now in any case by Fatou's lemma (the application of which is not necessary if \M\ < oo), liminf Q{fn,gn) > V lim inf $(|/n(z) - gn(y)\)K(d(x, y)) n—>oo » n—>oo = £ $(|/#(x)-ff#(j/)|)A'(4x,y)) = g(/#,ff#). From this and (2.12) we obtain (2.3) as desired. Define f0 = f and g0 = g. We must construct fn and gn for n > 1. Before we proceed to do this, I first claim that if we have any real function F on M and p G 3 , then M^)>Mn (2-13) For, let m = M(F), so that F|[mj = ^*|[m]- ft suffices for us to show that Fp|jmj = F\[m]-But first fix X\ G [m]. Choose any x2 G M such that x\ -< Z2- I claim that we then have F(zi) > F(x2). For, if x2 £ [m] then F(xi) = F*(x1) > F*(x2) = F(x2). On the other hand, if x2 fi [m] then we can use the equimeasurability of F and the definition of F * together with the fact that F|jm] = -f7*!^ ] to prove the inequality F(x\) > F(x2). We can now show that Fp(x) = F(x) if x G [TO]. For, x must lie in in Fixp or in pSjp. Suppose first that x G Sjp. Then x -< px. Also, Fp(x) — max(F(a;), F(px)), and so Fp(x) = F(x) as F(x) > F(px) by the work of the previous paragraph (just let x\ = x and x2 = px). Now, if x G Fixp then Fp(x) — F[x) automatically. Finally, consider the case where x G p$)p. Then, px -< x and so Fp(x) = mm(F(x), F(px)). Since x G [m], we must likewise have px G [m] and so F(px) > F(x), upon letting X\ = px and x2 = x in the work of the previous paragraph. Thus, Fp(x) = mm(F(x), F(px)) = F(x) as desired, and so (2.13) has been proved in all cases. Now, suppose fn-\ and gn-\ have been defined and that n-2< tMU-^nign-!)) < 2\M\ - 1. (2.14) (If V(A*(/n-i),M(0n-i)) = 2\M\ - 1, then just let /„ = /„_! and gn = gn-i-) We either have p(fn-i) < p(gn-i) or p(gn-i) < p(fn-i). We only consider the first case, since the second 80 Chapter II. Discrete symmetrization can be handled by just using the first and interchanging / and g in the construction. Let m = p(fn-i). If we can construct fn and gn such that M / n ) > m a n d p(Sn) > M(5n-l), (2-15) then we will have ip(p(fn), p(dn)) > i>(fi>(fn-i), p(dn-i)), and (d) will be verified. To do this construction, let x be the (m + l)st element of M, counting with respect to -< (such an element exists since (2.14) implies that m < \M\). Since m = /i(/„_i), it follows that fn-i(x) / f*(x). The equimeasurability of / and / * together with the fact that /n_i|jm] = /*|jm] implies that there exists a y with x < y and /„-i(y) = f#(x); moreover, we will have /n-i(y) > fn-\{x)-Let p € 3 be an isometric involution swapping x and y. Then, since /n_i(y) > /n_i(x) while £ £ iop as a; -< y = pa;, it follows that (/„_i)p(a;) = /„_i(y) = f*(x). On the other hand, M ( / n - i ) p ) > " 1 by (2.13) so that ( / „ _ i ) p | w = f*\[my Since [m + 1] = [m]U{x} it follows that (fn-i)P\[m+1] = f*\[m+1] and so p((fn-i)p) > m+1. Furthermore, again by (2.13) we also have p((gn-x)p) > p{gn-i). Letting /„ = {fn-i)P and gn = (gn-i)P, we obtain (2.15). Condition (b) follows from the definition of (-)p, while condition (c) follows from Lemma 2.2. • 3. The general case of graphs The purpose of this section is to show that in the case of a graph, the full Assumption A from the previous section is implied by an apparently weaker variant. We first give a definition of a graph in the sense in which we will be using this term. D e f i n i t i o n 3 . 1 . A g r a p h G consists of a countable set VertG of vertices and a set EdgeG of edges, where each edge is a two point set {v, w} with v ^ w in VertG, and where we assume that each vertex lies on at most finitely many edges. Thus, our graphs are undirected, at most countable, with all vertices of finite degree, with no self-loops, and with no multiple edges between a pair of vertices. We will often identify a graph G with the collection VertG of its vertices, using the same notation G to denote both. 81 Chapter II. Discrete symmetrization D e f i n i t i o n 3 .2 . A sequence of vertices v\, v2, • • • , vn of a graph is said to be a p a t h o f l e n g t h n — 1 j o i n i n g v\ w i t h vn if is an edge for al l i € { 1 , . . . , n — 1}. The points vi and vn are referred to as the e n d p o i n t s of the path. For convenience, we consider a vertex v standing by itself to be a path of length 1 jo in ing v wi th v. D e f i n i t i o n 3 .3 . A graph is said to be c o n n e c t e d if for every pair v ^ w of dist inct vertices there is a path jo in ing v wi th w. Suppose that G is a connected graph. Let d be g e o d e s i c d i s t a n c e on G, i.e., for a pair of vertices v and w we let d(v, w) be the length of the shortest path from v to w if v ^ w. We put d(v,v) = 0. Moreover , the tr iangle inequali ty can be easily verif ied, so that (G,d) is indeed a metr ic space. A g e o d e s i c is defined to be any path jo in ing v and w whose length is precisely d(v,w). In general, there may be more than one geodesic jo in ing v and w. We say that the vertices v and w of a graph G are a d j a c e n t if {v, w} is an edge of G. Note that no vertex is adjacent to itself. We now give a few more definit ions which wi l l be useful later. D e f i n i t i o n 3 .4 . A f u l l s u b g r a p h H of a graph G is a graph such that V e r t / 7 C V e r t G and such that whenever v and w are vertices of H then {v, w} is an edge of H if and only if it is an edge of G. We wi l l often identify a ful l subgraph wi th its collection of vertices. D e f i n i t i o n 3 .5 . The degree S(v) = 8Q{V) of a vertex v in a graph G is the number of edges of G containing v. D e f i n i t i o n 3 .6 . A graph is said to be o f c o n s t a n t d e g r e e if every one of its vertices has the same degree. D e f i n i t i o n 3 .7 . If G and G' are two graphs, then a g r a p h i s o m o r p h i s m <f> of G onto G' is a bijective map from V e r t G to V e r t G ' such that whenever v and w are vertices of G , then {v, w} 82 Chapter II. Discrete symmetrization is an edge of G if and only if {<f>(v), <f>(w)} is an edge of G'. In such a case the graphs G and G' are said to be isomorphic. A g r a p h a u t o m o r p h i s m <j> of G is a graph isomorphism of G onto itself. If G and G' are connected graph and equipped wi th the metric d described above, then the notion of a graph isomorphism of G onto G' precisely corresponds to the notion of an isometry of Ver t G onto V e r t G ' . Suppose now that the connected graph G is equipped wi th a well-ordering -< such that each element has at most f initely many predecessors. The fol lowing result shows that when verifying Assumpt ion A for a concrete graph, we only need to check (2.2) for x, y and p such that d(x, py) = 1. P r o p o s i t i o n 3 . 1 . LetG be a connected graph with geodesic distance d and an ordering -<. Fix a graph involution p of G. Suppose that for every x and y in $)p if x and py are adjacent then x and y are either adjacent or equal. Then, for any x and y in Sjp we have d(x,y) < d(x,py). Proof. Proceed by induct ion on n = d(x,py). F i rs t , if n = 0 then x = py. Since p is an involut ion, we likewise have px = y. Now, x -< px as x G fyp, hence x -< y. O n the other hand, y -< py = x, l ikewise, and we have a contradict ion. Secondly, if n = 1, then we are done by our hypotheses, since d(x,py) = 1 means precisely that x and py are adjacent. Thus , assume that n > 1 and that the proposit ion has already been proved whenever d(x,py) < n. Let x — X Q , X \ , . . . , xn-i, xn = py be a geodesic jo in ing x wi th py, where X{ is adjacent to whenever 1 < i < n. Note that d{xi,py) = n - 1. (3.1) We split the rest of the proof into two cases depending on whether X \ lies in 9)p U F i x p or in pf)p. Suppose first that x\ G f)p U F i x p . I c laim that then d(xi,y) < n — 1. Assuming this 83 Chapter II. Discrete symmetrization claim, we have d(x,y) < d(x,Xi) + d(xx,y) < 1 + (n — 1) = n, as desired, and the proof is complete. To prove the claim, there are two subcases to consider, depending on whether x\ lies in Fixp or in Consider first the subcase of x\ £ Fix p. Then, d(xx, y) = d{px\, py) = d(xi, py) = n — 1 by (3.1), where we have used the fact that p is a graph automorphism fixing x i , and so the proof of the claim is complete in this case. Now, suppose that x\ £ 9)p. By (3.1) we have d(xi,py) = n — 1 < n and thus by the induction hypothesis we conclude that d{x\,y) < n — 1, as desired. def The remaining case is that of x\ £ pf)P- But in that case we have z = px\ £ 9)p. Now, x and pz = Xi are adjacent, so that by assumption we must have x and z adjacent or equal, so that d(x, z) < 1. But, d(x, y) < d(x, z) + d(z, y) < 1 + d(z, y) = 1 + d(pxu y) = 1 + d{xupy) = n, where in the second last equality we have used the fact that p is a graph involution, while the last equality was a consequence of (3.1). The proof is thus complete. • 4. The octahedron edge graph We now proceed to prove that Assumptions A and B of §2 are satisfied for a few concrete graphs. Our first and simplest example is the octahedron edge graph, a graph with only 6 vertices so that the value of the resulting inequalities is probably only didactic. Let Hs be the edge graph of an octahedron (see Figure 4.1). We shall use the ordering -< induced by the labels given in Figure 4.1. Theorem 4.1. The octahedron Hs with the ordering shown in Figure 4.1 satisfies Assump-tions A and B of §2. Proof. Assumption B is trivial. Now, let 3 be the collection of all involutions of H%. The easy verification of the transitivity of J is left to the reader. 84 Chapter II. Discrete symmetrization Figure 4.1: The edge graph Hs of the octahedron In light of Propos i t ion 3.1, it suffices to prove that if p is in 3 while x and y are in S)p wi th x and py adjacent, then x and y are either adjacent or equal. To obtain a contradict ion, suppose that we are given x and y in 9}p w i th x and py adjacent, but d(x,y) > 2. W i thou t loss of generality we may assume that x -< y (otherwise, s imply exchange x and y and note that the involut ive character of the automorphism p implies that we have d(x,py) = d(px,y)). Now, then, we have four vertices x, y, px and py in the octahedron. These vertices are dist inct, and moreover d(x,y) > 2 and d(px,py) > 2. Since x and y are in F)p, we must have x -< px and y •< py-Given any vertex X £ Hs, there exists a unique vertex X G Hs such that d(X, X) > 2. Note that d(X,X) = 2 and X = X. Thus , since d(x,y) > 2 and d(px,py) > 2, we must have y = x and py = ~px. Recal l that we have assumed that x -< y. Look ing at F igure 4.1 we see that 0 = 5, 1 = 4 and 2 = 3 (where we have identified the vertices wi th their labels). Thus , our assumption that x -< y = x implies that x £ {0,1,2} . Now, if x = 0 then y = 0 = 5. Bu t y -< py, and 85 Chapter II. Discrete symmetrization this is a contradict ion since there is no vertex bigger than 5. Suppose now that x = 1. Then y = 1 — 4. Since y -< py, we must have py = 5. Bu t d(px,py) > 2, so pa; = py = 5 = 0. Bu t then pa; -< x, contradict ing the fact that x -< pa;. The remaining case is when x = 2. Then y = 2 = 3. We have x -< px and pa; 7^ y; also, y -< py. Thus , pa; and py both lie in {4, 5}. Bu t px = py and so we have a contradict ion as 4 7^ 5. . • The author is grateful to Professor Greg Kuperberg for drawing the author 's attention to the octahedron by point ing out that there is a set of reflections in planes about the origin which is transit ive on the octahedron's vertices when the octahedron is inscribed in the sphere. (Note that this fact could be used to give another proof of Theorem 4.1, essentially by restrict ion of the spherical case of Baernstein and Taylor [15].) 5. The circle graphs Zn Before proceeding to the case of the regular tree, we give the easier case of the cycl ic group Z „ . Define d(m, TO') to be times the length of the shortest arc of the unit circle in the complex plane jo in ing the points e2mm/n w i th e 2 m m ' ln. Note that d agrees wi th the graph distance if we consider Z n to be the graph composed of n points joined together in a circle (see Figure 5.1). Order Z „ by 0 ~< 1 -< - 1 -< 2 -< - 2 -< • • •, where the "• • • " here indicates that we keep on going unt i l we exhaust al l the elements of Z „ . See F igure 5.2 for an example of how symmetr izat ion works on Z n . Now, let Z, be a line through the origin of the complex plane at an angle which is an integer mult iple of ir/n. Define p^m to be TO' where ml is chosen so that e2mm'ln is the reflection of e 2 7 " m / n in the line L. Let 3 be the collection of al l such p^. It is easy to see that J is a transit ive collection of isometric involut ions of Z m . Now, given L as above, if L 7^ R then let H be the component of C\L which contains the point 1 G C , and if L = K then let H be the open upper half plane. It is easy to verify by drawing diagrams that TO 6 9)PL if and only if e2mm/n g JJ Qnt f rom this and the definit ion of the metric on Z m it is easy to see that we 86 87 Chapter II. Discrete symmetrization have Assumption A satisfied. On the other hand, Assumption B is trivial. Hence, we see that (2.3) and (2.4) both hold for Z„ with the ordering as above. This is a discrete analogue of Theorem 1.6.4, minus the discussion of the case of equality. We summarize the findings of this section as follows. T h e o r e m 5 . 1 . The group Z n equipped with the ordering -< satisfies Assumptions A and B of §2. 6. Regular trees D e f i n i t i o n 6 . 1 . A path vi, v2,.. • ,vn in a graph is said to contain a b a c k t r a c k i n g if there is an i 6 { 3 , . . . , n} such that V{ — Vi-2. D e f i n i t i o n 6 .2 . A t r e e is a graph such that between every pair of vertices there is at most one path without any backtrackings. It follows that in a connected tree, geodesies are unique. D e f i n i t i o n 6 . 3 . The p - regu la r t r e e Tp, where p £ {2, 3,...}, is any connected tree such that every vertex has degree exactly p. It is easy to see that a p-regular tree Tp exists, and that any two such trees are isomorphic as graphs. Note that T2 can be naturally identified with Z. See Figure 6.1 for a subset of T 3 . The unique path without backtracking joining vertices v and w will necessarily be a geodesic. Its length (i.e., the number of edges in it) shall be denoted by d(v,w) as before; we will write [v, w] for the set of vertices (including the endpoints) lying on this path. We write [u, v] = {v}. 88 Chapter II. Discrete symmetrization Figure 6.1: The ordering on the tree T 3 . The portion T3,A of the tree is shown. Note that h(0) = 0, h(l) = h(2) = h(3) = 1, h(4) = ••• = h(9) = 2, h(l0) = ••• = h(21) = 3 and h(22) = ••• = h(45) = 4. 89 Chapter II. Discrete symmetrization In Figure 6.1, for instance, we have [5,24] = {5,1,4,11,24} and d(5,24) = 4. In Figures 6.5 and 6.6, below, two more examples of geodesies are shown. We distinguish one vertex O which we shall call the r o o t of Tp. Given a vertex v of Tp, we write h(v)d=d{v, 0) for the h e i g h t of this vertex (see Figure 6.1, where we have let 0 — 0.) For k > 1 there are precisely p(p — l)k~1 vertices which have height exactly k. We write TPtk for the full subtree of Tp defined by all vertices whose height does not exceed k. Note that the subtree pictured in Figure 6.1 is T 3 )4. The subtree TPik can be called a "geodesic ball" in the regular tree. Given a vertex v, we say that a vertex to is a d e s c e n d a n t of v and that v is an a n c e s t o r of w providing v is contained in the geodesic from O to w. We say that w is a c h i l d of v or that v is the p a r e n t of w providing that w is a descendant of v which is adjacent to v. Note that in general each vertex other than O has a unique parent. Every vertex other than O has precisely p—1 children; the vertex O has p children. Write Descv for the set of all descendants of a vertex u; note that we always have v € Descu. Write Children v for the set of all children of a vertex v; note that Children v C (Desct>)\{u}. Note also that if the vertices v and w are adjacent then either v is the parent of w or w is the parent of v. To illustrate these definitions, note for instance that in Figure 6.1 we have T 3 ) 4 n Desc 4 = {4,10,11, 22,23,24, 25} and Children 4 = {10,11}, while the parent of 4 is 1. D e f i n i t i o n 6 .4 . We say that a well-ordering -< of Tp is s p i r a l - l i k e providing the following conditions hold for all vertices vr w, vi, v2, w\ and w2: (a) if h(v) < h(w) then v -< w (b) if h ( v i ) = h(v2) while v\ -< v2 and Wi is a descendant of V{ for i = 1, 2, then W\ -< w2. 90 Chapter II. Discrete symmetrization Such an ordering certainly exists, and may be chosen by induction (see Figure 6.1, where the spiral-like ordering -< is induced by the standard ordering < on the integers forming the labels). A spiral-like ordering is unique up to isomorphism. (I.e., if -<i and -<2 are spiral-like well-orderings of Tp then there is a unique graph automorphism a of Tp such that v -<i w if and only if av -<2 ctw- The inductive construction of a is quite easy and left to the reader.) Given such an ordering, we can form the symmetric decreasing rearrangement # with respect to it as in §1. Note that if G C Tp is finite then there is a unique k £ Z^ such that TPtk C G* C TPtk+i- Thus G* is always in some sense close to being a geodesic ball. If for some k we have |G| = \TP}k\ then in fact G* is precisely equal to the geodesic ball Tp^-6.1. The master inequality on regular trees Our main result about regular trees is that they, like the octahedron and the circle graphs, satisfy Assumptions A and B of §2 and thus also satisfy a master inequality. T h e o r e m 6 . 1 . The tree Tp equipped with any spiral-like well-ordering satisfies Assumptions A and B of §2. Before we prove this, we need a crucial result about spiral-like well-orderings and geodesies. L e m m a 6 . 1 . Let Tp be equipped with a spiral-like well-ordering Then for each geodesic Vi, u 2 , • • • , w n_x, vn joining vi with vn such that n > 4 and v\ -< vn, we have u 2 -< vn-\. Note that in the setting of the Lemma we have n = 1 + d(v\, vn). Proof of Lemma. Let a geodesic vx,v2,... ,vn-i,vn be given with v\ -< vn. We must have /i(?j2) = h(vi) ± 1 for some choice of ±. Suppose first that / i(u 2) = h(v\) + 1. Then since we have a geodesic it follows that h(v3) = h(v2) + 1, and h(v^) = h(v3) + 1 and so on, so that it follows that h(vn-i) > h(v2) and so v2 -< u n _ i by property (a) of spiral-like orderings. Now 91 Chapter II. Discrete symmetrization suppose that h(v2) = h(vi) — 1. If h(vn) > h(vi) then, since / i ( u „_ i ) = h(vn) ± 1 for some choice of ± , it follows that h(vn-i) > h(vi) — 1 = h(v2), and so v2 -< vn-i, again by property (a) of spiral- l ike orderings. Thus , the remaining case is when h(vn) < h(vi). Aga in by property (a) we must in fact have h(vn) = h(v\). The non-backtracking property of geodesies then guarantees that h(vn-x) = h(vn) — 1 and that vi and vn are children of v2 and respectively. Moreover, h(y2) — h[yn-x). Since n > 4 we have v2 ^ If we had i>„_i -< v2 then by property (b) of spiral- l ike orderings we would have vn -< vi, a contradict ion. Thus , we must have v2 -< u „ _ i , as desired. • The rest of this section wi l l be occupied wi th the proof of Theorem 6.1. Proof of Theorem 6.1. It is clear that Assumpt ion B holds. Thus it suffices to verify Assump-t ion A . Let 3 be the collection of al l involut ive graph automorphisms of Tp. It is easy to verify that 3 is t ransi t ive. B y Propos i t ion 3.1 the only other th ing that we must verify is that for p € 3 and v and w in 9)p, if v and pw are adjacent, then v and w are either equal or adjacent. We shall in fact prove that they are always equal. For , assume that u / w . Suppose v and pw are adjacent. Likewise, then, pv and w are adjacent since p is an involutive graph automorphism. Note that the four points v, pw, pv and w are al l dist inct because of the various assumptions above and as p is an involut ive automorphism. In general it is easy to see that given four points a\, a2, a 3 and a± on a tree, wi th a\ and a2 adjacent and wi th a 3 and also adjacent, it follows that there exists an i € {1,2} and a j G {3,4} such that al l four points a\, a2, a 3 and lie on the geodesic [a,-,aj]. App ly ing this to our s i tuat ion, we see that our points v, pw, pv and w must all lie on one of the geodesies [u,pu], [pw,pv], [v, w] and [pw, w]. In fact, we may reduce the cases even further. Let P = {v, pw,pv, w}. If P C [pw,pv], then likewise we must have P = pP C p[pw,pv] = [tw,u], since p preserves geodesies. Conversely, if P C [pw,pv] then P C Now, if two points x and y are contained in some geodesic [x', y'] then [a;, y] C [x', y \ It follows from the above, then, that if P C [pw,pv] then P C [ip,v] C [pw,pv], and if P C [w,v] then P C [pw,pv] C u]. 92 Chapter II. Discrete symmetrization Hence, in either case we have [v,w] = [pv, pw], which implies that the sets {v, w} and {pv,pw} are equal, whereas we know that al l four points in P are dist inct, a contradict ion. Moreover , we need not concern ourselves wi th the case P C since upon exchanging w and v the result in this case wi l l follow f rom the result in the case P C [w,pw] = [pw,w], as our assumption that v and pw be adjacent is symmetr ic in v and w as p is an involut ion. Hence we need only consider the case where P C [w,pw]. Wr i te [w,pw] = {wi,W2,... ,wn}, where {wi,Wi+i} is an edge of Tp whenever 1 < i < n — 1, and where w\ = w and wn = pw. Since v and pw are adjacent and v £ [w,pw], we must have w n _ i = v. Now, p is a graph isomorphism and it swaps the endpoints of [w,pw], so that in fact, since [w,pw] is a geodesic while geodesies are unique on a tree, it must map [w, pw] onto itself, w i th pw{ = t o n + i _ t - . Hence, pv = pwn-i = W2- Now, w\ -< wn by definit ion of 9)p as w\ = w £ $jp and wn = pw\. The geodesic [w\,wn] contains at least four points since P contains four dist inct points. Thus , by L e m m a 6.1 it follows that pv = wx -< wn-i = v, contradict ing the fact that v £ Sjp. • 6.2. The Faber-Krahn inequality for subsets of regular trees 6.2.1. Statement of the Faber-Krahn inequality If D is an open set in Kn and A = 5Z^=1 j r r is the ordinary continuous Lap lac ian , then let v\(D) be the smallest str ict ly positive eigenvalue of the operator —A act ing on functions / on D w i th the Dir ichlet boundary condit ion that they vanish on dD. Then , in the case n = 2, Lord Rayle igh [91, §210] conjectured that where D® is a Eucl idean ball of the same area as D. Th is is known as the Faber -Krahn inequali ty [47, 68]. It is in fact true in al l dimensions. We shall prove an analogous inequality where D is a subset of the p-regular tree, while D# is defined in terms of a spiral- l ike ordering and takes the place of D®. 93 Chapter II. Discrete symmetrization D e f i n i t i o n 6 . 5 . G iven any graph G, define the d i s c r e t e L a p l a c i a n A = AQ on G by A/(t,) = -/(«) + -L E /(»). 11 ' weN(v) for a funct ion / on G, where N(v) is the set of al l vertices of G adjacent to v. Thus on Tp the discrete Laplac ian is given by Af(v) = -f(v)+1- E /(»)• V weN(v) O f course in this case \N(v) \ = p for all v. Given a finite non-empty subset G of Tp, let v\(G) be the smallest str ict ly positive eigenvalue of —A on G w i th the Dir ichlet boundary condit ion that our functions vanish outside G. Standard eigenvalue methods (see Theorem 6.3, in §6.2.2, below) let us also compute V\{G) v ia the expression MG)= inf 11(f), (6.1) /€5)(G) where 'D(G) is the set of al l real functions / ^ 0 which vanish everywhere outside G, while is the Rayleigh quotient for G. (We shall verify (6.1) formal ly in the next section.) Note that W ) > since E-/wA^(u) = ^ E (/(«)- /H)(^)-^H), where E = Edge T p . (To verify this last identity, by l inearity it suffices to prove it if the supports of / and g have one point each, in which case the result is easy.) The main result of this section is as follows. The result was inspired by Fr iedman [54, Conjec-ture 4.3]. 94 Chapter II. Discrete symmetrization Theorem 6.2 (Faber-Krahn inequality for subsets of regular trees). Let G be a finite non-empty subset of Tp. Then, See Figure 6.2 for the extremal subsets G* of cardinalities from 1 to 12. In Figures 6.3 and 6.4 on pages 104 and 105, one may find two other extremal subsets together with the corresponding eigenfunctions. Recall that Theorem 6.1 guarantees the validity of (2.4) on Tp. In §6.2.3 we shall use this fact to prove (6.2). Only after that, in section §6.2.4, will we prove the condition for equality. The proof in §6.2.3 will show that (6.2) is valid for subsets of any constant degree graph on which we have the convolution-rearrangement inequality (2.4). However, it is not known, even given an appropriate convolution-rearrangement inequality, whether for a more general graph we can make a classification of the case of equality similar to the one given in Theorem 6.2 for the case of the regular tree. See Problem 7.3 in §7. 6.2.2. Some useful well-known results Let G be a non-empty finite subset of a constant degree graph H. Let 1) be the collection of all real functions on H which are zero outside G but do not vanish identically. Let v\ be the first strictly positive eigenvalue of the operator —A acting on *D. Write (6.2) where G* is defined with respect to any spiral-like well-ordering on Tp. Equality holds if and only if there is an automorphism of Tp mapping G onto G*. 71(f) = E , e G - j » A / » for fev. The following result is very well known, but we give a proof for completeness. 95 Chapter II. Discrete symmetrization 1 2 3 4 5 6 7 Figure 6.2: The extremal subtrees G * of T 3 wi th cardinalit ies from 1 to 12. 96 Chapter II. Discrete symmetrization Theorem 6.3. Assume that G is a non-empty finite subset of a constant degree graph H. Assume that given any vertex v in G there is a vertex w in H\G and a path from v to w in H. Then we have v\ = min 7Z(f). Moreover, the minimum of this functional is achieved at f if and only if f is an eigenfunction of —A with eigenvalue v\. If f is an eigenfunction with eigenvalue v\, then so is \ f\. We have v\ £ (0,1]. Finally, if there exist vertices v\ and v2 of G which are adjacent, then vi(G) < 1 — < 1, where p is the degree of each of the vertices of the constant degree graph H. Proof. Let A = inf 71(f). For / a function on G, write 1/2 ll/l^fe/W) W G / Let S be the collection of all functions / £ D such that | | / | | 2 = 1. Since 7Z(cf) = 71(f) for all c 0, it is easy to see that then \= mi 71(f) = Y,~f{v)Af(v), and that / £ D minimizes 71 over D if and only if //||/||2 minimizes it over 5. Now, S is also easily seen to be homeomorphic in a natural way to the - l)-dimensional sphere, and the function 7Z is continuous in the induced topology, so that by compactness the infimum is attained. Hence there exist minimizers for 7Z over S, and thus also over V. Suppose that / £ T) minimizes A. We shall prove that / is an eigenfunction of —A on G with eigenvalue A. For, fix w £ G. Without loss of generality we may assume that ||/||2 = 1 (else replace / by //H/H2O Let e be the function which is 1 at w and 0 elsewhere. It is clear that for h a sufficiently small real number we have / + he £ 1), and that d dh 7Z(f + he) v (6.3) h=0 97 Chapter II. Discrete symmetrization is well-defined. B y minimal i ty, this derivative must vanish. Bu t , by first-year calculus, this derivative is also equal to ( ( / , A e ) + (e, Af)) (/, /) - (/, Af) (2 (/, e)) (/,/) def where = X^eGcn(v)(3(v). Hence, sett ing (6.3) to zero and using the assumption that ||/||2 = 1 as well as the definit ion of e, we find that £ f(v)Ae(v) + Af(w) = 2 / H f(v)Af(v). (6.4) vEG VEG Bu t if ||/||2 = 1 then YIVEG f(v)^f(v) = —11(f) and in our case IZ(f) — A by minimali ty. O n the other hand, Ae(w) = —1, while clearly Ae(v) = 1/p for v G N(w) and Ae(u) = 0 for v fi {w} U N(w). Thus , J2f(v)Ae(v) = -f(w) + - Y / ( « ) = A / H . vEG P W£N(V) Thus , (6.4) becomes 2A /W = - 2 A / H . Since w € G was arbi trary, it follows that / is indeed an eigenfunction of —A on G wi th eigenvalue A. I now c la im that A > 0. For , we have X ; - / ( « ) M « ) = ; £ (f(v)-f(w))(g(v)-g(w)), ' (6.5) vEG ^ {v,w}EEdgeG for any / and g, so that 71(f) is always non-negative. (We have already noted the displayed identi ty in the case of the tree, and said that it is best proved by first verifying it for / and g whose supports have one point each, and then using l inearity for the general case. Th is works just as well for any constant degree graph.) In fact, 71(f) must be str ict ly posit ive, for the above identity shows that if it is equal to zero then f(v) = f(w) whenever v G G and w is adjacent to it. I c laim that this implies that if 71(f) — 0 then / = 0 if / vanishes outside G. For , f ix v G G. Let v = v\, u 2 , . . . , vn be a 98 Chapter II. Discrete symmetrization path in H such that vn £ H\G. Such a path exists by our assumptions. Shortening the path if necessary, we may assume that vi,... ,vn-i £ G. Then , i terat ing an observation made at the beginning of the paragraph, we see that f(v) = f(vi) = f(v2) = • • • = f(vn). Bu t vn fi G so that f(vn) = 0, and so f(v) = 0 as desired. Hence 1Z(f) = 0 and / £ D are incompatible assumptions. Thus , indeed, we conclude that A > 0. Now, let / be an eigenfunction of — A wi th eigenvalue A'. We have F rom this and the fact that if 1Z achieves its min imum at / £ T) then / is an eigenfunction wi th eigenvalue A, we conclude that indeed A = u\ and that eigenfunctions corresponding to v\ coincide wi th the minimizers of TI over X). The statement that | / | is also an eigenfunction if / is an eigenfunction follows from the ob-servation that 7^(|/|) < TZ(f) (this observation is clear f rom (6.5) and the triangle inequality) which implies that if / is a minimizer of TZ then so is | / | . To show that vi < 1, choose any w £ G. Let e be the indicator function of {w}, as before. C lear ly / £ D . Then Ae(w) = -1 so that vx < 11(e) = 1. Now suppose that v\ and v2 are vertices of G which are adjacent. Let / be the indicator function of {vi,v2}. C lear ly / £ D . Then , -Af(v) = X'f(v). Thus , nf) = = A'. '-LJ \ ui — > ) P for i £ {1, 2}. Let p = 1 - (1/p). Note that \\f\\j = 2. Thus , Af(Vi) = - 1 + -= T = P> as desired. • 99 Chapter II. Discrete symmetrization Definition 6.6. Let G be a graph. Define the relation ~ on the vertices of G by writing v ~ w whenever v is connected by a path in G to w. This is clearly an equivalence relation. Define a connected component of G to be an equivalence class under ~ . Definition 6.7. Let / be a function defined on a graph H, and let G be a subset of H. Then / is said to be superharmonic on G if for all v e G. Definition 6.8. Let C C Vert 17. Define C = C U dC, where dC is the set of vertices v of H such that there exists a v' £ C with v and v' adjacent. The following well-known result is known as the minimum principle for superharmonic functions. It does not require constancy of degree. Theorem 6.4. Let G be a finite subset of a graph H. Let f be non-negative on H and super-harmonic on G. Assume that there exists a vertex w 6 G at which f vanishes. Let C be the connected component of G containing w. Then f vanishes everywhere on C. Proof. We shall prove that if f(v) = 0 for v £ G then f(v') = 0 for every v' adjacent to v. This will suffice to prove the result in light of the definition of C. Now, the condition A/(u) < 0 implies that '• v'eN(v) But f(v') > 0 for v' £ N(v). This immediately implies that /(«') = 0 for all v' £ N(v), as Af{v) < 0 desired. • Finally we give the following also well-known result. 100 Chapter II. Discrete symmetrization Corollary 6.1. Let G be a connected non-empty finite subset of a constant degree graph H. Then the eigenvalue v\ has multiplicity 1, and the nontrivial eigenfunctions corresponding to it do not vanish anywhere on G and have constant sign on G. Proof. Let / be an eigenfunction corresponding to v\. F i rs t we prove that / has constant sign. For , assume that f(w) > 0 for some w £ G. I c laim that / > 0 everywhere on G. For , otherwise the funct ion g = \ f\ — f is not identical ly zero. Bu t , by Theorem 6.3, if / is an eigenfunction corresponding to v\, then so is | / | . Hence, so is g, since the difference of two eigenfunctions wi th the same eigenvalue is also an eigenfunction wi th the same eigenvalue. We then have -Ag = vxg on G. Bu t g > 0 since | / | > / , so that it follows that g is superharmonic and positive. Moreover, f(w) > 0 so that g(w) — 0, which is an immediate contradict ion to Theorem 6.4 if g does not vanish almost everywhere. (Here we have used the connectedness of G). Thus, indeed / has constant sign. Moreover, just as we argued wi th g, we can also use the min imum principle (applied to / if / is positive and to —/ otherwise) to see that / vanishes nowhere. Now, suppose that / i and f2 are nontr iv ia l eigenfunctions corresponding to v\. We must prove that fi = c/2 for some constant c. Since /1 and f2 have constant signs, we may assume that they are both positive. F i x any v £ G. We then have f\(v) > 0 and f2(v) > 0. Choose c so that fi(v) = cf2(v). Let g = /1 - cf2. If g vanishes everywhere, then we are done. Hence, suppose g does not vanish identically. Then g is a nontr iv ia l eigenfunction corresponding to v\, and by our work above, g cannot vanish anywhere. Bu t g(v) = 0, and so we have a contradict ion. • 6.2.3. Proof of not necessarily strict inequality in Theorem 6.2 Define the operator A on the set of real functions on Tv v ia Af(v) = /(«)+ £ f(w). w£N(v) 101 Chapter II. Discrete symmetrization Then, — (1 + p~l)f + p~lAf = A / , so that we can write P \ P J J2veof (v) But, clearly, Af(v) = E K(d(v,w))f(w), (6.7) •weTp where K is the function defined by K(t) = 1 for t < 1 and A'(i) = 0 for t > 1. By (2.4) we thus have J2f(v)Af(v)<J2f*(v)Af*(v) vEG V£G for positive / . On the other hand, by the equimeasurability of / and f* we have E / 2 w = E(/#(-))2-veG vEG Thus, in general, ^( / )>^( | / | )>^( | / | # ) . (6.8) But if / <E T)(G) then clearly we must likewise have | / | # € T)(G*). Hence (6.2) follows from (6.1). 6.2.4. Proof of condition for strict inequality in Theorem 6.2 The reader is warned that this section is perhaps the most difficult and involved in the whole thesis. In order to prove the condition for equality in Theorem 6.2, we first examine the properties of the functions / extremal for the Rayleigh quotient on the domain G*. (We say that / is extremal for 1Z if the minimum of 7Z is achieved at /.) By Theorem 6.3, an extremal / does exist, and may be taken to be positive. From now on we assume that / is a positive eigenfunction corresponding to v = i>\(G*). Then, as in the work of the previous section, we see that nf*)<n/)-102 Chapter II. Discrete symmetrization Bu t by extremal i ty of / and Theorem 6.3 we conclude that / # is also an eigenfunction corre-sponding to z/i(G#), and so by Coro l la ry 6.1 we see that / is a mult iple of / * . Since / and / # are equimeasurable, it follows that / = Hence / is symmetr ica l ly decreasing wi th respect to our spiral-l ike ordering. See Figures 6.3 and 6.4 for examples of two extremal subtrees G* and the corresponding eigenfunctions / . We need to improve the symmetr ic decrease condit ion to some sort of strict decrease condit ion. In the case where G * = TUik for some k, this,condit ion wi l l in effect say that if h(v) < h(w) < k then f(v) > f(w) > 0. In general, however, we cannot hope for this statement, since it fails, e.g., if G has precisely two points, or in the case of the tree in Figures 6.4. Henceforth, suppose that G* contains at least two points (otherwise Theorem 6.2 is tr ivial) and that / is posit ive. In light of Theorem 6.3, we have 0 < v < 1. We may rewrite the equation —Af = vf on G*, then, as / ( « ) = - W e G # , (6.9) wgJV(v) where K = (1 — v)~l > 1. It is clear that a part icular consequence of (6.9) holding on G* for a posit ive / and K > 1 is that / is superharmonic on G#, since in that case we have —Af — v'f where v' = 1 — Hence, if a posit ive funct ion / € X>(G#) solves (6.9) for some K > 1, then we necessarily have / everywhere str ict ly posit ive on G * . We now give our character izat ion of the strictness of the decrease of our eigenfunction / . Th is character izat ion wi l l in fact work for any posit ive solution / 6 £>((?*) of (6.9), for any K > 1, not just for K = (1 — v)-1. Proposition 6 . 1 . Suppose that f 6 £>(G#) is positive and symmetrically decreasing and that there is some K > 1 such that (6.9) holds. Suppose that v and w are points of Tp such that v -< w, v G G* and f(v) = f(w). Then either v = O and h(w) = 1, or else w is not a descendant of v. In either case, there is an involution p of Tp interchanging w with v such that f o p = f, and, moreover, f is constant on (Chi ldren v) U (Chi ldren w). 103 Chapter II. Discrete symmetrization Figure 6.3: An extremal subtree G* C T 3 with cardinality 21 and the eigenfunction / cor-responding to the first non-zero eigenvalue of —A. Note: The numerical values of / were computed with Maple and have been rounded off to two decimal places. However, it can be proved that for all the pairs of x and y that can be seen in the two displayed figures where f(x) and f(y) agree to two decimal places, we in fact have exactly f(x) = f(y). 104 Chapter II. Discrete symmetrization Figure 6.4: A n extremal subtree G* C T 3 w i th cardinal i ty 14 and once again with the eigen-funct ion / corresponding to the first non-zero eigenvalue of — A . See the note attached to the previous figure. 105 Chapter II. Discrete symmetrization The reader is encouraged to examine what this result asserts about the functions shown in Figures 6.3 and 6.4. C o r o l l a r y 6 . 2 . Suppose that f £ '£>(£#) is positive and symmetrically decreasing and that there is some K > 1 such that we have (6.9). Then the maximum of f is attained on a set of cardinality at most 1. Proof of Corollary. The max imum of / is clearly attained at O. Suppose f(0) = f{w) wi th O ^ w. Set v = O, and let p be as in Propos i t ion 6.1. Now every point w' wi th f(w') = f(0) must have h(w') = 1 by Proposi t ion 6.1, and thus must be adjacent to O. Hence, if there exist at least three dist inct points at which the max imum of / is attained, there must be at least two dist inct points adjacent to O at which the max imum of / is attained. Bu t fop — / , and p preserves adjacency and satisfies pO = w, so that it follows that there are at least two distinct points adjacent to w at which the max imum of / is at ta ined. Bu t then at least one of these two points must be exact ly a distance 2 from the root of Tp, which contradicts the fact that each point w' w i th f(w') = f(0) has h(w') = 1. • We now give the proof of Propos i t ion 6.1, broken up into four claims. F i rs t we note that by Coro l la ry 6.1, / cannot vanish anywhere on G # , since G # is clearly connected. Moreover if w fi G * then f(w) = f(v) = 0, and since v £ G * it follows that / = 0, which we assumed was not the case. Thus we may assume that w £ G * . C l a i m A l . Suppose that w is a descendant of v. Then v = O and h(w) = 1. Proof of Claim A l . Let v' be the parent of w. We then have v •< v' •< w so that f(v) > f(v') > f(w). Since f(v) = f(w) we must also have f(v') = f(w). Replacing v by v' if necessary, then, we may assume that v is the parent of w. Let wx,..., w p _ i be the children of w. If we can prove that v = O then we wi l l likewise have h(w) = 1 since w is a child of v. Hence, to obtain a contradict ion assume that v ^ O. Let v\,... ,vp-i be the children of v; note that 106 Chapter II. Discrete symmetrization w € {ui,..., vp_i}. Then, by (6.9), /(«) = +E ' (6-10) where x is the parent of v. Also, / H = sp"1 + E /(^j • (6-11) But the symmetrically decreasing character of / ensures that f(v) < f(x) (6.12) and maxf(wn) < min f(vm), (6.13) n m since /i(u) > h(x) and /&(io„) = /i(w) + 1 > h(w) = h(vm), for all rc, and m. Yet, since /(u) = /(w), it follows from (6.10)-(6.13) that we must in fact have f(w) = f(v) = f(x) and f(wn) = f(vm), Vn,mG{l , . . . ,p- l} . Moreover, w € {vi,..., so we have <* = f(v) = Hx) = / H = / K ) = /(um) for all n and m. Then, by (6.10), a = /(f) = np~lpct = KOJ, and so a = 0 as K > 1. But we have already seen that / cannot vanish anywhere on G* and so we have a contradiction as desired. • It now suffices to show that in general if v -< w and f(v) = f(w) then there is an involution p with the desired properties, and that / is constant on (Children v) U (Children w). 107 Chapter II. Discrete symmetrization C l a i m A 2 . We either have h(w) = 1 + h(v) or h(w) — h(y). Proof of Claim A2. If h(w) > 1 + h(v), then let w' be the parent of w. Then h(v) < h(w') < h(w) so that v -< w' -< w. We then have f(v) > f(w') > f(w) as / is symmetric decreasing. But f(v) = f(w), so it follows that f(w') = f(w). By Claim Al , we see that w' = O. But this is impossible as v -< w'. Hence, h(w) < 1 + h(v). On the other hand h(y) < h(w) as v -< w, so that h(w) must equal either h(v) or 1 + h(v). • We now make a few definitions. First, let A be the point of [v, w] minimizing h(A). If h{w) = h(v) then let B = A, and let A' and B' be those unique vertices of [A, v] and [A, w], respectively, which are also children of A. See Figure 6.5. If h(w) = 1 + h{y) then let B be the unique vertex of [A, w] which is a child of A, and set A' = A and B' = B. See Figure 6.6. Note that in any case we have d(A'', v) = d(B', w). C l a i m A 3 . Let x £ Desc A' and y £ Desc B', with d(x, A') = d(y, B'). Then f(x) = f(y). Completion of proof of Proposition 6.1 assuming Claim A3. Assume that the claim is just. The reader is advised to try to follow the proof along by looking at Figure 6.5 if h(w) — h(v) and at Figure 6.6 if h(w) = 1 + h(v). Note now that if h(w) = 1 + h(v) then B is a descendant of A, A £ Desc A', B £ DescB' and d(A, A') = d{B, B') = 0, so that by the claim we have f(A) = f(B). By Claim Al it follows that A = 0. In any case, let H = (Desc A') U (DescB') U {A}. It is easy to see (considering the cases h(w) — h(v) and h(w) — 1 + h(v) separately) that there is an involution p of H which interchanges v and w. Then, our involution p interchanges A' 108 Chapter II. Discrete symmetrization 3 Figure 6.5: Def in i t ion of A, B, A' and B' in the case where h(w) = h(v). 109 Chapter II. Discrete symmetrization 40 Figure 6.6: Def in i t ion of A , B, A' and B' in the case where h(w) — 1 + h(v) 110 Chapter II. Discrete symmetrization with B' and Desc A ' wi th Desc B'. Let x £ Desc A ' . Let y = px. Then , d(x, A') = d(y,B') since p is an isomorphism and pA' = B'. Thus , by C l a i m A 3 we have f(x) = f(px). On the other hand if a: £ Desc B' then px € Desc A ' so that by the above we have f(px) — f(p2x) = f(x) since p is an involut ion. Hence / o p = / on Desc A ' U Desc B'. If A € Desc A ' U Desc B' then we conclude immediately that / o p = / on H. Otherwise, we must have h(v) = h(w) and it is easy to see that p must then f ix A , so that f(pA) = / ( A ) , and thus we also have / o p = f on all of H. Now suppose that h(w) = 1 + h(v). Then as noted above, we have A = O and so H = Tp since A' = A in this case while D e s c O = Tv. Thus , / op = / everywhere on Tp. On the other hand, suppose that h(w) = h(v). Then , A = B and it is easy to see that A G Fixp as p swaps A' and B' which are both adjacent to A . Ex tend p to al l of Tp by setting px = x for x fi H. It is not difficult to see that p is sti l l an involut ive graph isomorphism and satisfies / o p = / everywhere on Tp. Moreover, in any case, if X\ and y\ are children of x and y respectively then d(x\,A') = d(x,A') + 1 = d(y,A') + 1 = d{yi,B'), and so f(xi) = f(yi) by C la im A 3 , as desired. • In order to prove C l a i m A 3 we first formulate yet another c la im. C l a i m A 4 . Suppose O < V < W and f(V) = f(W) for some V G G*. Then f(X) = f(Y) where X and Y are the parents of V and W respectively. Moreover, f is constant on (Chi ldren V) U (Chi ldren W). Proof of Claim A 4 . If W fi G* then f(W) — 0 and so f(V) = 0, which is impossible as / does not vanish in G*. Hence, W G G*. We now employ an argument simi lar to the one given in the proof of C la im A l . Note that X < Y so that f{X) > f(Y). Let V 1 } . . . , Vp_i and Wu . . . , Wp_ i b e t h e children of V and W 111 Chapter II. Discrete symmetrization respectively. Then for al l m and n we have Vm -< Wn and so f(Vm) > f{Wn). Bu t , by (6.9) we see that f(V) is K t imes the average of / ( A - ) , / ( V i ) , . . while f(W) is K, t imes the average of f(Y)J(Wl),...J(Wp_1). Now f(V) = f(W), and since f{X) > / ( Y ) and f(Vm) > f(Wn) for all m and n, it follows that we must have equali ty in al l these inequalit ies so that f(X) = / ( Y ) and f(Vm) = f(Wn) for al l m and n as desired. • F ina l l y we can now prove C l a i m A 3 and thus finish our proof of Proposi t ion 6.1. Proof of Claim A 3 . App l y i ng C l a i m A 4 we see that f{v') — f(w') where v' and w' are the parents of v and w respectively. If v' ^ O and v' ^ w' then we may apply C la im A 4 again to see that f(v") = f(w") where v" and w" are the parents of v' and w' respectively. Iterating this procedure, we wi l l eventually conclude that f(A') = f(B') since d(A',v) = d(B',w) and A' £ [0,u] while B' £ [0,w]. To complete the proof, we proceed by induct ion. Suppose that it has been shown that x < v. with x £ Desc A ' and y £ DescB' and d(x,A') — d(y,B') = n implies f(x) = f(y)- Indeed, this has been shown if n = 0. We shall show that then the desired relation holds if d(x, A') = d(y, B') — n + 1 > 0, and by induct ion we wi l l have completed the proof of the claim and thus of the Propos i t ion . B u t , if d(x, A') = d(y, B') — n + 1 > 0 and x -< y, then let X be the parent of x and Y the parent of y. We wi l l then have X -< Y , and by the induct ion hypothesis f(X) = f(Y) as d(X, A') = d(x, A') - 1 = d(y, B') - 1 = d(Y, B'). If X £ G* then it follows by C l a i m A 4 that / is be constant on (Chi ldren X) U (Chi ldren Y ) , and since x and y fal l into this set, we are done. B u t if X £ G* then Y ^ G # , and the symmetry of G * then implies that x and y also fai l to be in G * so that f(x) = 0 = f(y) and we are done. • 112 Chapter II. Discrete symmetrization Given Proposition 6.1, we will now proceed to prove the condition for equality in Theorem 6.2. Again, we shall do this by breaking the proof up into several claims. If G is isomorphic to G* then it is clear that we have equality as desired. Now, assume that there is no automorphism of Tp which maps G onto G#. We must prove that v\(G) > vi(G&). Let g G ^>(G) be a minimizer of the Rayleigh quotient for G. Since 1Z(g) > 7£(|<7|) we may assume that g has constant sign, replacing g by —g we may assume that g > 0. Let / = If / did not minimize the Rayleigh quotient for G# then Vl{G*) < 11(f) <K(g) = vl(G), where the second inequality used (6.8), and so we would be done as we would have a strict inequality as desired. Hence assume that / does minimize the Rayleigh quotient for G# and thus satisfies (6.9). Then, since the supports of / and g are G# and G, respectively, by our assumption there cannot be an automorphism cj) of Tv such that / = g o cp. Let 5 be a subset of G* of maximal cardinality with the property that every ancestor of every element of S is also in S and there exists an automorphism <f> of Tp with f\s — (gocp)\s. It is easy to see that S is non-empty, and in fact O G S. Moreover, S ^ G * since we have seen that there is no automorphism tp of Tp with / = g o tp. Replacing G by cp[G] and g by go<p (where <p is an automorphism of Tp such that / | 5 = (g o<p)\s), we may assume that f\s = g\s while for no set S' C G* containing ancestors of all of its points and of strictly larger cardinality than S is there an automorphism tp such that f\s, = (g o tp)\s,. Claim B l . There exist points v, V, w and W of Tp and an involution p G 3 such that the following conditions are satisfied: (a) V is the parent of v (b) if w is a descendant of v then W is a child of w; if w is not a descendant of v then W is the parent of w 113 Chapter II. Discrete symmetrization (c) v < w and V -< W (d) pv = w and pV = W (e) g(v) < g(w) and g(V) > g(W). The proof of this shall be given later. C l a i m B 2 . Assume that p, v, V, w and W are as in Claim B l . Let <!>(£) = t2. Define K{t) = 1 for t < 1 and K(t) = 0 forty 1. T/jen, Q{g,g;$,i<) > Q(gp,gp;$,K). Proof of Claim B 2 . L e m m a 2.2 guarantees that Q(g,g;^,K)>Q(gp,gp;^,K). Let x — v and y = V. We shall show that condit ions (i) and (ii) of L e m m a 2.2 are satisfied. Then the desired str ict inequal i ty wi l l follow since $(t) = t2 is str ict ly convex. The facts that g(v) < g(w) = g(pv) and g(V) > g{W) = g{pV) imply that (ii) holds. Now K(d(v,V)) = K(l) = 1. If we could show that d(v,pV) > 1 then it would immediately follow that K(d(v, pV)) = 0 and (i) would necessarily hold, so that the proof of the claim would be complete. Hence, to obtain a contradict ion, suppose that d(v,pV) < 1. Now, pV = W. B y condit ion (b) we have v ^ W. Thus , the only way we can have d(v,W) < 1 is if d(v, W) = 1, i.e., if W is either the parent of v or a chi ld of v. Suppose first that W is a child of v. Then , since v ^ w and W is adjacent to w, it follows that to is a descendant of v. Bu t were w to be a descendant of v then W would have been a child of w, which would have made it impossible for W to be adjacent to v, since » / w. Suppose now that W is the parent of v. Bu t then W = V, which is impossible since V •< W. Thus in both cases we have a contradict ion and the claim is proved. • 114 Chapter II. Discrete symmetrization Cont inue to assume C l a i m B l . Proof of the condition for equality in Theorem 6.2. Fol lowing the proof of Theorem 2.2, we see that our str ict inequal i ty Q{g,g;<f>,K) > Q(gp,gp;$,K) implies the str ict inequali ty ]T g(x)K(d(x,y))g(y) < £ gp(x)K(d{x,y))gp{y). x,yeTp x,yeTp Bu t , on the other hand, Theorem 2.2 says that J2 gP(x)K(d(x,y))gp(y)< £ (gp)* (x)K (d(x,y))(gp)* (y), x,y€Tp x,yeTp and the equimeasurabi l i ty of gp wi th g implies that (gp)* = g* = f'• Hence, Y g(x)K(d(x,y))g(y)< ]T f(x)K(d(x, y))f(y). x,y£Tp x,yeTp Bu t by (6.6) and (6.7) it then follows that vi(G) = K(g)>K(f) = v1(G*), as desired. • A l l that remains to be proved is C l a i m B l . Let v be the smallest (with respect to -<) element of G * \ 5 . Let w be an element of G&\S such that f(v) — g{w) (such a w exists because / = g& are equimeasurable while / | 5 = g\s)- We have f(v) / g{v) since if f{v) = g(v) then we could set S' = S U {v} and we would have / | s , = g\s,. The minimal i ty and choice of v would then ensure that \S'\ > \S\ and that al l ancestors of elements of S' are in 5 , thereby yielding a contradict ion. Then , we must have v -< w (we cannot have v = w since f(v) / g{v) and we cannot have w < v because of the min imal i ty of v). Claim B3. We have f{v) > f(w). 115 Chapter II. Discrete symmetrization Proof. To obtain a contradiction, suppose instead that f(v) < f(w). But v -< w so that f(v) > f(w) and so f(v) = f(w). Then, by Proposition 6.1 there exists an involution p swapping v and w, and satisfying / o p = f. Then, f(x) = f(px) = g(px) for every x G pS since g\s = f\s and p = p~l. Moreover, g(pv) = g(w) = f(v). Then, (go p)\s, — f\s,, where S' = {v} U pS. We have \S'\ > \S\ since v^pSaspv = w^S. If we can prove that the ancestor of every element of S' lies in S' then we will have obtained a contradiction to the maximal cardinality of S. (It is clear that we must have pS C G* since / does not vanish on S, hence / = / o p does not vanish on pS, while the support of / is precisely G*.) We now prove the above statement about ancestors of elements of S'. Let x G S'. First consider the case where x G pS so that px G S. We must show that if x ^ O and X is the parent of x, then X G pS. But, if X £ pS then pX £ S. Since X is adjacent to x, we have pX adjacent to px. As the parent of px must lie in S, it follows that pX is not the parent of px, but must instead be a child of it. Thus, f(pX) < f(px). But / = fop so that f(pX) = f(X) and f(px) = f(x), while, since X is the parent of x, we have f(X) > f(x). Thus, f(pX) = f(px) — f(X) = f(x). From Proposition 6.1 it follows that X = O and h(x) = 1, and likewise that px — 0 and h(pX) = 1. Since / evidently thus attains its maximum at x and at pX and also at O, while O £ {x,pX}, it follows from Corollary 6.2 that x = pX. Thus, pO = p2x = pX £ S. Since, f(pO) = f(0) is the maximum of / while pO £ S, it follows from the minimality of v that v -< pO so that f(v) = f(pO) likewise. But since the maximum of / is attained on a set of cardinality at most 2 and v ^ O, it follows that in fact pO = v. Hence, O = pv = w, which contradicts the choice of w G G*\S since O G S. It only now remains to show that the parent of x lies in S' if x = v. Let V be the parent of v. IfV^pS then pV £ S. The minimality of v shows that we have v •< pV. Thus, f{v) > f(pV). But f(pV) = f(V) as / = fop. Thus, f(v) > f(V). Since V is the parent of v it follows that f(v) < f(V) so that f(v) = f(V), and then Proposition 6.1 implies that V = O. But we have assumed that f(v) = /(to) and there are at most two points at which / attains its maximum, 116 Chapter II. Discrete symmetrization while it evidently attains it at V = O, v and w, so that w = v or w immediately yields a contradict ion. = O, and either option • Completion of proof of Claim B l . Let V be the parent of v. Note that V £ S by minimal i ty of v. If w is a descendant of v then let W be any child of w; otherwise, let W be the parent of w. Thus condit ions (a) and (b) are satisfied. I c laim that V ^ W. If w is a descendant of v then this is easy. Otherwise, suppose that V is the parent of both v and w. Then it is easy to see that there exists an involut ion p which fixes S (use here the fact that ancestors of elements of S lie in S) but swaps v and w. Then , we have f\su{v} = (d ° P)\su{v}- B u t the parent of v lies in S by minimal i ty of v, and so we have a contradict ion to the max ima l cardinal i ty of S. Hence indeed V ^ W. It is clear that from the construct ion of V and W we obtain the fact that V -< W as v < w. Thus, V <W and so condit ion (c) is satisfied. I further claim now that we must then have f{V) > f(W). To prove this, note that since V -< W, we must have f(V) > f(W). If W is the parent of w and if equality holds here, then by Proposi t ion 6.1 we likewise have f(v) = f(w), since v and w are children of V and W, respectively. Bu t we have seen that f(v) > f(w) and so f(V) cannot equal f(W) in this case. On the other hand if is a child of w then h(W) > h{w) > h(v) > h(V) and so if f(W) = f(V) then f(W) = f(v) since f(W) < f(v) < f(V). Since in our case is a descendant of u, it follows f rom Proposi t ion 6.1 that v = O, a contradict ion. Hence indeed / ( V ) > / ( F F ) . Now since V is the parent of v, the min imal i ty of v £ G*\S implies that V £ S so that g(V) = f{V). I now claim that g(V) > g(W). If FF £ 5 then g(V) = f{V) > / ( F F ) = g(W) and we are done. Suppose thus that FF fi S. Since / | s = g\s and / = g&, there must be a FF ' fi S such that / ( F F ' ) = g(W). B y min imal i ty of v and the choice of V we wi l l have V ^ v < FF ' . If g(W) > g{V) then since g(V) = f(V) > f(v) > / ( F F ' ) = g(W), we must in fact have f(V) = f(v) = / ( F F ' ) . Bu t v is a descendant of V and so by Proposi t ion 6.1, then, we must in fact have V — 0. Then , f(0) — f(v) = / ( F F ' ) , and so by Corol lary 6.2 we have 117 Chapter II. Discrete symmetrization W = v. Thus g(W) = f(W) = g(w) = g(0), since g(w) = f{v) and g(0) = f{0). Bu t the points 0, w and W are dist inct. Hence g attains the value f(0) at three or more vertices of Tp. B y equimeasurabi l i ty, / also attains the value f(0) at three or more vertices, which contradicts Coro l la ry 6.2. Hence, indeed, g(V) > g(W). On the other hand it is easy to see that g(v) < f(v) by the choice of v and the fact that / = g*. Bu t f(v) = g(w) so that we have g(v) < g(w). We have thus verified condit ion (e). If to is a descendant of v then let p be an involut ion of Tp which interchanges W and V; considering p restricted to [V, W] we see that it must also interchange v wi th w. If w is not a descendant of v then let p be an involut ion of Tp which interchanges w and v. It can be seen in this case that p must interchange W wi th V since in this case we must have both W and V contained in [y, w]. Thus in any case condit ion (d) is satisfied. • 7. Some open problems and two counterexamples We may define a number of classes of graphs depending on which, if any, of the inequalit ies and condit ions considered in §2 hold. Throughout this section, when we speak of an "order ing" we shall mean a "well-ordering such that every element has at most f initely many predecessors." Let G be a graph wi th an ordering -<. Let Ki(t) — 1 if t < 1 and let K~i(t) — 0 otherwise. Consider the fol lowing properties of G under the ordering (A) Cond i t ion A holds (B) Cond i t ion B holds (C) For al l posit ive / and g we have (2.4) for al l decreasing positive K (D') For all positive / and g which are simi lar ly ordered we have (2.4) for all decreasing positive K (D) For all positive / and g — f we have (2.4) for all decreasing positive K 118 Chapter II. Discrete symmetrization (M) The master inequality holds (c) For all positive / and g we have (2.4) for K = K\ (d1) For all positive / and g which are similarly ordered we have (2.4) for K = K\ (d) For all positive / and g = / we have (2.4) for K — K\ (m) Inequality (2.1) holds for all real / and g, and all convex increasing providing K = K\ Let <£>p be the collection of all graphs for which there exists an ordering -< under which condition P holds. In light of the results of §2 and some trivial implications, we have: ®A C (5M, (7.1) ®Mn®B c 0c c eD, c eD, (7.2) S m n 0 B C ( S c C <8rf, C 0 d (7.3) and <SQ C <5„ (7.4) where Q is C, J9', 73 or M, respectively, while q is c, cf', d or m, respectively. (The first inclusion in (7.3) does not follow directly from Theorem 2.2 but rather from its proof.) It is not known which, if any, of the inclusions in (7.1)—(7.4) can be reversed. In the few examples known to the author, either the graph has all of the properties (A)-(m) or it has none, but the author suspects that this will not be true in general. Open Problem 7.1. Which, if any, of the inclusions in (7.1)-(7.4) are strict, and which are not? Open Problem 7.2. Classify the graphs (or at least all finite graphs) lying in the various classes (Sp, where P ranges over the properties (A)-(m). In particular, determine what graphs lie in <8A H (5B; also, determine what graphs lie in <8d-119 Chapter II. Discrete symmetrization Cond i t ion (d) is of some interest in that it implies the existence of a Faber -Krahn type inequality of the form of inequali ty (6.2) of §6.2.1. However, we do not know whether the condit ion for equali ty given in Theorem 6.2 has universal!val idi ty whenever (d) holds. O p e n P r o b l e m 7 .3 . G iven a graph H wi th an ordering -< such that (d) holds, and given a subset G such that v\{G) = ^ i ( G # ) , must we then have G and G* isomorphic? W h a t if we addi t ional ly require that some of the more stringent condit ions from among (A ) - (m) hold? One may also ask the above questions restricted to the class of constant degree graphs, or even to the class of regular graphs (a graph is said to be r e g u l a r if its automorphism group is transit ive on vert ices). Note that one can find examples of very nice regular graphs which do not have any ordering under which property (d) holds. T h e o r e m 7 . 1 . Let G be either the cube Z 2 or the ternary plane Z§. Then there exists no ordering of the graph G under which G has property (d). The cube graph Z 2 is defined to have as vertices all tr iples (a, b, c) wi th a, b and c in Z 2 , where two vertices are defined to be adjacent if and only if they differ in precisely one coordinate. The ternary plane graph Z | is defined to have as vertices all pairs (a, b) wi th a and b in Z 3 , where two dist inct pairs (a, b) and (a1, b') are said to be adjacent providing a — a' or b = b'. (See F igure 7.1.) Remark 7.1. Let G be one of the two graphs in Theorem 7.1. Then clearly G G <5B- B y (7.1)-(7.4) it follows that there is no ordering under which G has any of the properties (A)-(m) other than (B). One could presumably produce a formal paper-and-penci l proof of Theorem 7.1. Instead, how-ever, the author programmed a computer to generate basically al l possible orderings on the given graph, and then for each ordering the computer produced enough randomly generated funct ions / to give a counterexample to property (d). Mo re details wi l l be given below. It 120 Chapter II. Discrete symmetrization 121 Chapter II. Discrete symmetrization is to be emphasized that, assuming the correctness of the author's simple computer program c u b e t e r n . c given in Appendix A, the results are exact, and in principle a human being could check that each counterexample is indeed a counterexample. In the next section we will outline how this program works. Open Problem 7.4. Let P be one of the properties (A)-(m) other than (B). Consider the problem of determining whether a given finite graph G is in (5p. Is this problem NP-complete? 7.1. How the computer proved Theorem 7.1 Let G be one of the two graphs from Theorem 7.1. In order to prove Theorem 7.1, we must show that for any ordering -< on G there exists a positive function / on G (depending on the ordering) such that J2f(v)Nf(v) > £/ # WiV/ # («), (7.5) v£G v6G where w£N(v) and where # is the rearrangement induced by the ordering -< while N(v) is the collection of all vertices adjacent to v. To see that this suffices, it is only necessary to note that in such a case if K\ (t) is 1 for t < 1 and 0 otherwise, then ]T f(v)K1(d(v,w))f(w)= J2 f(v)Nf(v) + J2f(v). Since the second summation on the right hand side of this expression is invariant under replace-ment of / with / * by equimeasurability, it will follow from (7.5) that J2 f(v)K1(d(v,w))f(w)>J2f*(v)I^(d(^w))f*H, v,wEG VEG which says precisely that (2.4) fails for g = f and K — K\, as desired. If G - 1\ then let O = (0, 0,0) and if G = 1\ then let O - (0, 0). What one must prove is that for every ordering -< there exists a function / such that (7.5) holds. However, because both of 122 Chapter II. Discrete symmetrization our graphs G have the property that all vertices are equivalent (i.e., both graphs are regular so that their automorphism groups are transitive on vertices), it follows by this symmetry property that we need only examine orderings -< such that O is the -<-initial element. There will be (\G\ — 1)! such orderings, which number equals 5 040 in the case of the cube graph and 40 320 in the case of the ternary plane. The computer program cubetern.c (see Appendix A) proceeds by looping through all of the (\G\ — 1)! orderings mentioned above. For each ordering, the program generates pseudorandom functions / (via the built-in Borland Turbo C++ 3 . 0 random number generator r a n d ( ) , seeded at the beginning of the program with the arbitrary value 317) with values in {0,1,..., 19}. Then, the program checks whether (7.5) holds. (This is an exact computation since the functions are integer valued.) If it does hold, then we have the requisite counterexample for the current ordering. If it does not hold, then we simply keep on generating more pseudorandom functions / as above, until one is found for which (7.5) holds. Of course, in principle one might never find such a function and in such a case we neither have a proof of Theorem 7.1 nor of its negation. However, as it turned out, the program did find such a counterexample for every ordering (for both choices of G) and this shows that Theorem 7.1 is just (assuming correct functioning of the software and hardware). In the case of the cube, the largest number of tries to find an / satisfying (7.5) happened to be 799. In the case of the ternary plane, this happened to be 777. On a 4 8 6 s x / 2 0 system under MS-DOS 5 . 0 , compiling the code in the t i n y model under Borland International's Turbo C++ 3 . 0 with all speed optimizations enabled gave a run time of 8 seconds for the case of the cube and 60 seconds for the case of the ternary plane. 8. Discrete Schwarz and Steiner type rearrangements 8.1. Basic definitions and results Let X and Y be countable sets equipped with counting measures and let # be a rearrangement from the power set 2X to the power set 2Y. The term "measure preserving" shall refer to 123 Chapter II. Discrete symmetrization count ing measure. We wish to study some general types of discrete rearrangements. D e f i n i t i o n 8 . 1 . The rearrangement # is said to be of ( d i s c r e t e ) S c h w a r z t y p e if it is measure preserving and A # = B* whenever \A\ — \B\ and A, B C X. Example 8.1. A prototypical example of a discrete Schwarz type rearrangement is the decreasing rearrangement * on Z Q defined in Examp le 1 . 2 . 2 . Remark 8.1. Let / and g be equimeasurable functions on X. Then , / # = g* if # is of Schwarz type. For , ( / # ) A = ( / A ) # = (gx)* = (g*)x-(The first and last equalities follow from Theorem 1 . 2 . 1 . The middle equality follows from the fact that fx and gx have the same cardinal i ty because of equimeasurabil i ty, and thus have the same rearrangement because of the Schwarz property of P r o p o s i t i o n 8 . 1 . A Schwarz type rearrangement # from 2X to 2Y induces a unique well or-dering -< onY with the properties that every element ofY has at most finitely many predecessors and if yo -< y\ -< y2 -< • • • is an ordered enumeration of Y then S* = {Vi : i < \S\] for every S C X. Proof. Suppose that we have two such orderings -<x and ' -< 2 , which induce the enumerations yJQ < y[ < y2 < • • • of Y for j = 1, 2 , respectively. To obtain the identity between -<x and -<2 it suffices to show that {yj :i<n} = {yf : i < n} for all n < \Y\ = \X*\ = \X\. Bu t to do this for some fixed n, choose a set S C X wi th cardinal i ty n, and then the two above quantit ies wi l l necessarily be equal, since they wi l l be both equal to 5 * . Hence we now need only show the existence of such an ordering. We define the ordering as follows. F i x a non-negative integer i < \X\. Let S be a subset of X wi th cardinal i ty i + 1 1 2 4 Chapter II. Discrete symmetrization and let x G S. Let S' = S\{x}. Then S' C S so that (5')# C S # , and the latter inclusion is proper since the former is proper and since S and S' are finite while # is measure preserving. Moreover, |(S')*| — |S*| — 1. Thus there is a unique element y ; 6 S*\(S')*. Because S* and (5')* depend only on the cardinality of S and S', it follows that whenever 5 and 5' are arbitrary sets of cardinalities i + 1 and i respectively, then S*\(S')* = { y 8 } -I now claim that if S is a finite subset of X then S* = {y,- : i < \S\}. (8.1) \s\ This is trivial if \S\ = 0. Hence assume that |5| > 1. Write S = U;-_05;, where the set Si has cardinality i for i = 0,..., |5| and where So C Si C • • • C S|s|- Then (S,-+i)*\(Sj)* = { y s } because of the remarks in the previous paragraph and because of the cardinalities of S;+i and Si. Since # preserves inclusions, it follows that (So)* C (Si)* C ••• C (S\s\)*- Thus, since (So)* = 0 , we must have |S| S* = (5|5|)# = UK5«+i)#\(5'-)#] = {w>,yi,...,y|s|-i} as desired. We now verify that (8.1) also holds if S is an infinite subset of X. For, then, let Si C S2 C S 3 C • • • be a collection of subsets of S whose union equals S and which satisfy \Sn\ = n for all n G Z + . By the definition of a rearrangement we have 00 s*=(J(sn)*. 71=1 But {Sn)* = { y 0 , . . . , y n - i } -Thus, S* = {yo,yi,2/2, • • •} as desired and (8.1) holds. Applying (8.1) with S = X, we see that yo,yi,--- is an enumeration of Y. Defining the well-ordering -< by yo -< yi -< y.2 < • • • we are done. • 1 2 5 Chapter II. Discrete symmetrization Definition 8.2. The rearrangement # is said to be of (discrete) Steiner type if it is measure preserving and there exists a countable collection A of disjoint nonempty subsets A of X such that: (i) L M = * (ii) for any A £ A the rearrangement # restricted to 2A is of Schwarz type (iii) if A and B are two distinct elements of A then A* and B* are disjoint. We call A the fibres of #, and we call any set A € A a fibre. The following proposition describes how Steiner rearrangement decomposes into Schwarz rear-rangements. Proposition 8.2. Let # be of Steiner type and let A be its fibres. Then for every S C X we have S= (j (Sn A)*. (8.2) In particular, Y=\jA*. AeA Proof. Let T = \JA^A (SCiA)*. First we note that T C S* since (S C\ A)* C S* as S D A C S and # preserves inclusions. We now verify the opposite inclusion in the case of a finite set S. If S is finite, by condition (i) in the definition of Steiner type we have a finite collection A\,...,An of distinct (hence, disjoint) elements of A such that S = (SnA 1 )u - - -u (SnA n ) . Let T'= (Sn A i ) # u - - - u (Sn An)*. 126 Chapter II. Discrete symmetrization The sets (5 f~l A i ) # , . . . , (5 n An)# are disjoint since A f , . . . , A * are disjoint. Moreover, the cardinality of S D Ai equals the cardinality of (Sn A;)#. Hence, the cardinality of S equals the cardinality of T", while of course the cardinality of S* equals that of S. But T' C T C 5*. Since |T'| = |5 # | < oo it follows that T' = T = 5 # , as desired. Now consider an infinite set 5. Write 5 = U^Li $n, where each Sn is finite and where S\ C g 2 C •••. We then have oo s * = U (*«)*. 71=1 since by definition rearrangements preserve countable increasing unions. But (Sn)*= \J(SnnA)*. AeA Thus, CO s*= U U ( 5 « n A ) # -Ae^ra=i But Si fl A C 52 H A C • • • and the union of these sets is 5 D A, so that oo \J(SnnA)* = (SnA)*, 71=1 by the increasing union preservation property of rearrangements. Thus, S#= | J (SnA)# AeA as desired. The last statement of the Proposition follows from applying the above with S = X and recalling that X* = Y by Definition 1.2.2. • Definition 8.3. A subset S of Y is said to be symmetric if there is a subset T of X such that T# = 5. A function / on Y is said to be symmetric if there exists a function g on X with f = g#. Corollary 8.1. Let # be of Steiner type, and let A be its fibres. Then a set S CY is symmetric if and only if for every A € A the set S fl A * is symmetric. 127 Chapter II. Discrete symmetrization Proof. Suppose S is symmetr ic . Then , S = T* for some T C X. Then , by Proposi t ion 8.2, since the A* are disjoint, we have S n A* = T* n A* = ( T n A)*, and so S f l A* s symmetr ic . Conversely, suppose that for each A £ A there exists TA Q X such that T* = SDA*. We must necessarily have TA Q A because of the disjointness of the A*. Let T = {JAEATA-Then , by Propos i t ion 8.2 we have T*= ( J ( T n A ) # = ( J T*= \J{SnA*) = S, AeA AeA AeA so that S is symmetr ic as desired. • The fol lowing result shows that the fibres of a Steiner type rearrangement are uniquely deter-mined. P r o p o s i t i o n 8 .3 . Let # be of Steiner type. Let A and B be collections of disjoint nonempty subsets of X such that conditions ( i)-( i i i) in the definition of a Steiner type rearrangement are satisfied both for A and for B. Then A — B. Proof. It clearly suffices to show that A C B, since the opposite inclusion then follows upon interchange of A and B. Let AeA. Suppose that A ^ B. Now, A C X = \JB. Thus, we can cover A wi th sets f rom B. Since A £ B, it follows that one of the fol lowing two possibilit ies must hold: (a) there is a set B £ B such that A n 5 / 0 and B A (b) there is a set B £ B which is a str ict subset of A. Suppose first that (a) holds, and let B be a set as in (a). Let x £ An B and let y £ B\A. Consider the set S = {x,y}. Since y £ A and A covers X, it follows that there exists A' £ A 128 Chapter II. Discrete symmetrization such that y € A'. In light of Proposition 8.2 applied with the collection A, we see that S* = (S n A)* U (5 n A')* = {x}* U {y}* (since 0# = 0, all the terms in the union (8.2) drop out except for S fl A and S fl A'). Moreover, since S has cardinality 2 and our rearrangement was assumed to be measure pre-serving, it follows that 5* has cardinality 2. On the other hand, {x} and {y} are both subsets of B G B, and by condition (ii) in the definition of a Steiner type rearrangement we must have {a;}# = {y}# since a Schwarz rearrangement of a set depends only on the cardinality of a set, while |{x}| = 1 = \{y}\. Hence, {x}# U {y}* has cardinality 1, contradicting the previously noted fact that \S*\ = 2. Now assume that (b) holds. Then since B covers X and hence also A, it follows that there is a set B' £ B distinct from B and also containing some element of A. Let x € A fl B and y S A fl B'. Applying condition Proposition 8.2 to the collection B, this time we see that if S = {x, y} then S* = (S n B)* U [S n B')* = {x}* U {y}#. As before, the cardinality of the left hand side is 2. On the other hand, {x} and {y} are both cardinality 1 subsets of A, and since # restricted to 2A is of Schwarz type it follows that {x}# = {y}# so that U {y}*\ = 1, a contradiction. Hence, both possibility (a) and possibility (b) leads to a contradiction, and our proposition has thus been proved. • Definition 8.4. The Steiner order -< on Y for a Steiner type rearrangement #: 2X —>• 2Y is given by defining x -< y for x, y € Y if and only if there is a fibre A such that x,y G A and x -< y according to the well-ordering induced by Proposition 8.1 applied to the Schwarz rearrangement # on 2 .^ Definition 8.5. A function / on a set Y is said to be decreasing with respect to a partial ordering -< providing f{x\) > /(a )^ whenever x\ -< x2. 129 Chapter II. Discrete symmetrization D e f i n i t i o n 8 .6 . A subset S of Y is said to be s y m m e t r i c if there is a subset T of X such that T* = S. A funct ion / on Y is said to be s y m m e t r i c if there exists a function g on X with f = g*. B y Theorem 1.2.1, a symmetr ic funct ion has symmetr ic level sets. P r o p o s i t i o n 8 .4 . The following are equivalent for a Steiner type rearrangement 2X —)• 2Y and an extended real function f onY: (i) / is symmetric (ii) the level set f\ is symmetric for every real A (iii) / is decreasing with respect to the Steiner order on Y. Proof. Because of Propos i t ion 8.2, we may consider the problem separately for each restriction I\A# of / to a set A * such that A £ A, and it is easy to see that / is symmetr ic if and only if each of these restrictions is symmetr ic wi th respect to the Schwarz type rearrangement given by the restr ict ion of # to 2A. The above reasoning shows that , replacing X by A if necessary, we may assume that # is a Schwarz type rearrangement. F i rs t we show that (ii) implies (ii i). For , fix x and y in Y with x -< y. Then x = y; and y — y} wi th i < 7 , in the notat ion of Proposi t ion 8.1. Let A = / ( y 2 ) . O f course we have Vi i fx- Now, if the set f\ is symmetr ic and if it contains yy then, since j > i, it follows from Proposi t ion 8.1 that it contains y,- as well . Hence, since y; ^ f\, we likewise have yj £ f\ so that / ( y ) = f(yj) < A = / (y , ) = f(x), as desired. We now prove that (iii) implies (i). To do this, let g be any function on X equimeasurable with / (such a funct ion exists since | X | = |AT # | = \Y\). I c laim that g* = f. For , fix A € R. We must prove that (<?*)A = fx- Bu t (g*)\ = {gx)* by Theorem 1.2.1. Let N be the cardinal i ty of g\. If N = oo then g takes on values str ict ly bigger than A infinitely often, and so does 130 Chapter II. Discrete symmetrization f. Since / is -^ -decreasing, it follows that / > A everywhere, so that fx = Y. On the other hand, by Proposition 8.1 it follows that if N = oo then (gx)* = {yo, yi,...} — Y, too. Suppose now that N < oo. Then g takes on values strictly bigger than A precisely ./V times. Since / is equimeasurable to g, the same is true of it. Since / is -^ -decreasing, it follows that But if the cardinality of gx equals N, it follows then from Proposition 8.1 that (gx)* = {yo, yi,..., yyv-i} = fx- Hence, we have proved that g& = f so that / is symmetric. Finally we show that (i) implies (ii). For, fix A £ R. Since / is symmetric, there exists g such that g# = / . Then fx = (gx)* by Theorem 1.2.1 so that it follows that fx is symmetric, as Via condition (iii), we immediately obtain as a corollary the following result. Proposition 8.5. Let f and g be two symmetric functions on Y with respect to a Steiner type rearrangement 2^ -4 2 y . Then f + g is symmetric. Moreover, if f and g are positive, then fg is symmetric. The following result is a very useful property of (discrete) Steiner type rearrangements. In a sense, it lets us "reverse" a Steiner type rearrangement. Proposition 8.6. Assume that 2X —> 2 y is a discrete Steiner type rearrangement with fibres A. Let f be any positive function on X such that A n fx is finite for every A > 0 and A £ A. Then there exists a one-to-one map d>: Y —> X with the following properties: (ii) if g is any positive symmetric function on Y and if g is the function on X such that fx = {y0,yi,...,yN-i}-noted before. • (i) f* = fo4> g(x) = < g(rx(x)), ifxe<j>[Y] 0, ifxfi 4>[Y] 131 Chapter II. Discrete symmetrization then (g)* = g, (g • /)# = g • f* and £ f f ( * ) / ( s ) = X > ( y ) / # ( y ) (8.3) ocEX yeY (iii) 4> maps A* onto A whenever AeA (iv) if either A n supp / is finite or f is strictly positive on A for a fibre A 6 A, then <p maps A* into A. Moreover, if S C A is a fixed subset of some fixed fibre A and if we have inf / > sup / , s A\S then we may choose <p with the additional property that whenever g is a positive function with suppy C S* then suppy C S. Proof. Because of Proposition 8.2, we may restrict all our functions on X to A € A and all our functions on Y to A*, and then construct <f>: A* —>• A, and prove that all the above results hold. Then, we may piece these things together via Proposition 8.2 and via the fact that the {A*}AeA are disjoint as are the {A}^e^. But # restricted to subsets of A 6 A is of Schwarz type. Hence, it suffices to prove Proposition 8.6 for rearrangements of Schwarz type. Thus, assume that # is of Schwarz type. Then A = {X}. Without loss of generality, the support of / is non-empty (otherwise the result is trivial, since we can let <fi be any bijection then). Write Y = {yo, y\,...}, where the yt are as in Proposition 8.1. Using the fact that the fx are finite for all A > 0 we may construct a sequence x0, x\, x2,... with the property that f(x0) = maxx / and f[xi) = max{/(a;) : x € A^ \{a;o, Z i , . . . , a > ; - i } } for i > 0. Moreover, if supp / is finite, we may easily ensure that XQ, X \ , x2, • • • is an enumeration of X (since after having chosen those X{ which form the support of / , we then choose all the 132 Chapter II. Discrete symmetrization other X{ arbitrarily in such a way that we enumerate all of X.) Let X' = {xo,xi, x2,.. •}• It is clear that because of the finiteness of fx for all A, the set X' contains at least all the points x of X such that f(x) > 0. Hence, if either / > 0 everywhere or supp/ is finite, we have X' = X. Now define for i € Z Q . It is clear that <f> is a bijection from Y onto X'. Since A — {X}, condition (iii) is trivial. Since <p[Y] = X', we have already verified (iv). Condition (i) is also easy to verify. To do this, fix A e R . We must prove that (/ #)A = {focj>)x. But (f*)x = (fx)* by Theorem 1.2.1. Suppose first that A < 0. Then, fx = X since / is positive, so that (fx)* = Y. Likewise .(/ ° 4>)x = y then, since / o d> is also positive. Suppose now that A > 0. Suppose that the cardinality of the set fx is i. Consider first the case where i = oo; necessarily we have A = 0. Then, / takes on infinitely many strictly positive values. The choice of X' implies that / is strictly positive on X', hence (/ o cj>)x = Y. But (fx)* = Y as well, since we have a Schwarz rearrangement, and the Schwarz rearrangement of an infinite subset S of X must equal Y since X# = Y and |X| = \S\. Hence, in that case we have verified the desired result. Suppose now that i < oo. Then, (/A)# = {yo,yi,---,Vi-i} by Proposition 8.1. But fx consists of points at which / takes on its i largest values. By the choice of the x{ we have fx = { a ; o , a ; i , . . . , a ; j _ i } . Thus, (fo 4>)x = rl[fx] = {yo, yi, • • = (/A)#, as desired, where the second last equality follows from the definition of <f>. We have thus verified (i). We must now verify (ii). I claim that g is equimeasurable with g. For, again fix A € R. If A < 0 then \gx\ = \gx\ = oo as desired. So now suppose that A > 0. 133 Chapter II. Discrete symmetrization Then , g\ = <t>[g\] since g\ C X' for A > 0 as g vanishes outside X'. Since cf> is one-to-one it follows that \g\\ = \g\\. Hence our claim is proved. Now, g is symmetr ic so that g = h# for some h. Moreover, g and h are equimeasurable, hence g and h are equimeasurable so that g# = h* since we have a Schwarz rearrangement (Remark 8.1). Now, g(xi)f(xi) is decreasing wi th respect to i (since g(yi) is decreasing wi th respect to i by symmetry ) , just as / was, and vanishes outside X', just as / d id . App ly ing the same method which we used to show that / * = / o <f> we find that (g • /)# = (g • f) o cj) = g • as desired. Since # is measure preserving it follows immediately that (8.3) holds (Proposi t ion 1.2.1). To verify the "moreover" , suppose that we are given such a set S. The condit ion inf / > sup / , S A\S shows that we may choose the X{ satisfying the further constraint that if X{ £ S for some i then 5 C {XQ, .. (I.e., we may preferentially at each stage choose an element of S over an element of A\S.) If we do this, then it is clear that if g is a symmetr ic function whose support lies in S # , hence in {y; : i < \S\}, then the support of g wi l l lie in {xi : i < \S\}. • 8 . 2 . R e a r r a n g e m e n t o n a p r o d u c t s e t We are st i l l working in the discrete sett ing of countable sets wi th counting measure. D e f i n i t i o n 8 .7 . Let # be a rearrangement from 2X to 2 y . Let Z be any countable set. Then the ^ - p r o d u c t r e a r r a n g e m e n t # f r o m 2ZxX t o 2ZxY is defined by: S*=(J[{z}x({x:(z,x)eS}#)l for any S C Z x X, where the "# " on the right hand side of the above displayed equation is the #-rearrangement for subsets of X. Remark 8.2. It is clear that the Z-product rearrangement is indeed a rearrangement in the sense of Def in i t ion 1.2.2 if # : 2X —>• 2Y is. Moreover, the Z-product rearrangement is measure preserving if # : 2X —> 2Y is. 134 Chapter II. Discrete symmetrization Remark 8.3. Suppose that #: 2X —> 2Y is a Steiner type rearrangement with A being its fibres. Then the Z-product rearrangement is also of Steiner type, and its collection of fibres is given by Az = {{z}xA:zeZ,AeA}. The verification of this fact is almost immediate. In particular, the product construction lets us start with a Schwarz type rearrangement for subsets of X and obtain a Steiner type rear-rangement for subsets of Z X X. This will let us construct a number of interesting examples of Steiner type rearrangements. 8.3. Symmetrization and preservation of symmetry We recall the following definition. Definition 8.8. A rearrangement # mapping a cr-pseudotopology T into itself is said to be a symmetrization if ( A # ) # = A* for all A G T. Let K be a positive function on X X X. Given a positive function g on X, define Kg(x) = J2K(x>y)9(y), yex and K*g(y)=J2s(x)K(^y)-x£X The following result is very similar to Theorem 2.3. Theorem 8.1. Suppose that #: 2X —> 2X is a discrete Steiner type symmetrization. Assume that f(x)K(x,y)g(y)< £ f*(x)K(x,y)g*(y) x,y£X x,yeX for all positive functions f and g on X. Then Kg and K*g are both symmetric whenever g is a positive symmetric function. The above result should remind us of Corollary 1.6.2. 135 Chapter II. Discrete symmetrization Proof. It suffices to prove that Kg is symmetric when g is symmetric, since the other assertion def follows upon replacing K(x,y) by L(x, y) = K(y, x). For any positive / and symmetric g we have £ f(*)K(*,y)9(y)< £ f*{x)K{x,y)g{y), x,y£X x,yEX since g* = g as 5 is symmetric and # is a symmetrization. Rewriting the above displayed equation, we see that for any positive / we have xex xex The desired result then follows from Proposition 8.7, below. • Proposition 8.7. Let # be a discrete Steiner type symmetrization. Then, an extended real function h on X is symmetric if and only if xex xex for every positive f on X for which both sums make sense. Before we outline the proof of this, we need the following result. Lemma 8.1. Let # be a discrete Steiner type symmetrization. Then each fibre A £ A is symmetric. Proof. Suppose that A, B e A. Then, (A* (IB)* = (A*)*DB* = A*DB* (8.5) by Proposition 8.2 and by the fact that # is a symmetrization. Suppose that B e A is different from A. Then A* (1 B* — 0 by Definition 8.2(iii), so that (8.5) implies that A* D B = 0. Since A covers X, it follows that A* C A for every AeA. Now, to obtain a contradiction suppose that A* is a proper subset of A G A. Since the {B* : B e A} disjointly cover X (Definition 8.2(iii) and Proposition 8.2), it follows that there 136 Chapter II. Discrete symmetrization is a B# distinct from A# such that B e A and fl A / 0. But by what we have already proved, we have B& C iJ. Hence, B n A ^ 0 . Hence B = A, which contradicts the assumption Proof of Proposition 8.7. Suppose first that h is symmetric. Then h# = h and so (8.4) follows from Theorem 1.2.3. Suppose now that (8.4) holds for all positive / . To obtain a contradiction, suppose that / is not symmetric. Because of Proposition 8.2, we may decompose X into the fi-bres A of Definition 8.2, and consider (8.4) on each of them separately, since the rearrangements can be computed on each separately. By this argument, we may assume that our symmetrization in fact is of Schwarz type, since it is of Schwarz type on each A € A and since A = A* for all A G A. However, it is easy to see that any Schwarz type symmetrization of functions on a countable set is equivalent to decreasing rearrangement of functions on Zjy=={n G Z Q : n < N} for N equal to the cardinality of our countable set; to see this, use the method of proof of Proposition 8.4 and the mapping yi i -> i sending X onto Zjy, where the y, are as in Proposition 8.1. But a result equivalent to Proposition 8.7 on Z^ v is contained in [58, Thm. 369]. • Finally, we state a useful generalization of Proposition 8.7. Proposition 8.8. Let # be a discrete Steiner type symmetrization. Then, an extended real function h on X is symmetric if and only if for every finite S C I . Proof. In light of Proposition 8.7, it suffices to prove that if (8.6) holds for every finite S then (8.4) holds for every positive g. But, given a positive g with finite support we may write that S# should be distinct from A*. Hence, indeed, A* = A. • (8.6) xes# xex J U xegx 137 Chapter II. Discrete symmetrization in l ight of Remark 1.2.2. App l y i ng (8.6) we see that (8.4) holds for our g. A l imi t ing argument then shows that it likewise holds for any g for which both sides of (8.4) holds. • Remark 8.4- It is easy to see that in fact it suffices to prove (8.6) for all 5 which lie completely wi th in a single fibre A £ A, because of Proposi t ion 8.2 and L e m m a 8.1. 9. Haliste's method for exit times, discrete harmonic measures and discrete Green's functions Throughout this section assume that we are given a discrete set X and a Steiner type sym-metr izat ion # : 2X 2X. 9.1. Definitions and statement of results for generalized harmonic measures and Green's functions 9 . 1 . 1 . T h e k e r n e l a n d t h e a s s u m p t i o n s o n i t Let V = Z x X. Let K: V X V —> [0,1] be a funct ion such that for al l z £ V we have £ # ( * , « ; ) = 1. (9.1) w6V We call such a funct ion K a k e r n e l . We now consider the fol lowing assumptions on a funct ion L: V X V —> [0, oo). A s s u m p t i o n 9 . 1 . The fol lowing inequali ty is valid whenever / and g are positive functions on X while m and n are in Z : J2 f(x)L{(m,x),(n,y))g(y)< ]T f*(x)L{(m,x), (n,y))g*(y). (9.2) x,yEX x,y£X Wri te Lg{y) = J2wev L(vi w)d{w). P u t L*(v,w) = L(w, v) so that L*f(w) = 2~2vev f(v)L(v> w)-From Theorem 8.1 we obtain the fol lowing result. P r o p o s i t i o n 9 . 1 . Assume Assumption 9.1. Both (Lg)(n,-) and (L*g)(n,-) are symmetric functions from X to [0, oo) for each n £ Z providing g(m, •) is a symmetric positive function on X for every m £ Z . 138 Chapter II. Discrete symmetrization A s s u m p t i o n 9 .2 . For every fixed TO and n in Z , the quant i ty yex is independent of the choice of x G X. Fina l ly , we make the fol lowing assumption on our kernel K. A s s u m p t i o n 9 .3 . There exists A G [0, oo) such that Ld=K + XS satisfies Assumpt ions 9.1 and 9.2, where S(z, w) — l{z-wy. For D CV, let D* = \j[{i}x({x:(i,x)eD}*) . This is of course the Z-product rearrangement construct ion (Definit ion 8.7) based on the rear-rangement # for subsets of X. A s noted in 8.2, D >->• D # is a measure preserving rearrangement. We shall cal l D* the ( d i s c r e t e g e n e r a l i z e d ) S t e i n e r s y m m e t r i z a t i o n of D. If D = D* then we say that D is S t e i n e r s y m m e t r i c . See Figure 9.1 for an example of Steiner sym-metr izat ion in the case of the discrete cyl inder where X = Z n , and Figure 9.2 for an example where X = Z . 9 .1 .2 . T h e k e r n e l i n o u r m a i n e x a m p l e s Our pr imary example wi l l have X a constant degree graph satisfying the master inequality wi th respect to some well-ordering wi th each element having finitely many predecessors. The rearrangement # on X wi l l be defined wi th respect to this well-ordering as in §1. Such examples which we have proved in this chapter to satisfy the master inequali ty are the linear graph Z (with edges {j,j + 1} as j ranges over Z ) , the circular graph Z n (with edges {j,j + 1 } as j ranges over Z n ) , the octahedron edge graph Hg and the p-regular tree Tp. Given X a constant degree graph satisfying the master inequality wi th each vertex of degree p, 139 Chapter II. Discrete symmetrization Figure 9.1: Symmetr izat ion on Z x Z n . The symmetr ized and unsymmetr ized sets in question are indicated by black ellipses. 140 Chapter II. Discrete symmetrization let p-1, if x and y are adjacent, Kx{x,y)=l 0, otherwise. It follows that Y2yex K(xi v) = 1 f° r all x £ X. Then, let A' z be any kernel on Z, i.e., any function K1: Z 2 —>• [0,1] such that ]Cyez KZ(X, y) = 1 f° r all z G Z. Two interesting cases are the simple random walk kernel K$(x, y) = | whenever |a: — y| = 1 and K§(x, y) = 0 otherwise, and the trivial kernel Kf(x,y) = Then, given a kernel i f z on Z and the kernel Kx on X, we may define a kernel on V. For r G [0,1], let (Kz <g> K*)((m, »), (n, y)) = rA'z(m, rz)^ + (1 - r)A"*(a:, y)<Sm, n. It is not difficult to verify that K"1 ® Kx is a kernel. P r o p o s i t i o n 9 .2 . Let X be as above satisfy the master inequality. Then Assumption 9.3 is satisfied by A' z <g> Kx for any r G [0,1]. Proof. Let A = 1 and set L = (Kz <g) Kx) + 6. First we verify Assumption 9.1. To do this, fix positive / and g on X and m, n G Z. Suppose first that m ^ n. Then, for x and y in X we have L((m,x), (n,y)) — 8x>yrKI'(m,n), since 8((m, x), (n, y)) = 0 and by the definition of (g> Kx. Hence, f(x)L((m, x), (n, y))y(y) = r/i'z(m, n) £ /(*)$(*) x,y£X x£X < rKz(m, n) £ f*(x)g*(x) xex = £ f(x)L((m'x)'(n,y))9{y), x,yeX as desired, where we have used the Hardy-Littlewood inequality (Theorem 1.1 of this chapter or Theorem 1.2.3). Suppose now that m = n. Then, for x and y in X we have L((m, x), (n,y)) = (1 - r)Kx{x, y) + (1 + rA'z(m, m))SXty. 141 Chapter II. Discrete symmetrization If we can prove that (1 - r)Kx(x, y) + (1 + rKz(m, m))Sx,y = K{d{x, y)) for some decreasing funct ion K, then by Theorem 2.2 we wi l l obtain (9.2). Bu t if d(x,y) — 1 then x and y are adjacent so that (l-r)Kx(x,y) + (l + rKz(m,m))SXty = (1 - r ) ^ 1 < 1. If x = y then (1 - r)Kx(x, y) + (1 + rKz{m, m))SXiV = 1 + rKz{m, m). If d(x, y) > 1 then (1 - r)/sr*(x, y) + (1 + rKz{m, m))Sx>y = 0, as x and y are then neither equal nor adjacent. Hence, if we let K(x) = 1 + rKz(m,m) for £ < 1, K(x) = (1 — r ) p _ 1 for 1 < x < 2 and A'(a;) = 0 for x > 2 then we wil l have (1 — r)Kx(x, y) + (1 + r i ( z ( m , m))SXjy = K(d(x, y)) as desired. Thus Assumpt ion 9.1 holds. On ly Assumpt ion 9.2 remains. Suppose first that m ^ n. Then , J2K((m,x),(n,y)) = rKz(m,n) yex which is clearly independent of x. On the other hand, suppose that m = n. Then , Y^K({m,x),{n,y)) = l + rK'z{m,m) + (l-r) £ p~l = 1 + rKz(m,m) + 1 - r, yex yeN(x) where N(x) denotes the set of al l vertices of X adjacent to x so that |A^(a;)| = p, and hence again we have independence of x. Hence Assumpt ion 9.2 is val id. • 9.1.3. The random walk on V We now consider the M a r k o v process {Rn} on V for n G Z Q wi th the transit ion probabil i t ies P(Rn+1 = w | Rn = z) = K(w, z). 142 Chapter II. Discrete symmetrization Because in our examples this process will always be some kind of random walk, we shall refer to it as our "random walk". We call K the kernel of the walk {Rn}. We shall write Pz(-) and Ez[-] for probabilities and expectations, respectively, where the random walk is conditioned to have the starting value Ro = z. T Remark 9.1. Consider the kernel K = K% <g) Kx mentioned in §9.1.2. Then, R n can be characterized at follows. At each time step, while sitting at the point (TO, X) £• V, a weighted coin is flipped. With probability r a move is taken with x fixed, but with m changed according to the transition probability (in the case where A' z = K§ we move to (m± 1, x) with equal probability, and in the case Kz = Kf we remain put at (TO, X).) With probability 1 - r, the first coordinate TO is left fixed, but x is changed according to the transition probability Kx, i.e., a random point adjacent to x is chosen and the walk moves to it. Suppose now that X = Z„ for n > 3 or X = Z, that r = \ and that Kz = A ' f . Then V is a discrete cylinder Z X Z n and the walk Rn at each times step can be easily seen to have an equal probability \ of moving to any of the neighbouring points in Z X Z„ or Z x Z. (I.e., with equal probability \ , we can move from (a;, y) to any of the four points (x + 1, y), (x — 1, y), (x, y + 1) and (x,y - 1).) 9.1.4. Generalized harmonic measure Instead of simply considering harmonic measure for {Rn}, we shall consider a slightly more complicated situation for which the proofs, however, are no more difficult. Let {X„}" = 0 be a sequence of independent and identically distributed random variables uniformly distributed over (0,1]. For s: V —>• [0,1], we define TS = inf{n > 0 : Xn > s(Rn)}. This is a stopping time with respect to an appropriate filtration. We may interpret the situation as having 1 — s(z) indicate the probability of the random walk being killed while standing at z, so that s(z) is a survival probability; then, TS indicates the amount of time for which the random walk survives. We may call the situation described above "a random walk with dangers." 143 Chapter II. Discrete symmetrization The usual case of harmonic measure will be recovered if we let s be the indicator function ID def . of a set D C X, since then rs coincides with the exit time TD — inf{rc > 0 : Rn fi D}. For S C I and z G V, we write ui(z, S; s) = Pz(TS < oo and RTs G S). Technical remark 9.1. In some cases the condition r s < oo is redundant, since if the walk {Rn} is recurrent and visits every point, then we have TS < oo with probability one providing there exists a z G V with s(z) < 1. 1/2 Remark 9.2. Let X — Z n or X = Z. Consider the case of K = A'f <g> Kx. Then, Rn is a simple random walk on Z x X. If s — ID and S is a subset of the boundary of D (the boundary being defined as the set of points of V\D which are one simple random walk step away from D), then CJ(Z, S; s) is just the ordinary discrete harmonic measure at z in D of X; it is simply the probability that if the random walk started at z ever exits D, its first exit from D lands it at a point of S. Thus, in light of the connection between Brownian motion and harmonic measure (Theorem 1.5.5), u(z,S;s) is quite analogous to the classical harmonic measure u(z,S;D) for S C dD and D C C. Note that Z X Z„ is a tube, and thus is the discrete analogue of the tube R x T . On the other hand, the latter tube is conformally equivalent to the punctured plane C\{0} via an exponential map. The effect of a general choice of s is that of making the domain D somewhat "fuzzy." Note that if s = ID then s# = 1D# so that symmetrization of s corresponds to symmetrization of a domain. Finally, we give a certain partial ordering on the set of positive functions on V. For f,g:V-t [0, oo), we write / -< g providing that for every i G Z and every convex increasing function $ on [0, oo) we have iex jex 144 Chapter II. Discrete symmetrization By [ 5 8 , Thm. 1 0 8 ] , this is equivalent to the condition sup Y]f(i,j) < sup Y]g(i,j) ( 9 . 3 ) being valid for every k G Z Q and each i G Z , where the suprema are taken over all k element subsets I of X. In particular if / < g then supjeX f(i,j) < sup j e^ j). If, moreover, g is Steiner symmetric and X — Z then the right hand side of ( 9 . 3 ) becomes Y 9(hJ), 3 = -l if k = 21 + 1, and / £ ff(*.j). j=-/+i if A; = 2 / , while on the other hand sup j^^ g(i,j) = y ( « , 0 ) . Definition 9 . 1 . Let X be a non-empty set equipped with a Schwarz type symmetrization Then the initial element of X is defined to be the unique element O of {a;}* where a; G L (This definition is unique because of the Schwarz type of #.) If g is Steiner symmetric and # is of Schwarz type on X with initial element O, then sup g(i,j) = g{i,0) jex for all i. Our result on the effect of symmetrization on harmonic measure is as follows. Theorem 9 . 1 . Let {Rn} be a random walk associated to a kernel K which satisfies Assump-tion 9 . 3 . Fix k G Z . Let s: V —> [ 0 , 1 ] be a function vanishing on {k} X X. Let S C {k} X X. Then, u(-,S;s)<u;(;S*;s*). Moreover, u(-, S*; s*) is Steiner symmetric. 1 4 5 Chapter II. Discrete symmetrization In particular, if # is a Schwarz type symmetrization on X with initial element O, then for every i £ Z we have supu((i,j),S;s) < oj((i,0),S*;s*). 1/2 Remark 9.3. In the case of X = Z n , K = K§ ® K X , and s = ID, Theorem 9.1 is an analogue of (a trivial generalization of) a result of Baernstein [7, Thm. 7] (stated in the present thesis in Theorem 1.6.2). Of course, Baernstein's result is concerned with circular symmetrization. However, our space Z X Z „ is a discrete analogue of E x T, and the latter is conformally equivalent to C\{0} under an exponential map. Using this exponential map, we can define a circular rearrangement on IR x T in a natural way by pulling back the rearrangement on C. Our rearrangement # on Z x Z n is a discrete analogue of that circular rearrangement on R x T. Indeed, it might even be possible to obtain Baernstein's result on IR X T as a limiting case of results on Z x Z n (we would have to take n to infinity and use some sequence of sets Dn C ZxZ„ which in some way approximates a domain D C R x T). Strictly speaking, our Theorem improves on a discrete result of Quine [90], but in the special case of the simple random walk, still under the assumption that s = Ip, it is actually a consequence of Quine's result on the subharmonicity of the discrete *-function [90]. An example of an application of our Theorem with X = Z can be seen in Figure 9.2. The idea is that we have a random walk starting from the point S which continues until it either bumps into one of the black squares (which represent dangers, such as a dragon or a cliff) or it gets to one of the squares on the right hand edge where it safely exists. Then, our Theorem asserts that the probability of reaching the right hand edge in the original situation (top figure) is less than or equal to the probability of reaching it in the symmetrized situation (bottom figure). (Note that in the figure, the black squares correspond to the complement of the set we are allowed to walk on.) 146 Chapter II. Discrete symmetrization Unsymmetrized situation Y Y ^ ^ ^ Y Y Y ^ ^ ^ ^ ^ • 6 — ^ 0 ^ — v ~ — v ^ ^ / 6 Y ~ ^ Y — - Y V-^v^=^t Symmetrized situation Figure 9.2: Steiner symmetrization on Z 2 . The set being symmetrized is indicated by white squares. 1 17 Chapter II. Discrete symmetrization 9.1.5. Generalized Green's functions We may now define a Green's function for our "fuzzy" domain defined by the survival function s. Simply let gs(x,y) = Ex E 1{Rn=y} ,n=0 This represents the expected number of times that the random walk starting at x visits y before being killed by one of the dangers. Note that, depending on questions of transience, it is quite possible for gs to be infinite at some pairs of points. If s = lrj> then gs corresponds to the usual definition of the discrete Green's function go for D. We say that an extended real function G on V2 is Steiner bi-symmetric if for every pair of integers i and j we have E G((i,a),{j,fi))< E G((i,a),(j,(l)), for all subsets K and L of X. Now, given two positive extended-real functions F and G on V2, write F <G providing sup V F((i,a),(jJ))< sup V G((i,a),(j,P)), |Aj=fc,|L|=/ a€j^eL |A-|=fc,|L|=; a e K M L for all integers i and j and all positive integers k and /, where the suprema are taken over all subsets K and L of X containing precisely k and / elements, respectively. If G is in addition Steiner bi-symmetric, then this condition is equivalent to saying that for every pair of integers i and j and every K C X we have J]$(^F(( ! , A ) , ( J , /3) )J < E $ f E G((i,a),(j,(3)) peX \a£K ) /sex \aeK# , for every convex increasing $ on [0, oo]. In particular if K is singleton and # is of Schwarz type on X with initial element O, then it follows that for every fixed a £ X we have E a), (j,/?))] < E O ) , pex pex for any convex non-decreasing $ and for every pair of integers i and j. 148 Chapter II. Discrete symmetrization T h e o r e m 9 .2 . Consider a random walk {Rn} with a kernel K satisfying Assumption 9.3. For any m 6 Z + U {oo} and any s: V —> [0,1] we have 9s <9s*, and, moreover, gs# is Steiner bi-symmetric. In particular, if # is of Schwarz type on X with initial element O, then for any integers i and j , any y £ X and any convex increasing function $ we have pex pex Remark 9.4- Th is is a discrete version of a generalization of a result of Baernstein [7, T h m . 5] (see Theorem 1.6.1 of the present thesis). In the special case of the simple random walk on Z x Z m and s — I D , this can probably be proved by using Quine's result [90] on the subharmonici ty of the discrete ^-funct ion. Our proofs, like those of Bore l l [25], are based on the probabi l ist ic method of Haliste [56, proof of T h m . 8.1]. 9.2. Reducing to the case A = 0 in Assumption 9.3 We shall show that if we can prove Theorems 9.1 and 9.2 for K satisfying Assumpt ion 9.3 with A = 0, then we can prove them for K satisfying Assumpt ion 9.3 wi th general A > 0. For , suppose that K satisfies Assumpt ion 9.3 wi th some A > 0. Define L = c(K + XS), where c > 0 is chosen so that (9.1) holds wi th L in place of K, and where S(v, w) is 1 if v = w and 0 otherwise. It is easy to see that c = (1 + A ) - 1 . It is clear that L satisfies Assumpt ions 9.1 and 9.2. Use superscripts K and L to dist inguish quantit ies denned in terms of the random walk {R^ } and the random walk {R^} defined by the transi t ion kernels K and L respectively. Professor Gregory Lawler would cal l R^ "the walk R^ wi th geometric wait ing t imes" . 149 Chapter II. Discrete symmetrization Assume now that Theorems 9.1 and 9.2 have been proved for the kernel L. We shall prove them for the kernel K. To do this, proceed as follows. We may describe the random walk {R%} slightly differently from before. A step of this random walk consists of first flipping a coin and with probability p = A/(l + A) staying put, while with probability 1 — p = 1/(1 + A) taking a step with transition probabilities defined by the kernel K. Let Sn be the event that the flip of the coin was such that we took the step with transition probabilities defined by K. Now, note that the walk R^ will eventually take a step defined by the kernel K, and that the distribution of R^+1 conditioned on the event Sn is the same as the unconditioned distribution of R^+i • Moreover, the probability that the random walk R^ will survive until one of the events Sn happens is equal to (1 - p)s(z) + (1 - p)p(s(z))2 + (1 - p)p2(s(z)f + •••= {lZpSS{^ = where <f>(t) = ^Zpj*- Note that <j> is a strictly increasing function mapping [0,1] onto itself. The above shows that R% with survival probabilities s behaves much as R^ with survival probabilities (ho s. This observation shows that uK(z,X;(f>os) =uL(z,X;s), or, equivalently, uK(z,X;s) =uL(z,X;(h-1 os), both for any choices of z G V, S C V and s: V —>• [0,1]. Since d> is strictly increasing it follows that (0 _ 1 os)# = (f)"1 os* (Theorem 1.2.2). These observations show that Theorem 9.1 for the walk {R^} with kernel L implies an analogue for the walk {i? }^. Now, if the random walk {R„} is at a point w then the probability that it survives at least k contiguous steps before one of the events Sn happening (i.e., before taking a step in accordance with the kernel K) is The expected number of contiguous steps that it survives without taking a step according to 150 Chapter II. Discrete symmetrization K is then equal to CO / \ E ^ ) V - 1 = r ^ ^ = ( i -p)-VK™)) . On the other hand, the expected number of contiguous steps that the random walk {R^} equipped with survival probabilities <j> o s will survive and stay at w before taking a step according to K is <fi(s(w)), if we condition on it starting at w. Combined with the results of the previous paragraph, we can easily convince ourselves that the identity 9$os(z> "0 = (l - p)~lgs(z> w)> must hold. Using the monotonicity of d> and Theorem 9.2 applied with the kernel L, we can see that Theorem 9.2 must also hold for the kernel K. 9.3. Exi t times and proofs Our proofs are based on the methods of Haliste [56, Proof of Thm. 8.1]. Assume that {Rn} is a random walk associated with a kernel K. Let ix\ : V —> Z be defined by ni(i,j) = i and let 7 r 2 : V —>• X be defined by 112(1, j) = j-Let Q be the <7-field generated by {ixiRn}^^ on our underlying probability space. Theorem 9.3. Let I,JCX. Fix i 6 Z. Let the kernel K satisfy Assumption 9.3 with A = 0. Then J 2 p i h j ) ( T s > N and TX2(RN) <= J | Q) (9.4) < ^ P{h3)(rs > N and TX2(RN) G J* I G), jei# almost surely for every non-negative integer valued random variable N which is measurable with respect to Q. Setting J — X and taking an (unconditional) expectation we will obtain the following theorem. 151 Chapter II. Discrete symmetrization Theorem 9.4. Let I C X. Fix an arbitrary i £ Z. Let the kernel K satisfy Assumption 9.3 with A = 0. Then, Y,P{i'j)(Ts >N)<J2 P{i,3)(rs* > N), (9.5) jei jei# for any non-negative N. Remark 9.5. Note that if we apply (9.5) with s* in place of s, then we easily obtain the fact that z i—^ PZ(Ts# > N) is Steiner symmetric for every ./V via Proposition 8.8. Moreover, (9.5) implies that in general (z ' y Pz(TS > AO) < (z ^ Pz(TS# > AT)). Proof of Theorem 9.1. Assume A can be taken to be zero in Assumption 9.3. (The case of A > 0 then follows from the work of §9.2.) We shall prove that J 2 p i l ' j ) ( T s < 0 0 a n d Rrs G S) < E PihJHTs# < oo and € S*), (9.6) iei jei# for S C {k} X X. This will imply Theorem 9.1 (the requisite Steiner symmetry will follow by applying the result with s = s* and using Proposition 8.8). We shall now prove that J 2 p ( i ' j ) ( T s < oo and RTs £ X \ Q) j e I (9.7) < J2 p ( h 3 ) i T s # < 0 0 a n d R r s # ex\g). jei# Inequality (9.6) will then follow by taking an unconditional expectation. Now, to prove (9.7), first suppose {ft\(Rn)} never visits A;. Then both sides of (9.7) vanish and we are done. Hence, suppose that {TTi(Rn)} visits k and let N = inf{n > 0 : 7rx(i?n) = k}. In this case we clearly have P(TS < 0 0 | Q) — P(TS# < 00 \ Q) = 1 since s vanishes on {k} x X. For convenience, set 5(z) = s(z) for z ^ {k} x X and let s(z) = 1 for z £ {k} x X. Then, (9.7) is equivalent to the assertion that E p ( i J V s > N and RN e X \Q) < ^ P(i'j)(r-S# > N and RN £ X* \ Q). But this inequality holds by Theorem 9.3 since N is ^ -measurable. • 152 Chapter II. Discrete symmetrization We now prove Theorem 9 . 2 . Proof of Theorem 9.2. It suffices to prove that whenever K and L are finite subsets of X and i and j are integers, then we have We proceed once more by conditioning on Q and then taking an unconditional expectation. In this way, using the definition of the Green's function, we see that it suffices for us to prove that a£K,peL n=0 aEK#,0eL# n=0 By Fubini's theorem this is equivalent to the assertion that oo E E E(t'a)[kRn=(m^ra>n}\G] N = ° * ™ G L ( 9 . 8 ) < E E ^(!'a)[1{Rn=(i,/?) and T s # > n } I G]-Let T be the random set {n £ Z Q : Ki(Rn) = j}- Then, ( 9 . 8 ) is equivalent to the assertion that E P{i'a)(MRn) e £ and r s > n \ Q) < E E p ( % , a ) M R n ) € i # and Ts#>n \ G). neTa&K* But this follows immediately from Theorem 9 . 3 . • We now proceed to prove Theorem 9 . 3 itself. In order to do this, we first state a lemma giving the inequality that lies at its heart. This inequality is a generalization of the idea of the proof of Haliste's [ 5 6 , Lemma 8.1] and can be called an "iterated convolution-rearrangement inequality". We now need an assumption on a function L: X x X [0, oo). 1 5 3 Chapter II. Discrete symmetrization Assumption 9.4. The following inequality is valid whenever / and g are positive functions on X: f(x)L(x,y)g(y)< £ f* (x)L(x, y)g*(y). x,y£X x,y€X Write Lg(v) = Y^wev ^(u> w)d(w) a n d L*f(w) = Y,vev f(v)L(v, w). By Theorem 8.1 we obtain the following result. Proposition 9.3. Assume Assumption 9.4. Then the functions Lg, L*g: X [0,oo) are sym-metric whenever g: X —> [0,oo) is symmetric. Lemma 9.1. Let L°, L1,... , Lk~x he a sequence of functions on X X X all satisfying Assump-tion 9.4- Let S°, S1,... , Sk also be a collection of positive functions on X. Let *k(S, L) = Y' • S°(u0)L°(u0, i/O • ••Sk-1(vk-1)Lk-1(vk-Uvk)Sk{vk), for k > 0, where all sums are taken over X, and let ty0(S,L) = S°(UQ). Then for each k 6 Z Q we have yk(S:L)<Vk(S*,L), where (5#)*'(j) =\S*)* ( j ) . Assuming this lemma, we now give a proof of Theorem 9.3. Proof of Theorem 9.3. Let f(n) = 7TI (JR„) so that Q = ff({/(n)}~0). Let T be the a-field generated by {RN}^=1. Let N be a (/-measurable random variable with values in Z Q . Let P,(i,j,N) = P^3){TS > N and K2(Rn) eJ\G). We shall now give a formula for ps. To do this rigorously, let g(n) = TT2(RN). Then, ES{N) = {TS > N and n2{RN) € J} / N \ (9.9) = {9(N)e J } n f){XN < s(f(n),g(n))} . \n=0 / 154 Chapter II. Discrete symmetrization Thus, P(ES(N) | T) = l{g{N)eJ} • s(/(0),5(0))S(/(l), <7(1)) • • • s(f(N), g(N)). (9.10) Let Ln(x,y) = P(g(n)=y\g(n-l) = x,G), for 1 < n < N. Let LN+1(x,y) = SXiy. Put SN+2 — lj. Then, taking the conditional expectation of (9.10) given G, we find that we have to basically average G over all functions g with transition probabilities Ln, and we obtain = £ £ - E v i t x »2ex vN+\ex (9-11) sU^),i)L\j)vMf^)^i)L\yuy2)---s{f{N),vN)LN+\vN, vN+1)SN+2(vN+1). Now, let L°(x,y) = Sx>y. Set S° = 1/. Let 5n(i) = s(f(n - for 1 < n < N + 1. Then, it follows from (9.11) that Y,Ps(hJ,N) = VN+2(S,L), (9.12) where is defined as in Lemma 9.1. Thus the left hand side of (9.4) equals \P;v+2 {S, L). Henceforth, implicitly condition all our reasoning on G-Now, we have (s(i, •))* = (s#)(i, •), the first quantity being defined in terms of the rearrange-ment on X and the second in terms of the Steiner rearrangement on V = Z X X induced by the first. Tracing through the above work we see that the right hand side of (9.4) thus equals ^N+2(S*, L). Hence, the conclusion of our theorem will follow from Lemma 9.1 as soon as we prove that all our kernels Ln satisfy Assumption 9.4. First of all, L° and LN+1 both satisfy Assumption 9.4 in light of the Hardy-Littlewood inequality (Theorem 1.1). Thus it suffices to verify that Ln satisfies Assumption 9.4 for 1 < n < N + 1. 155 Chapter II. Discrete symmetrization More precisely, we shall prove that there is a set A G Q such that P(A) = 1 and such that (given Q) we have Assumption 9.4 holding on A. Then via Lemma 9.1 we will obtain the inequality i$N+2(S, L) < ^Ar+ 2(5*,L) on A, and hence almost surely (given Q), as desired. Define kl{m,n) = £ A'((m,i),(n,j)), jex for i € X and m and n in Z. Note that in fact kl(m,n) does not depend on i in light of Assumption 9.2; we shall sometimes omit the superscript i and at other times we shall retain it for emphasis. Let Af B=\J{k(f(n-l)J(n)) = 0}. n=l It is clear that B has probability zero by definition of / and the random walk K. Let A be the complement of B. This is then a Q measurable event which happens almost surely. Henceforth, we assume that A happens. Then, Ln(x, y) = P{g{n) = y\g(n-l) = x, Q) = P(Rn = (f(n),y) | Rn-! = (f(n-l),x),g) = K((f(n -l),x), (f(n),y))/kx(f(n - 1), f(n)) = Kttf(n-l),x),(f(n),y))/k(f(n-l)J(n)). But now on A the quantity k(f(n — l),/(n)) is nonzero, and it is independent of x as noted before. Hence, Assumption 9.4 follows from Assumption 9.1 and Proposition 9.1, as desired. • Our proof of Lemma 9.1 is a generalization of the proof of Haliste's [56, Lemma 8.1]. Proof of Lemma 9.1. Proceed by induction on fc. If k — 1 then Lemma 9.1 is equivalent to Assumption 9.4. Hence assume that k > 1 and that Lemma 9.1 has been proved for all smaller values of k. To reduce clutter, let T - S*. Put C M = E • • )^ • ••Tk-1(uk.l)Lk-1(uk.1,uk)Tk(uk), 156 Chapter II. Discrete symmetrization and BM=52S0(v0)L°(v0,v)S1(v). Since L e m m a 9.1 holds for A; — 1 it follows that The left hand side of this inequali ty equals ^k(S,L). Hence, I c la im that Y,B*{yx)c{yx)<^k(T,L). (9.13) If this c la im is just then we are obviously done. Now, we may assume that S1 has finite support (the general case follows by approximat ion v ia the monotone convergence theorem and L e m m a 1.2.1.) A p p l y Propos i t ion 8.6 to the funct ion f — B which has finite support if S1 has finite support. Then , B* = B o <f>. Define c(v) = c(<f>-l{y)) and d(u) = Sl{v)c(v). Then , £ B # M C ( ^ ) = £ B K ) ^ i ) = J2 S°{v0)L0{vo,vi)d{M (9.14) < ^ T ° M £ ° K ^ I ) ^ # ( ^ ) , by Assumpt ion 9.4. Let E(v) = YdT°{yQ)LQ{vQ,u): B y Propos i t ion 9.3, the funct ion E is symmetr ic . I c laim that Y,E{vi)d*(Vl) < Y,E{vi)Tl(vMvi)- (9-15) 157 Chapter II. Discrete symmetrization If this claim is true then we are done since the right hand side of this inequality is precisely equal to ^k(T, L), so that (9.13) would immediately follow from (9.14). Now, since S 1 has finite support, it follows that d has finite support. Applying Proposition 8.6 with / = d, let ip be the function u<p" given by that Proposition. Then, d#{v) = d(ip{i>)) = S1(ip(v))cip(u)). Then, the left hand side of (9.15) equals J2E(vi)S\v)c(v), where E{v) = E(ip~x{v)). Applying Theorem 1.2.4, we see that YJ E ( V 1 ) S 1 ( P ) C ( V ) <J2E(^)T\u)c(u), as desired, where we have used the fact that T1 = (S1)* and the facts E& = E and c# = c which came from our two applications of Proposition 8.6, since E and c are symmetric. The symmetry of E was already noted. The symmetry of c, on the other hand, is also not hard to prove. To prove it, one only needs to inductively apply Proposition 9.3 as well as Corollary 8.5. • 10. A discrete Beurling shove theorem 1/2 In this section we are working with the simple random walk Rn defined by the kernel K§ <S> KZm on Z x Z m for m > 3 (see §9.1.2), and all quantities (Green's functions, etc.) should be interpreted with respect to it. Given D C Z x Z m , we write u>(-, •; D) for w(-, •; l c ) and go for 9iD-Let D = Z " x Z m , where Z " = {-1, -2,...}. Let T — {0} x Z m . Then the following result is a discrete analogue of Beurling's shove theorem [23, pp. 58-62] (an account of this theorem and some generalizations in the continuous case will be given in §IV.8). Theorem 10.1. Let H be a finite non-empty subset of Z _ x {0}, and set U = D\H. Let U° = D\H', where H' = {-\H\,-\H\ + 1,... , -1} x {0}. Then, u>((t,0),T;U)<u((t,0),T;U«), (10.1) 158 Chapter II. Discrete symmetrization whenever t < M{t' : (t',0) £ H}. Remark 10.1. One may conjecture a number of generalizations of this. One such would be to consider a survival funct ion s instead of U, such that s and s° vanish identically on V\D and are identical ly 1 everywhere on D except possibly on Z _ X {0}, while s°(-,0) is the decreas-ing rearrangement of s( - ,0) . Then the conjecture of course is that the analogous inequality continuous to hold. Th is conjecture appears to be nontr iv ial even in the case of m — 1 (with appropriate definit ions which collapse the random walk to a one-dimensional walk) , although in that case it is true (Theorem IV.9.1) . The proof of Theorem 10.1 wi l l be done almost exactly as in the continuous case, as soon as we establish two lemmas and discuss the notion of a discrete harmonic funct ion. Let U C V = Z x Z M . Wr i te dU for the collection of points of V\U which lie precisely one simple random walk step away f rom U, i.e., dU = {(x,y)eV\U: (x-l,y)eU or (x+l,y)eU or (x, y - 1) G U or (x, y + 1) G U}. Wri te U = U U dU. Let / be any funct ion on U. Define A / ( a r , y) = \[f{x + 1, y) + f(x -l,y) + f(x, y+l) + f(x, y-1)]- f(x, y), for (x,y) G U. Then A is a discrete Lap lac ian. Moreover, for any i G Z Q we have Af(z) = E[f(Rt+1) | Rt = z]- f(z), (10.2) where {Rn} is our simple random walk on V. We say that a function / is a (d i sc re te ) h a r m o n i c f u n c t i o n o n U if it is defined on U and satisfies A / ( z ) — 0 for all z G U. Example 10.1. The funct ion z H-> gz>(z,w) is a discrete harmonic function on D\{w} for any fixed w G V. Th is is easiest seen direct ly f rom the definit ion of go — giD and from equa-t ion (10.2). Example 10.2. The funct ion z >->• u>(z, A; D) is a harmonic function on D for A C 3D. Th is is also easy to see f rom (10.2) and the definit ion of u(z, A; D) = u(z, A; l r j ) . 159 Chapter II. Discrete symmetrization Clearly sums and scalar multiples of harmonic functions are harmonic. A central result about discrete harmonic functions is the following very well-known maximum principle. T h e o r e m 1 0 . 2 . Let f be a harmonic function on U C V which is bounded above and let C be a real constant such that f(z) < C for all z £ dU. Assume that U ^ V. Suppose that there exists w £ U such that f(w) > C. Then f is constant on U. We now state our two lemmas which provide the keys to the proof of Theorem 10.1. L e m m a 1 0 . 1 . Leth{t\,t2) = 0), (t2, 0)). Then for fixed t2 £ Z~ the function h(-,t2) is increasing on (—oo, t2] n Z~, and decreasing on [t2, — 1] fl Z _ . Similarly, for fixed t\ £ Z _ the function h{t\, •) is increasing on (—oo, t{\ fl Z~, and decreasing on [t2, —1] fl Z~. Proof. Fix t2 £ Z~. First suppose t\ > t2. We shall show that h{t\ - l,t2) > h(ti,t2). Let Di = {ti — 1, ti,... , —1} X Z m Then, it is easy to see that Hh,t2)= J2 ^{(h,0),{(ti-l,a)};D1)gD((t1-l,a),{t2,0)). But gjj is bi-symmetric because of Theorem 9.2 so that go{{ti — 1, a)i (^2, 0)) < 5z?((ii — l,0),(i2,0)). Thus, h(h,t2) KgDi^-1,0), (t2,0)) w((ti,0),{(ti- l,a)};Di) = w((tx, 0), {ti - 1} x Zm}; 7Ji)/i(ii - 1, t2) < /l(t! - l,t2). The inequality + l,t2) > h(ti,t2) in the case t\ < t2 is proved very similarly. The case of t\ fixed can be handled just as above (or else it can be noted that it follows from the fact that h{t\,t2) = h{t2,t\) for the simple random walk.) • Now, for a subset S of Z x Z m , let r s = inf {n > 0 : Rn fi S} 160 Chapter II. Discrete symmetrization and fs = inf {n >0:Rn(£S}. The following lemma then is valid for any random walk, not just the simple random walk. It extends in an easy way to a number of situations. Lemma 10.2. Let <p(z) = 1 - u(z,T;U). Then <t>{z) = ^ SD(z,w)ip(w), for a positive function tp on H. More precisely, we may take Tb(W) = PW{fu = fD). Proof. For ij)(w) = Pw(fu = Tp), by the definition of the Green's function and by Fubini's theorem we have ^2QD(Z, w)ih(w) weH weH CO 71=0 Pw(fu = fD) = J2J2 pz(Rn = w a n d t d > n)p" ;( ft/ = fD) n=0 w£H oo = J2pZ (R™ eH,rD>n and (Rk G U, V& G {n + 1,... , T D - 1})). n=0 But it is easy to see that the events within the Pz(-) are disjoint for distinct values of n since H C Uc. Moreover, it is easy to see that the union over n £ Z Q of these events is the event {3n G Z Q . (Rn G H and n < Tp)}. But clearly the probability of this event if the random walk starts at z is precisely u>(z, H; D) = (f>(z). • Proof of Theorem 10.1. For conciseness, given a subset L of Z Q X {0}, write inf Id=inf{/ : (Z,0) G L}. Let t0 = inf H. We proceed by induction on N — \t0\ - \H\. First, if iV = 0 then H — H' and we are done. Suppose now that the result has been proved whenever \to\ — \H\ < N and that 161 Chapter II. Discrete symmetrization H = •10,0) (-9,0) • • t to (-8,0) (-7,0) (-6,0) • • • -5,0) (-4,0) (-3,0) (-2,0) (-1,0) • • • • • (-10,0) (-9,0) (-8,0) (-7,0) (-6,0) (-5,0) (-4,0) (-3,0) (-2,0) (-1,0) • • • • • ' • • • • • Figure 10.1: An example of the sets if and H\ (indicated with B's) in a case where to = -9, ti = —7, |if | = 6 and N = 3. The symbol Aj indicates a point contained in the set Aj. N > 1. Let tx = inf{t G Z~ : t > t0,t fi if). Since N > 1, we have tx G {t0 + l,..., 1}, and moreover {t0,..., ti - 1} C H. (See Figure 10.1.) Define Hi = ( # n [ * i , l ] ) U { * o + l , . . . , i i + l}. (See Figure 10.1.) It is easy to see that \Hi \ = \H\ and that i i i is in fact just H with the hole at ti deleted. Moreover, | inf Hx\ = t0 + 1 so that | inf Hi\ — \H\ < N as t0 < 0. Thus, if we form (iii)' from iii in the same way that H' is formed from H, by our induction hypothesis we will have w((t, 0), T; ZAffO < w((0, t), T; D\(ffi)'), whenever i < inf Hi, and in particular whenever t < inf i i . But \Hi\ = \H\ so that (iii)' = H'. Thus, the desired inequality (10.1) will follow as soon as we establish the fact that c((t,0),T;D\H)<u((t,0),T;D\Hi) (10.3) whenever t < inf H. Write H = AiUA 2 where A x = {t0,..., ti - 1} and A 2 = {*i + l , • • .1}C\H. Let <?!>(z) = u{z,T; D\H). Then, by Lemma 10.2 we have cj){z) = (f)i(z) + fo(z), 162 Chapter II. Discrete symmetrization where <t>i(z) = Y 9D(z,w)ib(w), u>€A, for i = 1,2. For (x, y) 6 ZQ X Z M , let 0(a;, y) = fa (x — 1, y) + <^ 2(a:, y)- I claim that faz) > l-u{z,T;D\Hx) (10.4) for all z G ZQ X Z M . Suppose for now that this claim is just. Then, for t < inf H we have co((t,o),T;D\Hx) > l - M * - i , o ) - & ( t , o ) . But 0 x(i — 1,0) < ^i(i, 0) since #£)((£,0), («, 0)) is increasing in t for i < u (Lemma 10.1) and since tp is positive while Ax C [—to, -1] X {0}. Thus, co((t, 0), T; Z A # i ) > 1 - &( i , 0) - fa(t, 0) - 1 - 0(i, 0) = u((t, 0), T; D\#), which is precisely what we were supposed to prove. Thus, we need only verify (10.4). But 4> is a bounded3 discrete harmonic function on D\HX since fa and fa are harmonic on D\AX and J D\A 2 , respectively, (since the fa are sums of Green's functions to which we can apply Example 10.1), and u(-,T; D\HX) is harmonic in D\Hi (Example 10.2), while Hx = {(1,0) + A1)UA2. Thus, the maximum principle (Theorem 10.2) implies that to show (10.4) it suffices to verify that (10.4) holds on d(D\Hx) = Hx U H2 and T. But on T, the inequality (10.4) holds trivially as its right hand side vanishes while the left is positive. Suppose now that z € Hx = ((1, 0) + Ax) U A2. Then the right hand side of (10.4) equals 1. There are two cases to consider. First suppose that z 6 ((1,0) + Ax). Then, z — (x + 1, 0), where (x, 0) G Ax. We have 4>{z) = fa(x +1-1,0)+ fa(x + 1, 0) = fa(x, 0) + fa(x + 1, 0). 3The function <f> is bounded since <f> is bounded which implies that <f>\ and <f>2 are both bounded. 163 Chapter II. Discrete symmetrization But whenever w £ A2 and (x, 0) G A l 5 we have go{{x + 1, 0), w) > gD{{x, 0), w) by Lemma 10.1 so that (f>2(x + 1, 0) > <f>2(x, 0), and so 4>(z) > 4>1(X^) + Mx^) = (f>(x^)-Now, (rc, 0) G H so that cf>(x, 0) = 1 - u((x, 0), T; D \ / f ) = 1, and so (10.4) is verified. Suppose now that z = (x,0) £ A2. Then, But whenever w G A i and (rc, 0) G A 2 we have gn((x — 1, 0), w) > yo((a;, 0), w) by Lemma 10.1, so that 4>\{x — 1,0) > 4>\(x, 0), and thus ^(*) > ^i(a;,0) + </»2(x,0) = cf>{x,0) = 1, since (x,0) G H as before, so that again (10.4) is verified. • Remark 10.2. The methods of this section apply equally well to a random walk on ZQ X G where G is one of the constant degree graphs for which a master inequality holds under some well-ordering -< with each element having at most finitely many predecessors, and where 0 G Z M is replaced by the initial element O with respect to For instance, in the above work we may replace the graph Z M by the regular tree Tp. 11. A general rearrangement method for difference equations In this section we shall present a quite general rearrangement method for difference equations. Our method is a modification of the method of Baernstein [11]. Let X and Y be countable sets equipped with counting measures and let # be a Steiner type rearrangement from the power set 2X to the power set 2Y. Let A be the fibres of this rearrangement. Let K be a positive function on X x X. For a function / on X, define Kf(x) = J2K(x^)f(y) vex 164 Chapter II. Discrete symmetrization and /C/(y) = £ / ( z )K (z ,y ) . Let L be a positive function on Y X Y, and define Lf and L*f analogously to Kf and K*f. We make the following assumptions on K and L. Assumption 11.1. For every x 6 X we have X>(* ,y)< l . Assumption 11.2. For every pair of positive functions / and g on X we have J2 f(x)K(x,y)g(y)< ]T f*(x)L(x,y)g*(y). (11.1) Assumption 11.3. If / is a positive symmetric function on Y then so is L*f. Remark 11.1. If X = V, = £ and # is a symmetrization, and if (11.1) holds for all positive / and g, then Assumption 11.3 follows from Theorem 8.1. The setting of X = Y,K — L and # a symmetrization is very close to the setting of the previous section. Example 11.1 (Trivial example). Let X = Y, K = L and let S* = S for all S. This is a Steiner type symmetrization (just let the fibres be A = {{x} : x € X}), such that all functions are symmetric and all our assumptions are trivial. We define the operators Df=(K-l)f=(Kf)-f and Ef=(L-l)f = (Lf)-f. In many interesting cases these will essentially be discrete Laplacians. If / is a real function on X while g is a real function on Y then we write / < g if s s 165 Chapter II. Discrete symmetrization for all symmetric sets S. Since level sets of symmetric functions are symmetric, it easily follows that / < g if and only if for every positive symmetric h we have Y Y Let M. be the collection of all positive functions symmetric f on Y such that Y Given a function g on Y, define the functional Y for / G M. It is clear that / <g if and only if aj#_g(h) < 0 for all h £ M. Given a countable set X, let be the collection of all real functions / such that | 5 n | _ 1 £ / ( * ) " > 0 whenever {Sn}^^ is a collection of subsets of X with | 5 n | —> oo. Lemma 1 1 . 1 . Suppose that X = Y and that # is a symmetrization. Let v be a function on X such that v <v. Then v is symmetric. Proof. Suppose that v and do not coincide. Then there exists x G X such that v(x) / v*(x). Choose the fibre A such that x G A. By Lemma 8.1 we have A — A*. Let -< be the ordering on A = A* induced by Proposition 8 .1, and let x 0 -< x \ -< • • • be an enumeration of A. Choose the smallest n G such that v(xn) ^ v*(xn). Then, v*(xi) — v(x{) for i G { 0 , . . . , n — 1} and v*(xn) > v ( x n ) since v * restricted to A is the -^decreasing rearrangement of v (this follows from Proposition 8 .1). Letting S = {XQ, ..., x n } , which is symmetric, we have Yv* = Yv = v*(Xn) ~ > °' s s contradicting the assumption that v < u. . • 166 Chapter II. Discrete symmetrization Proposition 11.1. Suppose that u<v. Then for every convex increasing function $ and fibre A £ A, we have £*(«(»))< Y xeA yeA* Proof. Considering this question only on the fibre A, we see that we may assume that we are working with a Schwarz type rearrangement. Since X and Y have the same cardinality, we may with no loss of assume that X = Y. But then we actually have a Schwarz type symmetrization, as can be easily seen. Now, then, it follows that X (equipped with the Schwarz order of Proposition 8.1) is order-isomorphic to a subset of ZQ and then our result follows from [58, Thm. 108]. • 11.1. Our assumptions Assume that # is a Steiner type rearrangement from 2X to 2Y. Now assume Assumptions 11.1, 11.2 and 11.3. Let be a subset of X. Suppose that u and v are positive functions on X and Y, respectively, satisfying -Du < <f>(u) — c-u + X, on fi (11.2a) and -Ev > cp(v) - d • v + n, on 0 # . (11.2b) Assume that cp is a real function on [0, oo) such that: cp is convex and increasing on [0,oo). (H-3) Assume that c and d are functions on and fi* respectively such that c > 0 (11.4a) d > 0 (11.4b) -c<-d. {11 Ac) 167 Chapter II. Discrete symmetrization (In interpret ing (11.4c), since c and d are only defined on Q and Q* respectively, we extend c and d to al l of X and Y respectively, by sett ing them to + 0 0 outside their domains of definition.) Assume that —d (after the extension) is symmetr ic . Assume that A and p are any real functions on Q and respectively, such that A o / x . (11.5) (Again , to interpret this, extend A and p to al l of X and Y, respectively, by sett ing them equal to — 0 0 outside their respective domains of definition.) F ina l ly , we need two boundary value condit ions on u and v. F i rs t , assume that inf u>supu, V A G A, (11.6a) A n n A\n where A is the set of the fibres of our Steiner type rearrangement. Moreover, assume that J2U- u ' V C / C X (11.6b) u \ n u#\n# Remark 11.2. In part icular, (11.6a) and (11.6b) wi l l be satisfied if u and v vanish outside Q and Q*, respectively. F ina l ly , we make the fol lowing assumption on the operator L and the domain Q * . A s s u m p t i o n 11.4. Let / be any posit ive funct ion wi th supp / nonempty. Define /o = / . Inductively, given fn, let / n + 1 = / „ + L*fn. Then there exists n £ Z + such that supp / Q * . Th is assumption essentially says that a random walk defined by L has a positive probabi l i ty of eventually leaving 1 1 . 2 . A discrete rearrangement theorem for difference equations or difference inequalities Our main discrete rearrangement result is encapsulated in the fol lowing theorem. 168 Chapter II. Discrete symmetrization Theorem 1 1 . 1 . Suppose that the assumptions give in §11.1 are made. Assume moreover that u is in w(X). Then u<v. The proof of Theorem 11.1 depends essentially on three main observations, which we shall label as Lemma 11.2, Proposition 11.2 and Proposition 11.3, respectively. Lemma 1 1 . 2 . Let F £ A4. Then, there is a sequence of positive numbers ai and symmetric finite non-empty sets Si such that for each i there exists A = A(i) £ A with Si C A*, with the property that E ^ ai < 1 while N i=l where fs, = \Si\'1ls,, and N £ Z+ U {oo}. Proof. First we do this for functions fs £ M. of the form fs = \S\~1ls for S a nonempty finite symmetric subset of Y. Let A' = {A £ A : AC\ S ^ 0}. This is a countable set. Let SA = SD A* and aA = | 5 | _ 1 | 5 n A\ for A £ A'. I claim that then fs = £ aAfsA. AeA' The proof of this is essentially trivial since the {A*} are a countable disjoint cover of Y. Moreover, Y « A = |5|- 1 |5| = 1. AeA1 Hence, we have proved our Proposition for functions of the form fs-Now, suppose that we have a general function F £ j\A. Let T = {f(x) : x £ Y,f(x) ^ 0}. Then, for each A > 0 the set Tfl [A, oo) is finite since otherwise F could not be summable while we had assumed that F £ M. It follows that we may find a strictly decreasing enumeration t0 > h > t2 > • • • of T. Let M = \T\. Let 6i = U - ti+1 for 0 < i < M. If M < oo, then let <W = tM-169 Chapter II. Discrete symmetrization Let Ti = {x : F(x) > U). The T are clearly symmetric if / is symmetric (this need not be the case for a general rear-rangement, but is easy enough to check for a discrete Steiner rearrangement by intersecting each T2 with a fibre and checking that the intersection is Schwarz symmetric much as in Propo-sition 8.4). Then, it is very easy to see that M i=0 Let Pi = \Si\Si. Then, M F{x) = YJfrfT„ where fx{ = | — 1 1 ^ . Moreover, YJy fx{ = 1 so that M M i=0 i=0 x£Y M xeY i=0 = !>(*) <i, xeY by Fubini's theorem and since F G A4. Now, fx, G M. is the kind of function for which we have already proved the Lemma so that hi = /E « k / s v AeA> for some countable set A\ and for positive alA with YIAEA1 AA — 1> a n d ^A a symmetric set contained within a single fibre A G A. Then, M *'=o AeA[ 170 Chapter II. Discrete symmetrization by Fubini's theorem. Moreover, also by Fubini's theorem we have M £ £ M < i . Renumbering the index set, the proof is complete. • Proposition 11.2. Assume that (11.6a) and (11.6b) hold for some pair of real functions u and v on X and Y respectively, with u € w(X). Assume that ou#_v attains a strictly positive maximum on M. at a function F € M.. Then supp F C fi*. Proof. Suppose that suppF £ Write a = ou#_v to reduce clutter. Let G = 1^ # • F. The function G is symmetric by Corollary 8.5. I claim that a(G) > o(F). (11.7) Suppose that this claim is just. Then, because supp F <Z fi# we have J2Y G < J2Y F- Thus, there exists A > 1 such that y j y AG < yjy F < 1 so that AG € Ai. But, cr(AG) = Aa(G) > Xo(F) > o(F), since <r(F) > 0, and thus we have a contradiction to the assumption that o attains a maximum at F. Hence, it suffices to prove (11.7). Now, by Remark 1.2.2 and Fubini's theorem we will have o(F)-o(G)= £ (u#(y)-v(y))F(y) yeY\n# fOO £ {u*(y) - v(y))lFx(y) dX (1Lg) y€Y\n# oo £ (u*(y)-v(y))dX. yeFx\n# Since F is symmetric, the set Fx is symmetric. Thus, it suffices to prove that for all finite symmetric sets V we have £ u*(y)< £ v(y), (11.9) y€V\n# y£V\n# 171 Chapter II. Discrete symmetrization since then (11.7) will follow immediately from (11.8). Since the A * for A € A form a disjoint cover of Y and V fl A * is symmetric for each symmetric V (Proposition 8.1), it suffices to prove (11.9) for V a subset of A * for some fibre AeA. Use the "moreover" in the Proposition with S = A fl fi, which is acceptable by (11.6a). Let g\ = ly- and g2 = lynp.#- Then, g\ is the indicator function of some set V, and g2 is the indicator function of some set W. By the "moreover", we have W C S. Clearly, also, W CV. Since, Yu(x) = £ w # ( y ) f f i ( y ) xev yeY and Y «(*) = Y u*(y)92(y), xew yeY it follows that Y u(x) = Yu*(y)(9i(y)-92(y))= Y u*(y)- (n-10) xev\w yeY yev\n# If we could prove that V\W = V\Q, then (11.9) would follow from (11.10) together with (11.6a) and the fact that V* — V since gf = g\. Hence, we must show that V\W = V\Q. Now, W C fi and V C A by Proposition 8.6(iii). We shall show that either V = WorW = An£l. In either case, it follows that V\W = V\Q. Suppose first that V ^ W. Then, g\ and g2 do not coincide, and hence V is not a subset of fi*. But V is symmetric and contained within the symmetrization a single fibre A * , and # is of Schwarz type on the fibre A , so that it follows that in fact fi* n A * is a proper subset of V, since of two Schwarz symmetric sets, one must always be contained in the other (this follows from Proposition 8.1). Thus, g2 = ln#nA#. Moreover, since V is finite, so is fi* f l A. Then, II^ I = | f i # n A # | and W is contained in 5 = finA. Since |5| = | f i * n A # | as f i#nA* = ( f i n A ) * (Proposition 8.2), it follows that in fact W = fi fl A, as desired. • Proposition 11.3. Make all the assumptions o /§ l l . l , except possibly for (11.6a) and (11.6b). Assume that cru#_v attains a strictly positive maximum over Ai at a function F with suppF C fi*. Then it also attains its maximum over Ai at L*F. 172 Chapter II. Discrete symmetrization Assume this proposit ion for now. Proof of Theorem 11.1. Wr i te a = cru#_v to reduce clutter. The relation u < v is equivalent to the relation o~(F) < 0 for al l F 6 M. To obtain a contradict ion, suppose that cr(F) > 0 for some F e M. For S a finite non-empty subset of Y, let fs — | 5 | _ 1 l s . P u t A / " = f { / 5 :3AeA.SCA*,S symmetr ic , 0 < \S\ < oo}. I c la im that supFeM a = s u p F e J v - a. To see this, f ix F 6 M. B y L e m m a 11.2, we then have i with the ai posit ive constants whose sum does not exceed 1 and wi th fs{ € Af. Then , a ( F ) = J > < 7 ( / 5 l ) i by Fubin i 's theorem. It is clear that if F is such that (r(F) > 0 then there exists i so that °~{fSi) > a(T)- Hence s u p ^ g ^ a < supp^a since we have assumed that s u p F G A / ( a > 0. Bu t J\f C M so that o~ has the same supremum over Af as it does over M. I c laim that it follows that a at tains a max imum over Af and hence over A4. For , let Sn be a sequence of sets such that | 5 r i | _ 1 l 5 r i G N and \Sn\~1cr(lsn) —> supM a. If the collection {AeA: 3n. Sn C A# } is finite then by passing to a subsequence we may assume that there exists AeA such that Sn C A for all n. In that case, since Sn is a symmetr ic subset of the rearrangement of a single fibre, by the Schwarz property of the rearrangement restricted to subsets of fibres, it follows that Sn = Sm if and only if | ,S n | = \Sm\. If \Sn\ takes on only f initely many values, by passing to a further subsequence we may then assume that all the | 5 „ | , and hence all the Sn, are equal, and it follows immediately that a attains its max imum over A4 (in fact at | I — 1 1 ) - Hence assume that \Sn\ takes on infinitely many values. Then , passing to a subsequence if necessary, we may assume that \Sn\ is a str ict ly increasing sequence 173 Chapter II. Discrete symmetrization tending to oo. Since the Sn are nested (they are symmetric subsets of a rearrangement of a fibre), it follows from the assumption that u G w(X) (which implies that u* G w(Y) by equimeasurability) that we must have limsup^^ \Sn\~1 ou# _v(lsn) < 0, so that sup^ o < 0, which we had assumed not to be the case, and so we have a contradiction. Suppose then that the collection {A G A : 3n. Sn C A*} is infinite. Then, passing to a subsequence, since the A* are disjoint, it follows that we may assume that the Sn are disjoint. Since sup^ o > 0, there exists e > 0 and TV G Z + such that for n > N we will have | 5 n | - 1 ( T ( l 5 j > e. Define Un = W e t h e n h a v e N+n N+n °(lun) = a ^ n ) > E £\Si\ = \Un\e. (11.11) i=N i=N It is clear that the Un are nested and satisfy |C/^ ,j —>• oo. Hence lim sup |C / n | _ 1 «T ( l r j „ ) < 0, which immediately contradicts (11.11). We have thus proved that indeed the maximum of a is attained over M. By our assumptions, this is a strictly positive maximum. Suppose it is attained at Fo G M.. By Proposition 11.2, the support of Fo lies inside fi*. Define Fn = | (F n _i + L*F n_i). Suppose that F„_i G M, that the support of F n _ i is contained in fi* and that a attains its maximum over Ai at Fn-\. By Assumption 11.3 the function L*F„_i is symmetric and by Assumption 11.1 it follows easily that y j y L*F n _i < YJY Fn-i < 1 since F n _ ! G Ai. Now | (F n _i + Z/*F„_i) is symmetric by Corollary 8.5. It is then clear that F n G At. We then have cr(F„) = \{o(Fn-i) + cr(Z,*F„_i)). By Proposition 11.3 we have cr(L*F„_i) = <7(F„_i) so that o(Fn) = <r(Fn_i). Since o attains its maximum at F„_i it follows that it must also attain this maximum at Fn. We have thus inductively proved that the Fn are a sequence of positive functions whose supports lie in fi*. Since these supports are non-empty (else our maximum of o would be negative), in light of the choice of F n we have contradicted Assumption 11.4. • 174 Chapter II. Discrete symmetrization Our proof of Proposition 11.3 is based on the methods of Baernstein [11]. The methods use a trick of Weitsman [99] to handle the 4>(u) — c • u terms. Proof of Proposition 11.3. Write a — cru#_v. I claim that it suffices to prove Proposition 11.3 for the special case of a function F of the form F = fs = \S\-1ls, where S is a finite nonempty subset of A* for some fibre A. For suppose we have done this, and that F is a general function. Apply Lemma 11.2 to write i with the ai positive, Yli ai 1 a r , d Si a finite symmetric subset of Af for A; a fibre. Then, by Fubini's theorem, a(F) = YdaicUsi). i Since o~(F) > 0 and ^ a8- < 1, and, finally, c(/st) < o~(F) (since a attains a maximum at F), it follows that <r(/sj = o~(/) for all F and that V_\ a,- = 1. Applying the result for the special case of the fst, we will see that a(L*fSt) = a(fSt) (11.12) whenever ai ^ 0. (Of course s u p p C suppF C Q# in such a case.) Then, L*F = J2«iL*fs, i by Fubini's theorem and it follows from (11.12) by another application of Fubini's theorem that a(L*F) = o-(F) as desired. Hence, it suffices to prove our result in the special case of F = fs, for a finite nonempty symmetric S C A * , where A £ A is a fibre. Let y0, J/i, • • • be the enumeration of A* induced 175 Chapter II. Discrete symmetrization by Proposition 8.1, since # is a Schwarz type rearrangement from 2A to 2A*. Then, S = {y t : z < N}, where iV = \S\ < oo. Since ti £ wpO it follows that «>, is finite for all A > 0. Use Proposition 8.6 to construct the function F from the function F such that F# = F, (F • = F • tio^=n* £ F ^ M * ) = £ F(y)u*(y) (11.13) and suppF C (£2 fl A), where A is the fibre such that suppF C A*. (Here we use the fact that supp F C fi# together with (11.6a). We write where Proposition 8.6 has ud>" and put S = 9, n A in the "moreover".) I claim that we then have -J2mE(u*-v)(y) yeY < - £ F ( x ) D U ( a O + £ ] %)£<%) (11.14a) xex yeY <^F(x){4>(u{x))-c(x)u{x) + \(x)) (11.14b) xex - F(y)(<f>(v(x)) - d(y)v(y) + p(y)) yeY < Y,ny)[4>{u*(y)) + (-d)(y)u*(y) (11.14c) yeY + n(y) - <f>(v(x)) + d(y)v(y) - n(y)] < F(ymu*(y)) - d(y)u#(y) (11.14d) yeY - [<h(v(y)) - d(y)v(y)]]. 176 Chapter II. Discrete symmetrization We now just i fy this chain of inequalit ies. F i rs t , note that £ F(x)Du{x) = F{x)Ku{x) - ]T F{x)u(x) xex xex xex <Y,F*(y)Lu*(y)-YF(x)u(x) y£Y x£X = J2ny)Lu*(y)-J2F(y)u*(y) yeY yeY = Y,F{y)Eu*{y). yeY The first equali ty followed by definit ion of D; the subsequent inequali ty followed from Assump-t ion 11.2; the subsequent equali ty was given by (11.13); the final equality followed from the definit ion of E. Hence, we see that the inequali ty in line (11.14a) is val id. L ine (11.14b) follows f rom (11.2a) and (11.2b) since s u p p F C and s u p p F C We now just i fy (11.14c) in three steps. F i r s t note that ^P(x)X(x)<^F*{y).X*(x) xex yeY = £ F ( y ) A # ( y ) ( 1 1 . 1 5 ) yeY <J^F(y)p(y), y^Y where the first inequali ty followed f rom the Hardy-L i t t lewood inequali ty (Theorem 1.2.3), the middle equali ty followed f rom the fact that F # = F, and the f inal inequali ty followed from the symmetry and posi t iv i ty of F together wi th condit ion (11.5). Now note that £ F(x)u(x)(-c(x)) < J2(F • u)*(y)(-c)*(y) xex yeY <J2(F-uf(y)(-d)(y) ( 1 L 1 6 ) yeY = Y,F(y)u*(y)(-d)(y). yeY 177 Chapter II. Discrete symmetrization Here, the first inequali ty followed f rom the Hardy-L i t t lewood inequali ty (Theorem 1.2.3). The second came from (11.4c) and the symmetry of (F • u)#. The final equality came from the relation (F • u)* = F -u*. Fina l ly , note that by the definit ion of F and by the relation w* = u o tp, we have J2 F(x)<f>(u(x)) = F{y)4>(u(^y)) (11.17) = YJF{y)<t>{u*{y)). y£Y Pu t t i ng (11.15) and (11.16) together, we obtain (11.14c). Tr iv ia l manipulat ion then yields (11.14d). Let R be the right hand side of (11.14d). We shall show that R < 0. For , F = N - 1 ! - ^ , . . , ^ - ! } so that N-l N-R= £ [ - % ; ) ( u # ( y t ) - v(yt)) + <t>{u*{yi)) - </>(v(yi))] i=0 N-l i=0 where <j>' is a one-sided derivative of our convex funct ion cp (the choice of side is irrelevant). Let tp(i) — —d(yi) + c/>'(u#(y4)). Since — d and u# are symmetr ic , —d{yi) and « * ( y j ) must be decreasing as i increases (Proposi t ion 8.4). Now d> is convex so <p' is increasing, and it follows that ^ is decreasing in i. For conciseness, let a(i) = w*(y;) — u(y 2) and put N-l W{n)= £«(*), i=n with W(N) — 0. B y summat ion by parts, we then have N-l N R<Y * ( 0 « ( 0 i=0 N-l = Y *(*)(w(*) - +1)) i=0 N - l = * (0 )VF(0 ) - - l)FT(iV) + Y (*(*) - *(* - !) W O i = l J V - l = « ( 0 ) W ( 0 ) + Y (*(*) - * ' (* - i ) ) ^ ^ ) -i'=l 178 Chapter II. Discrete symmetrization Now, \IJ(i) ——1) < 0 since $ is decreasing. Moreover, $(0) < 0 since <p is increasing. We will then be able to conclude that R < 0 as soon as we prove that W(i) > 0 for i £ {0,1,..., N — 1}. Of course W(0) = E^\u*(yi)-v{yi)) = Nv(F) > 0. Let Fn = n-H^^y. Then, Fn £ M. Thus, for n = 1,..., N — 1 we have a{Fn) < a(F). Note that FN = / . But <r(Fn) = n.^™^1 "(0- Thus, for n < N we have n - l n-1 iV-1 j=0 i=0 i=0 since VJi^o1 ~ Ncr(F) > 0. It follows from the above displayed equation that N-i W{i) = £ a[i) > 0 i=n for n = 1,..., N — 1, as desired. Hence we have indeed proved that R < 0. Thus, by (11.14a)-(11.14d) we have - £ F ( y ) £ ( u # - i ; ) ( y ) < 0 . yeY Now, = L — 1, so that £ F(y)(u* - v)(y) < £ F(y)L(u* - v)(y) = £ L*F(y)(u* - v)(y). yeY yeY yeY Hence a{F) < a(L*F), as desired. • 11.3. Applications 11.3.1. Monotonicity of the system Let X = Y and K — L. Let 5* = S for every subset S of X. This a Steiner type symmetrization with fibres {{x} : x £ X}. Every set and every function is symmetric. The condition / < g means simply that / < g everywhere. Assumptions 11.2 and 11.3 are trivial in this case. The content of Theorem 11.1 in this case is as follows. 179 Chapter II. Discrete symmetrization Corollary 11.1. Let fi be a subset of X. Suppose that u and v are positive functions on X satisfying —Du < 4>{u) — c • u + A, on fi and —Dv > 4>{v) — c • v + A, on fi, where cp is a real function on [0, oo) such that cp is convex and increasing on [0, oo), c is any positive function on fi and A is any real function on fi. Moreover, assume that inf u > sup u n x\n and that sup u < inf v. Finally assume that fi satisfies Assumption 11A (note that of course fi* = fi for any fi C X). Assume Assumption 11.1. Suppose that u £ w(X). Then u < v everywhere on X. 11.3.2. Generalized harmonic measures We show that Theorem 9.1 can also be seen as a consequence of Theorem 11.1. To see this, note that by the work of §9.2, we may assume that K is a kernel satisfying Assumpt ion 9.3 wi th A = 0. Let L = K. Then Assumpt ions 11.1-11.3 are satisfied. We first perform a simple approximat ion argument that allows us to assume that s has finite support . Let T be the cr-field generated by { i ? n } ^ L 0 . Adop t the notat ion of §9.1. Consider the event n Ms) = \J{Xm < s{Rn)}. 771=0 Wri te p = in f {n > 0 : Rn £ {k} x X}. 180 Chapter II. Discrete symmetrization Then, u(z, S; s) = EZ[P(p < oo and AP \ T and RP £ S)]. Now, conditioned on T and p < oo we have A P involving only finitely many of the numbers s(v). Now, let Vjv be an increasing sequence of finite subsets of V whose union is all of V. Define SN(X) = s(x) • lyN(x) and SN — S D VN- Then, we have P{p < oo and A p ( S J V ) and RP 6 SN \ T) f P(p < oo and Ap(s) and R P 6 S | T) with probability 1 as N —> oo, since almost surely P(p < oo and Ap(s) and RN \ depends on at most finitely many of the s(y). Hence, LO(Z,S;SN) t w(z,S;s). Let uN(z) = u(z,S;sN). Clearly UN has at most finite support, since if SN{Z) = 0 then UN(Z) = 0 while SN has finite support. Define T = {TO e Z : 3x £ X. P(m'x\p< oo) = 0}. Note that in fact T = {TO £ Z : Vz G X. p(m'x)(p < oo) = 0}, by Assumption 9.2. Note that if TO G T and a; G X then UN(m, x) = 0. Let fijv- = (suppSAr)\[({A;}ur) x X]. Let u(z) = u(z, 5*; s*). Write A = D = £' = /,C — 1. Then, for any function / on V we have Af(z) = Ez[f(R1)]-f(z), 181 Chapter II. Discrete symmetrization since K precisely gives the transition probabilities of the random walk {RN}. Fix 2 ? Q . Then, uN(z) = sN(z)Ez[uN(R1)]. For, UN(Z) (if z fi {k} xX) is the probability that we will survive while at point z and that from the next step we will survive until arrival at S; this latter probability is precisely EZ[UN(RI)]. Hence, uN(z) - sN(z)Ez[uN(R1)}. Thus, AuN(z) = Ez[uN(Ri)] - uN(z) = (sN(z))~1uN(z) - uN{z) - [(sN(z))~1 - l]uN(z). Let cpj(z) = ( S J V ( Z ) ) - 1 — 1- (This makes sense since if z € fi then SN(Z) > 0.) Then, —AUN = —CNUN on fi. Set CAT = +00 outside fi^v. Now, if sjy is strictly positive on fi, then is strictly positive on fi]y (this follows from the fact that rearrangements preserve inclusions, while (supp /)* = supp(/*) for any positive function / since for a positive function / we have supp f = f0 and can apply Theorem 1.2.1). Then, —Au = — d • v, on fi/v, where d(z) = (s#(z)) - 1 — 1. Set d(z) = +00 for every z such that s#(z) = 0. I now claim that —CAT <l —d. To see this, write c(z) = (s(z))~1 — 1 for every z such that s(z) > 0 and c(z) = 00 if s(z) = 0. Then, —c > — CN everywhere on V since s > SN everywhere. Hence, it suffices to show that —c <i —d. But in fact we have (—c)# = — d by Theorem 1.2.2 since —c = (f>os and — d = cp o s#, where ct> is the monotone function defined by cp{t) = I —00, if t < 0, 1 - i - 1 , if t > 0. Hence, the relation —c< —d follows trivially. 182 Chapter II. Discrete symmetrization Now, u is a positive funct ion. Let A be a fibre of V. Then , there is some TO G Z such that A C {TO} X X. (This is because each fibre of V is of the form of a cartesian product of a singleton {m} wi th a fibre of X.) If TO = k or TO G T then A does not meet Q^. Otherwise, A f l QM = A f l supp SAT, and (11.6a) (with UN in place of u and in place of f2) follows by posi t iv i ty of upi and the fact that UN vanishes wherever SN vanishes. To verify (11.6b) (with UN in place of u and fi^ in place of 0 ) , it suffices for us to verify it in the case of U being a subset of some fibre A of V. There are two cases. E i ther A and (T U {k}) x X are disjoint, or A is a subset of (T U {k}) x X. Consider first the former case. Then , U\QN lies outside the support of SN and hence UN vanishes there and (11.6b) is t r iv ia l as v is posit ive. Consider now the case when A is a subset of ( T U {k}) x X. Then , since k fi T it follows that A is either a subset of T X X or of {k} x X. In the former case, upi and v both vanish identical ly on T and we are done. Suppose then that A C {k} X X. Now, on {k} X X, we have UN equal to the indicator funct ion of SN and v equal to the indicator funct ion of S*. Now, then, on A , the funct ion is smaller than the indicator funct ion of S. Hence, (11.6b) is a consequence of the Hardy-L i t t lewood inequali ty (Theorem 1.2.3) applied wi th / = I 5 and g = u, restricted to the fibre A. Note that u has finite support (since s does), so that u G w(V). O f course Assumpt ion 11.4 follows from the choice of T and the finiteness of Q]y. Hence, the assumptions of Theorem 11.1 are satisfied. Hence, < V . Let t ing —> 00, we conclude that u < v (use L e m m a 1.2.1). This easily implies the conclusion of Theorem 9.1. For we can use condit ion (9.3) to prove that u •< v. Then , all that remains to be proved is the Steiner symmetry of v. Bu t this follows by noting that v < v by a second appl icat ion of the above work, and then using L e m m a 11.1. The crucial fact in the above work was the formula -Au(z, 5; s) = -[(s(z))-1 - l]u(z, S;s), valid outside on {z : s(z) > 0} and it is essentially this formula which together wi th Theo-rem 11.1 implies Theorem 9.1. 183 Chapter II. Discrete symmetrization Remark 11.3. We can use a similar technique for proving Theorem 9.2. There, the necessary formula is Agsu(z) = lu-[(s(z)r1-l}gsu(z), on {z : s(z) > 0}, where gfj{z) = Y^weU dizi w'is) f° r U C V. The above equation is also of a type that can be handled by Theorem 11.1. Since we have already given one rigorous proof of Theorem 9.2 and have shown how to use Theorem 11.1 to prove Theorem 9.2, we leave it to the interested reader to work out the details of a proof of Theorem 9.2 via Theorem 11.1. 11.3.3. Exit times By Haliste's method, we were also able to prove a result about exit times, namely Theorem 9.4. It is natural to ask whether that result can also be proved to be a consequence of Theorem 11.1. The answer is positive. Our method here is actually very simple: since we have already shown that Theorem 11.1 implies Theorem 9.1, we need only prove that Theorem 9.1 implies The-orem 9.4. To do this, proceed as follows. Assume that the hypotheses of Theorem 9.4 are verified. Let 53 = Z x V. Let # be the Z-product symmetrization on QJ obtained from the Steiner symmetrization # on V. Given the kernel K on V, let £((m,6), (m>') = 5m+limlK(v, v'). Then, the hypotheses of Theorem 9.1 are satisfied if we put .£ and 03 in the place of K and V, respectively. Let © = {N} x V. For m £ Z and v € V let s(m, v) = s(v), where s was our survival function on V. Then, it is not hard to see that u*({0,z),&;s) = Pz(rs > AT), (11.18) for z € V, where is generalized harmonic measure on QJ defined with respect to the kernel Let J = {i} X I. Then, by Theorem 9.1 we see that 5 > * ( ( 0 , j ) , © ; s ) < E>*((0,j),e#;s#). 184 Chapter II. Discrete symmetrization Now, @# = (5 by definition of the Z-product rearrangement on QJ, since V# = V. Moreover, s * ( m , v) = s*(v) as is easily seen, and 3# = {i} x 7*. Hence, by (11.18) (and its analogue for -^rearranged sets and functions) we see that £ pi(Ts>N)< £ P>(rs>N). )e{i}xT# But this is precisely the conclusion of Theorem 9.4, and so we see that Theorem 9.1 indeed implies Theorem 9.4. 185 Chapter I I I Chang-Marshall inequality, harmonic majorant functionals, and some nonlinear functionals on Dirichlet spaces Overview The purpose of the present chapter is to study the A $ and T$ functionals. We shall begin by defining the non-l inear functionals A $ act ing on functions on an arbi t rary finite measure space (§1.1). Then , in §1.2, we shall define various Dir ichlet spaces on which our functionals are to act; we shal l also review some very basic results about these Dir ichlet spaces. In §1.3, we shall define the r$ funct ionals and describe the connection between them act ing on domains of area at most VT and the A $ act ing on functions f rom the unit bal l of the holomorphic Dir ichlet space. In §2 we give the impor tant Chang-Marsha l l [32] inequality which in fact started our whole invest igat ion. We also give Essen's improvement [44] of the Chang-Marsha l l inequality, and state the Moser-Trudinger [78] inequali ty which motivated the Chang-Marsha l l inequality. Then , we prove that in a strong sense the Moser-Trudinger and Chang-Marsha l l (and a fortiori Essen) inequalit ies are unimprovable (Theorem 2.1). Section 3 wi l l be pr imar i ly devoted to the study of the A $ functionals on functions on an ar-bi t rary finite measure space. In §3.1 we shall give some results on the existence of extremal functions for A $ functionals on balls of Dir ichlet spaces by giving an important upper semi-cont inui ty result (Theorem 3.2) f rom the author 's joint paper wi th Matheson [75]. Th is result improves on a theorem of Matheson [73]. In §3.2 we examine the A $ functionals acting on 186 Chapter III, Functionals on a set of domains and on Dirichlet spaces the unit balls of Hi lber t spaces of measurable funct ions. We wi l l be able to obtain some gen-eral results which wi l l al low us to give a proof of C i m a and Matheson 's theorem [35] on the weak cont inui ty of the Chang-Marsha l l funct ional on the punctured unit bal l of the Dir ichlet space. O u r results wi l l also be useful for proving weak continuity results in the case of the Moser-Trudinger funct ional . Then , in section 3.3 we consider the general notion of a "cr i t ical ly sharp inequal i ty" . M u c h of the mater ia l in §3.3 comes f rom the author 's paper [88]. Par t icu lar examples of cr i t ical ly sharp inequalit ies wi l l be the Chang-Marsha l l and Moser-Trudinger inequalit ies. We shall prove that given a cr i t ical ly sharp inequal i ty for A $ we may perturb $ in such wise as to, depending on our wishes, either gain (Theorem 3.9) or destroy (Theorem 3.7) the existence of a function at which the max ima l value in the inequali ty is at ta ined. Our results are new even in the cases of the Chang-Marsha l l inequal i ty and the Moser-Trudinger inequality, al though in the former case the result was strongly suspected by C i m a and Matheson (personal communicat ion) and in the latter it was conjectured by M c L e o d and Peletier [77]. We also obtain a part ia l converse (Theorem 3.8) to the upper semicontinuity result of Theorem 3.2. In §4 we come back to the specific case of the A $ functionals on balls of Dir ichlet spaces. In §4.1 we shal l give a var iat ional equation for the extremals of our funct ionals. Th is is due to the author, refining a part ia l result of Andreev and Matheson [5], and is taken from a joint paper wi th Matheson [75]. In §4.2 we give a joint result of the author wi th Matheson [75] which shows that under appropriate condit ions on $ the extremals of the A $ functionals automatical ly satisfy some regularity condit ions. In §4.3 we give some important assumptions on the functions 4> to which our methods are appl icable. Then , in §4.4 we give some useful extensions of the results of sections 4.1 and 4.2. We shall use a variat ional equation from §4.4 later in §IV.10. We have noted that some of the results of the present chapter come from the paper of Matheson and Pruss [75], and the proofs are sometimes taken from that paper. When this is done, the proof and the result can be assumed to have been due to the author, unless otherwise noted (as in the case of Theorem 4.2). 187 Chapter III. Functionals on a set of domains and on Dirichlet spaces In §5 we return to symmetr izat ion theory, and study the symmetr ic decreasing rearrangement act ing on boundary values of real parts of functions from the Dir ichlet spaces. We prove a str ict symmetr izat ion theorem (Theorem 5.1) and use it to prove that if $(z) = <f>(Rez) for an appropriate funct ion <j>, then the extremal functions of A $ are univalent wi th Steiner symmetr ic image. F ina l ly , in §6 we discuss a rearrangement method due Baernstein [12]. Our main result is that the symmetr ized domain the method produces is always a subset of the Steiner symmetr izat ion of the original domain (Theorem 6.1). Hence, the method does not increase areas of domains, and this fact allows us to prove the existence of extremals for T$ where $(z) = rf>(Rez) for appropriate <f> by using the rearrangement to reduce the problem to the A $ funct ional , and then using the results of §3.1. O n first reading, the reader may wish to omit the proofs of the results in §3.2 and the material of §3.3.4. The proof of Theorem 6.1 can also be omit ted. 1. The A$ and T$ functionals and Dirichlet spaces 1.1. The A $ functionals on a finite measure space Let (I, p) be a finite measure space. Let B be a set of complex-valued or real-valued functions on I and let $ be a Bore l measurable funct ion f rom C or E , respectively, to E . Given / G B, let In reference to the A $ funct ionals, "bounded" shall mean "bounded above" . D e f i n i t i o n 1 .1 . The funct ional A $ on B is said to be b o u n d e d providing sup A $ ( / ) < oo. D e f i n i t i o n 1.2. A funct ion / is said to be extremal for A $ on B if / G B and A « ( / ) > A * ($) 188 Chapter III. Functionals on a set of domains and on Dirichlet spaces for all g e B. B y an abuse of notat ion, if <p is defined only the half-line [0,oo) then we define A 0 ( / ) = A * ( / ) , where <&(z) = <£(|.z|). 1.2 . D i r i c h l e t s p a c e s For a complex funct ion / on the unit circle T , write f(n) for its n th Fourier coefficient. For a real / on T , write cn(f) and sn(f) for its n th Fourier cosine and sine coefficients, respectively. We shall often identify a funct ion / on the (open) unit disc D wi th its nontangential boundary values on T . D e f i n i t i o n 1.3. For 0 < a < oo, let the ce-weighted h o l o m o r p h i c D i r i c h l e t s p a c e X>a be the Hi lber t space of al l functions / holomorphic on the unit disc D wi th / (0 ) = 0 and CO ii/iik=E i^/wi2<~-71 = 1 D e f i n i t i o n 1.4. Let the a - w e i g h t e d r e a l h a r m o n i c D i r i c h l e t s p a c e o a be the real Hi lbert space of al l real functions / harmonic on the unit disc D wi th / (0 ) = 0 and CO n/iiL = En a(^(/)+-»(/))< °°-71=1 Remark 1.1. It is clear that £ ) a has the inner product CO ( / , 5 ) = £ n « ( / ( n ) , ^ ) ) , 71=1 whereas u Q has the inner product CO (f,9) = M / ) C n ( < / ) + * „ ( / ) * „ ( < / ) ) • 71 = 1 Remark 1.2. The map / H-» Re / is an isometry sending X>a onto Ha. To see this, it suffices to note that c n ( R e / ) = R e / ( n ) and s n ( R e / ) = — I m / ( n ) . 189 Chapter III. Functionals on a set of domains and on Dirichlet spaces D e f i n i t i o n 1.5. The spaces D i and are known as the h o l o m o r p h i c and r e a l h a r m o n i c . . . . . def def D i r i c h l e t s p a c e s , respectively. We wi l l wri te 13 = 1)1 and D = 0i for short. It is easy to verify by expanding / in a series that = - ff \f'(x + iy)\2dxdy, * J J B |2 Is and 11/112 = " ff \Vf(x,y)\2dxdy. K JJ D Since for a holomorphic funct ion / the quant i ty \f'(x + iy)\2 is the Jacobian of the mapping / , it follows that 7r||/||ijj is the area of the image of D under / counting multiplicities, and if / is univalent (i.e., one-to-one) then 7r||/|||, is precisely equal to the area of the image of / . It is in the space 1) that our greatest interest lies. Technical remark 1.1. To place © in a larger picture, it should be noted that D contains un-bounded functions (any univalent map to an unbounded region of finite area wi l l be in D ) , and that not every funct ion f rom the disc algebra lies in X). Indeed, let CO / ( * ) = ^ 2 - m / V " . m=0 Evident ly , this series converges uniformly on T and thus / lies in the disc algebra. However, CO oo • £ n | / > ) | 2 = £ 2 ™ 2 - m = oo, n=l m=0 and so / ^ X). O n the other hand, X> C B M O A . One of the easier ways to prove this is to note that f rom the definit ion of the norm on Dir ichlet space it easily follows that if / € X> then oo / m — 1 \ s u p \f(mn + J) I <'oo. m > l 1 \ • , I - n=l \j=l j which is C . Fefferman's sufficient (and, if we have f(k) > 0 for al l k > 0, then also necessary) condit ion for a holomorphic funct ion / on D to be a member of B M O A [24, 97]. D e f i n i t i o n 1.6. Let 25, b, 2$ a and ba be the unit balls of D , o, D a and o a , respectively. 190 Chapter III. Functionals on a set of domains and on Dirichlet spaces These unit balls will be the ranges of the A $ functionals in which we will be interested. The following result will also be useful. The most useful case is where a = 0 and (3 = 1 and that case was already proved by Andreev and Matheson [5]. T h e o r e m 1.1. The inclusions XJ/3 C D a and D/3 C T)a are compact for 0 < a < (3 < 00. In particular, if fn is a weakly null sequence in T>a or oa for a £ (0, 00), then / „ —> 0 in L2(T) and hence fn tends to 0 in measure on T. ll/ll®o — II/IIL2(T) f° r a function / with mean 0 on T, The following lemma is doubtless well-known but is easier to prove than to attribute. L e m m a 1.1. Fix 1 < p < 0 0 . Consider the space £p(w), where w is a non-negative weight with wn —> 00 as n —y 0 0 . Let ak be a norm-bounded sequence in £p(w), converging pointwise to 0. Then ak —> 0 in P-norm. Assume this for now. Proof of Theorem 1.1. It suffices to prove that if fk -> 0 weakly in then fk —> 0 in T)a norm. The assertion for fk—>f weakly where / ^ 0 then follows by considering instead the sequence fn — f 1 while the assertion for the harmonic Dirichlet spaces follows from Remark 1.2. Now, let p = 2, and put (ak)n = nal2fk(n). Then, I I K ) l l ^ = I I M b Q . (1.1) Let w(n) = n^la. Note that w(n) —> 00 as n —>• 0 0 . Moreover, l l ( a * ) l l * ( u , ) = ll/fcllsv Hence, ak satisfies the conditions of Lemma 1.1, and thus ak —> 0 in I2 norm. In light of (1.1), we conclude that fk —> 0 in u a norm, as desired. The assertion that ||/||i,2(x) —>• 0 follows since ll/lloo = II/IIL2(T) f° r a function / with mean 0 on T. • 191 Chapter III. Functionals on a set of domains and on Dirichlet spaces Proof of Lemma 1.1. W i thou t loss of generality assume that ||afc||^P(w) < 1 for al l k. F i x e > 0. Choose N sufficiently large that wn > 1/e for n > N. Then , oo oo J2 Kafc)"lP ^ £ E wn\Mn\p < (e/2)\\ak\\ n=N n=N O n the other hand, for fixed N N-l iK)«r^° as k —> oo. It follows that l i m s u p ^ o Q ||OA;||#> < £• Since e > 0 was arbi trary, the proof is 1.3. The 1"$ functionals acting on domains and the A$ acting on holomorphic and harmonic functions We now establish the convention that when A $ acts on a holomorphic or harmonic function / on D, then A $ ( / ) is defined in terms of the normalized Lebesgue measure on T , i.e., where f(et$) as usual is short for n.t. l im f{elB). D e f i n i t i o n 1.7. Let B denote the collection of al l domains in the plane which contain the origin and whose area does not exceed ir. Given a domain D £ B and a Bore l measurable funct ion $ on C , let h(-; <h, D) be the solut ion of the Dir ichlet problem on D w i th boundary value o n dD. D e f i n i t i o n 1.8. G iven a Bore l measurable funct ion $ on C and D £ B, let providing the right hand side is defined. If &+\dD £ Ll{u>D) and $ \ d D € Lx(uD) then write r*(D) = oo; if $ + | a £ , 6 Ll{uD) and $ - | 8 £ ) g Ll(uD) then write T*(D) = - o o ; if ^ Ll(uD) and £ Ll{uD) then say that T$(D) is undefined. Say that the complete. • 192 Chapter III. Functionals on a set of domains and on Dirichlet spaces funct ional T$ is b o u n d e d providing it is defined (i.e., the Dir ichlet problem in question has a solution) and sup T$(Z)) < oo. DeB Say that a domain D is e x t r e m a l for T$ providing DeB and U(D)>U(D') for all D' 6 B. B y abuse of notat ion, if cf> is defined on [0,oo) then we write for T$ where * ( * ) = * ( | z | ) . The case that wi l l interest us the most is when $ is a continuous subharmonic funct ion, in which case r * (D) = LHM(0, D) (Theorem 1.5.4). The r $ funct ionals are actual ly closely related to the A$ functionals on 53. F i rs t note that the map / H-> / [B ] sending a funct ion to its image maps 53\{0} into B, since if / £ 53 then /[O] has area at most n (since the area of the image counting multiplicities is at most TT). The fol lowing result is essentially well known (the first part is impl ic i t in , e.g., [44]). T h e o r e m 1.2. Let f be any Nevanlinna class function whose image is contained in a domain D. Let $ be subharmonic on C and continuous at every point of dD. We then have A * ( / ) < r * ( D ) . /// is univalent and D = / [D] , then equality holds. Proof. P u t h(z) = L H M ( z , D). A s in [44], note that 2n ± j ' * * ( / ( r c w ) ) d0<± fj h(f(re«)) dO = h(f(0)) = fc(0), for every 0 < r < 1, since h > $ on D. Tak ing the l imit as r f 1 we conclude that A$(/) < /i(0) (use Fatou 's lemma together wi th the existence of radial l imits of / due to Theorem 1.3.5 and the assumption that / € N). 193 Chapter III. Functionals on a set of domains and on Dirichlet spaces If / is univalent then / is a uniformizer and we may apply Theorem 1.5.7 to obtain the desired equality. (Of course, the domain in this case is Greenian since all s imply connected domains are Greenian, while al l images of univalent functions are simply connected.) • C o r o l l a r y 1 .1 . Fix p £ [ l ,oo) . Let D be a Greenian domain containing the origin and let f: D D be a uniformizer for D. Then D £ Hp if and only if f £ HP(B). Proof. It is clear that if D £ Hp then / £ Hp. Conversely, suppose that / £ Hp. Let g be any holomorphic funct ion on D wi th image in D. Let w = g(0). Let U = —w + D. O f course, / is surjective so that there exists w' £ D such that f(w') — w. Let <j> be a Mob ius transformat ion of D onto itself such that <^>(0) = w'. Then , / o <p is easily seen to lie in Hp if and only if / lies in Hp. Let F(z) = f(4>(z)) — w'. Then , F is a uniformizer for U (since <j> is a conformal automorphism of D) wi th F(0) = 0. We have F £ Hp. Let G(z) = g(z) - w. Then the image of G lies in U and G(0) = 0. Let $ ( z ) = \z\p. We have A*(G) < r*(t/). Bu t by Theorem 1.5.7, we have r«(tf) = A * ( F ) . Bu t A $ ( F ) < oo, so that A$(G) < oo and hence G £ HP(B) so that g £ HP(D) as desired. • 2. The Chang-Marshall, Essen and Moser-Trudinger inequalities The Chang-Marsha l l [32], Essen [44] and Moser-Trudinger [78] inequalit ies are al l closely related. We now state them, using the notat ion of the previous sections of this chapter. E s s e n I n e q u a l i t y ( [44]). Let $(£) = e*2 for t £ [0,oo). Then T$ is bounded on the set B of all domains in the plane containing 0 and having area at most TT. C h a n g - M a r s h a l l I n e q u a l i t y ( [32]). Let $(£) = e*2 for t £ [0,oo). Then A$ is bounded on the closed unit ball 23 of the Dirichlet space. 194 Chapter III. Functionals on a set of domains and on Dirichlet spaces M o s e r - T r u d i n g e r I n e q u a l i t y ( [78]). Let $(£) = e*2. Then A $ is bounded on the set T of 1, where the integral in the definition of A $ is taken with respect to the measure dp(x) = e x dx on [0, oo). In other words, the Chang-Marsha l l inequal i ty states that Note that in l ight of Theorem 1.2 and its preceding remarks, the Chang-Marsha l l inequali ty is actual ly a consequence of the Essen inequal i ty while the Essen inequali ty restricted to U instead of on al l of B is a consequence of the Chang-Marsha l l inequality. The Chang-Marsha l l and Essen inequalit ies are also closely connected wi th the Moser-Trudinger inequal i ty since Marsha l l [72] has found a fair ly easy proof of the Chang-Marsha l l inequali ty using the Moser-Trudinger inequal i ty 1 , while Essen's proof of his inequali ty [44] also uses the Moser-Trudinger inequali ty in an essential way. For more work related to the Moser inequality, see, e.g., [1, 2, 6, 18, 19, 20, 21, 30, 31, 33, 34, 49, 52, 53, 69, 77, 80, 81, 86]. For more work related to the Chang-Marsha l l inequality, see [5, 35]. A l l three inequalit ies (Essen, Chang-Marsha l l and Moser-Trudinger) are unimprovable in that the funct ion <fr(t) = e*2 cannot be replaced by any faster growing funct ion. In the case of the Chang-Marsha l l inequality, the fol lowing result is an improvement of a result of Matheson and Pruss [75, T h m . 1]. T h e o r e m 2 . 1 . Let *f> be any nonnegative Borel measurable function on [0, oo) such that atleast one of the following three conditions is fulfilled: Of course Marshall's proof uses some rather specialized complex variable techniques, and thus may be considered more difficult if one is more comfortable with real variable methods. absolutely continuous real-valued functions f on [0, oo) satisfying / ( 0 ) = 0 and J0°° (f (x))2 dx < while the Moser inequal i ty states that 195 Chapter III. Functionals on a set of domains and on Dirichlet spaces (a) t i—^ (^e*) is convex, increasing and \P is continuous at 0 while T$ is bounded on B (b) A$ is bounded on 23 (c) A$ is bounded on T, in the setting of the Moser-Trudinger inequality. Then there exists a finite constant C such that ty(t) < Ce*2 for all t £ [0, oo). Proof in the case of (c). Suppose that sup^GJr A$ (/) = C < oo. Fix t £ [0,oo). As in [78], let f{x) — tm\n(x/t2,1). It is easy to see that / £ T. Define the measure dp(x) — e~x dx. Then, for x > t2 we have f(x) = t. Thus, roo roo C > A * ( / ) = / V(f(x))e-Xdx > / y(t)e~x dx = y(t)e-t2. Jo it2 Hence, < Ce*2. Since t was arbitrary, we are done. • Case (a) of Theorem 2.1 actually follows from case (b), in the same way as we have indicated that Essen inequality implies the Chang-Marshall inequality. (The stated assumptions guarantee that "I/(|z|) is subharmonic by Theorem 1.4.4.) Before we prove case (b), we need to define the Beurling functions, which also will be useful later, as well as the cut-off Beurling functions. Set Ba(z) = —====, for a £ O\{0} and z £ D, where the branch of the logarithm is chosen so that Ba(a) is positive. The denominator was chosen so that | |5 a||s = 1- The Ba will be called the B e u r l i n g f u n c t i o n s . They are normalized versions of the reproducing kernels for the space V. Note that the Beurling functions are univalent and in fact star-like (i.e., has star-shaped images). To see this fact, note that as is well known a function / holomorphic on D is star-like if and only if Re-TrY > 0 /(*) 196 Chapter III. Functionals on a set of domains and on Dirichlet spaces on D. To verify this identity is a simple elementary exercise. Now, set Ma = yAog yzj^ p• Given M > 0, let o(M) € (0,1) be such that Ma(M) — M. Define the domain DM = Ba(M)p]n{\z\<M}. Since BA^M) is starlike, it follows that DM is star-shaped and hence simply connected. Define the cut-off Beurling function BM as the Riemann map from D onto DM sending 0 to 0 with B'M (0) > 0. For convenience, define Bo = 0. The following Lemma is a variation on a result of Beurling [23, pp. 39-41]. Lemma 2.1. There exists an absolute constant c > 0 such that \{zeY:\BM{z)\ = M}\>ce-M2 for every positive M, where \ • \ indicates normalized Lebesgue measure on T. Assuming this, we can finish the proof of case (b) of Theorem 2.1. Proof of case (b) of Theorem 2.1. Suppose that sup^G(gA$(/) = K < oo. Fix M > 0. We then have K>A*(BM)= [ V(\BM\) JT >V(M)\{ze T:\BM(Z)\ = M}\ >^{M)ce~M\ so that \P(M) < Kc~leM2, which completes the proof in light of the arbitrariness of M. • Proof of Lemma 2.1. Clearly, by possibly adjusting the value of c, we need only consider the case of M strictly positive, et a = a{M). Set U = Ba[D]. Put w = \{z € T : R e £ a > M}\. Then, as in [23], an elementary computation shows that w>ce~M\ (2.1) 197 Chapter III. Functionals on a set of domains and on Dirichlet spaces for some absolute constant c > 0. But w is the harmonic measure at 0 of {z £ T : Re Ba > M} in D. By conformal invariance of harmonic measure under the univalent map Ba, the number w is also equal to the harmonic measure at 0 of {z £ dU : Re z > M} in the domain U. Let wi(z) = u(z,{z e dU :Rez > M};U), and w2(z) = u>(z, {z £ dDa : \z\ = M}; Da). We shall apply the maximum principle to conclude that w2 > wi on Da. For, in order to do this it suffices to verify this inequality on dDa, since both functions are harmonic on Da C U. Fix z £ dDa. Suppose first that \z\ = M. In that case w2(z) = 1 and since w\ nowhere exceeds 1, we are done. Suppose now that \z\ ^ M, so that w2(z) = 0. Then necessarily \z\ < M and z £ dU by definition of Da. Then, Rez < M. Hence, z £ 0*/\{.z £ dU : Rez > M}. Thus, w\(z) = 0 < 0 = ^ ( z ) - We have thus verified that w2 > w\. Hence, w2(0) > wi(0) > ce~M\ (2.2) Applying conformal invariance of harmonic measure under the map B~l together with Re-mark 1.5.5, we see as before that w2(0) = \{z£T:\Ba{z)\ = M}\, which clearly completes the proof in light of (2.2). • 3. General results on the A$ functionals on measure spaces In this section we discuss results on the A$ functionals acting on a collection of functions on a finite measure space. From time to time we shall give applications to the Chang-Marshall or Moser-Trudinger inequality. In all of this section, (I, p.) will be a finite measure space with p(I) > 0 and the functionals A$ will be defined by A*(f) = J$(f)dp, 198 Chapter III. Functionals on a set of domains and on Dirichlet spaces for $ on C and by A * ( / ) = j®{\f\)du, for $ on [0, co) . 3.1. Existence of extremals We are interested in general cr i ter ia for existence of an extremal funct ion. Our method for deriv ing this existence wi l l be to use a compactness argument. F i r s t we remark that the unit balls 25 a and ba of the Dir ichlet spaces are compact wi th respect to sequential convergence in measure on T providing a > 0. In the case of 23 = 23i , this was noted by Andreev and Matheson [5]. The easiest way to see this fact is to note that weak convergence in one of these balls implies L2(T) convergence by Theorem 1.1. However, these balls are al l weakly compact (Banach-Alaoglu) , so that they must be compact wi th respect to sequential convergence in measure on T . Now, if these balls are sequentially compact wi th respect to convergence in measure, then extremals for A $ exist provided that A $ is upper semicontinuous wi th respect to the topology of sequential convergence in measure, i.e., providing that l i m s u p A $ ( / n ) < A $ ( / ) 71—>CO whenever fn —> f in measure on T wi th / and the fn ly ing in the appropriate bal l . The fol lowing result is a direct consequence of Fatou 's lemma. L e m m a 3 . 1 . Let C —y [0,oo) be lower semicontinuous. Then, given a sequence {fn} of measurable functions on I converging in measure to f, we must have l im inf A * ( / n ) > A $ ( / ) . n—tco The fol lowing simple result is then due to Matheson [73]. We say that A $ is upper semicontin-uous along a sequence / „ —» / converging in measure providing l i m s u p A $ ( / n ) < A $ ( / ) . 199 Chapter III. Functionals on a set of domains and on Dirichlet spaces Theorem 3.1 (cf. Matheson [73]). Let /«—>•/ in measure on I, with f almost everywhere finite. Suppose that $: C - r R is upper semicontinuous and C —>• R is lower semicontin-uous with $ < \P everywhere. Then if A$ is upper semicontinuous along a sequence fn—*f converging in measure, so is A$. Proof. The function \P — $ is lower semicontinuous and nonnegative, so that by Lemma 3.1 we have lim inf^oo Aip_$(/„) > A$_$(/), and thus A<p_$ is lower semicontinuous along our sequence. But A$ = A$ — A$_$, and A$ is upper semicontinuous along this sequence, so A$ must be upper semicontinuous along it as well. • The above result shows that under some continuity assumptions on the functions the upper semicontinuity property of the A$ is preserved under pointwise majorization of the functions . • Corollary 3.1. The functionals EM defined for M > 0 by EM(f) = \{zeT:\f(u)\>M}\, where | • | is normalized Lebesgue measure on T, attain their maxima on Q5a for every a > 0. Proof. Let = 1{|z|>M} a n d ^(z) = 1 so that A$ = EM and A$ = 1. Then the conditions of Theorem 3.1 are satisfied along any sequence converging in measure on T, and since 2Ja is sequentially compact with respect to convergence in measure, it follows that if /„ is a maximizing sequence for E M on Q5a (i.e., lim^oo EM(fn) = sup/E<BA EM{f)) then we may, by passing to an appropriate subsequence, assume that it converges in measure on T and then we will have lim sup EM{fn) < £ M ( l i m / n ) , n—¥oo so that the maximum of EM will be attained at lim fn. • It is not known what the extremal functions look like for the EM on It would be of interest to determine this even if only for a = 1. 200 Chapter III. Functionals on a set of domains and on Dirichlet spaces We now state a result which we will use a number of times in the future. Our result is the author's generalization of a result of Matheson [73] and uses a different method of proof from that of Matheson. Recall that $+(z) = max($(z), 0) for all z. T h e o r e m 3.2 ([75]). Let fn —» / in measure on I, with f almost everywhere finite. Let <& be a finite-valued upper semicontinuous function on C, and let ^ be a non-negative Borel-measurable function on C such that is uniformly bounded on {fn} U {/}• Suppose that We shall give a (quite elementary) proof of this result at the end of this section. However, first we wish to go back to Dirichlet spaces and discuss some consequences of this result. Before we do this, we would like to state the following theorem. T h e o r e m 3 .3 . For f £ *Ba where a > 1 we have where ((a) = X ^ L j n 01 is the Riemann zeta function. For a £ [0,1) we have A $ bounded on <8Q, where $(*) = t2^1'0). $ + ( z ) = o(#(z)) uniformly in argz, as \z\ —> oo. Then lim sup. n—foo A * ( / „ ) < A * ( / ) . 11/11 £°°(T) Proof. To prove the first assertion, let a. •n = f(n) and use the Cauchy-Schwarz inequality to note that CO a. •n n=l oo Il/lb. ' a V^Rll/lk, 201 Chapter III. Functionals on a set of domains and on Dirichlet spaces To prove the second assertion, note that Di embeds in BMO (see Technical Remark 1.1) while Do is obviously a (closed) subspace of L 2(T). It follows by interpolation that D a embeds in L 2/( 1 _ a)(T) whenever a £ (0,1). This interpolation can be done in the complex method, using the Stein-Weiss interpolation theorem [22, Chapter 5] to interpolate between Do and Di (since these are essentially weighted f2 spaces) and a theorem of Fefferman and Stein [48, equation (5.1), p. 156]2 to interpolate between L2 and BMO. • Thus, we obtain the following result. The case where a — 1 and 3>(z) = <f>(\z\) for <f> monotone increasing is due to Matheson [73]. Theorem 3.4. Fix a £ (0, oo). Let $ be an upper semicontinuous function on C. If a < 1 then make the following additional assumptions: (i) If a £ (0,1) then $+(z) = o(\z\2^l-a">), uniformly in argz, as \z\ —>• oo. (ii) If a = 1 then 3>+(z) = o(elzl2), uniformly in argz, as \z\ —» oo. Then A $ attains its maximum on 2 J a . The proof of this follows by sequential compactness of <Ba with respect to convergence in measure, combined with an application of Theorem 3.2, where we choose \P as follows: (i) If a £ (0,1) then m{z) = \z\2Kl-a\ (ii) If a = 1 then V(z) = e^2. (iii) If a > 1 then $ is any finite-valued upper semicontinuous function such that 3>+(z) = o($(z)) as t -> oo. (Existence of such a function is easy; e.g., take *B(z) = (\z\ + $+(z))2.) 2Fefferman and Stein [48] work with spaces on 1" and not on T as we do. But, as is well known, the proofs in the case of spaces on T work in exactly the same way. A typographical error on p. 156 of [48] should be noted. The inequality "1 < p < oo" preceding equation (5.1) should instead read "1 < p < oo". The case p = oo is likewise true, but was not proved in [48]; it was only proved later by Jones [64]. In any case we do not use the case p = oo but only the case p = 2. 202 Chapter III. Functionals on a set of domains and on Dirichlet spaces In cases (i) and (iii) the boundedness of A $ on 2 J a follows f rom Theorem 3.3. In case (ii), the boundedness is precisely the Chang-Marsha l l inequality. Theorem 3.4 gives us a large supply of cases in which extremal functions exist. C o r o l l a r y 3.2 ( A n d r e e v a n d M a t h e s o n [5]). / / $ ( £ ) = tp for p G (0, oo) or $(*) = e a * 2 for a G (0,1) , then A $ attains its maximum on 23. However, the functions at which these max ima are attained are generally not known. It has been proved by A lec Matheson (unpublished) that if <&(£) = tp for p G (0,4] and t G [0, oo) then A $ attains its max imum at the identi ty funct ion. It is also known that this is not the case if p > 4; this is essentially due to Sakai [95] (see [75, discussion after Open Prob lem 4] for details.) O p e n P r o b l e m 3 . 1 . Does A $ at ta in its supremum over 23 when <&(£) = e < 2 ? The above problem was raised by Andreev and Matheson [5]. C i m a and Matheson [35] have shown that an aff irmative answer can be given if one restricts the problem to the subset of 23 given by the closed convex of the Beur l ing functions Ba. However, in general the problem remains open. We shall see in §3.3.3 that to answer this problem it does not suffice to use general arguments based on the asymptot ic behaviour of e < 2 but one must use more quanti tat ive arguments which also take into account the nonasymptot ic behaviour of this funct ion. It has been conjectured by Andreev and Matheson [5] that in fact A $ attains its max imum at the identi ty funct ion z. Since A $ ( z ) = e, this conjecture is equivalent to the conjecture that A * ( / ) < e for all / G 23. Th is conjecture has been numerical ly verified by the author for 40 mil l ion pseudorandomly chosen degree 6 polynomials / wi th posit ive coefficients and = 1 (see source code in Append ix B ) . (Note that it does in fact suffice to consider polynomials wi th posit ive coefficients; see Chang and Marsha l l [32, p. 1022] who at t r ibute this observation to J . Clunie.) 203 Chapter III. Functionals on a set of domains and on Dirichlet spaces We may now give the proof of Theorem 3.2. Proof of Theorem 3.2. Dropp ing to a subsequence, we may assume that we have almost ev-erywhere convergence. For , every subsequence of {fn} has an almost everywhere convergent subsubsequence, and it suffices to show that for every subsequence fnk along which A $ ( / n j f c ) converges the l imit is no greater than A $ ( / ) . Now fix e > 0. Let M be a uniform upper bound for A $ on { / n } U { / } . Since $+(z) = o(iS(z)) and ^ is nonnegative, we may choose a number T such that $(z) < (e/ '4M)\P(z) whenever \z\ > T. Fur thermore, we may ensure that our choice of T is such that p{x:\f(x)\ = T} = 0 (3.1) and |^(0)|A*{a: : |/(a:)| > T" - 1} < er/4, since / is almost everywhere f inite. Let U(z) = z if \z\ < T and U(z) = 0 if \z\ > T. Then Uofn —tUof almost everywhere outside the set {x : \f(x)\ — T}. B y (3.1) this latter set has measure zero, so we have convergence almost everywhere. Since $ is upper semicontinuous, we also have l im sup $ o U o fn < $ o U o / , (3.3) almost everywhere. Now the \U o fn\ are uniformly bounded by T, so that the $ o U o fn are uni formly bounded by some constant K independent of n since $ is bounded on the compact set B ( T ) , being upper semicontinuous and finite. Then using the fact that we have a finite measure space, and apply ing Fatou 's lemma to the non-negative funct ions K — $ o U o / „ we find that l imsup A$(C/ o / „ ) < / l im sup $ o f / o / „ < A < j ( [ / o / ) , (3.4) n—^oo Jl n—yco where the last inequali ty follows f rom (3.3). 204 (3.2) Chapter III. Functionals on a set of domains and on Dirichlet spaces Bu t by the definit ion of U we have A « ( / „ ) - A*(U ofn)= [ ( $ ( / „ ) - $(0)) J{x:\f„(x)\>T} < I [ ( £ / 4 M ) * ( / n ) - $ ( 0 ) M { > : | / n ( * ) | > T} \J{x:\fn(x)\>T} I < (e/AM)M - ${0)p{x : \fn(x)\ > T}, where in the last two inequalit ies we have used the choices of T and M, respectively. Further-more, by (3.2) and the fact that fn converges in measure to / , it follows that l im sup \$(0)\p{x : \fn(x)\ > T } < e /4 . n—Yoo Therefore, l i m s u P A $ ( / n ) - A $ ( C / o / n ) < e/2. (3.5) n—>-oo In much the same way we find that | A * ( / ) - A * ( t f o / ) | < e / 2 . (3.6) Then , put t ing (3.4), (3.5) and (3.6) together, we find that l im sup A $ ( / n ) - A $ ( / ) = l im sup ( A $ ( / n ) - A$(U o fn) TI—>oo n—too + A*(tf o / n ) - A * ( £ / o / ) + A * ( E / o / ) - A * ( / ) ) Since e > 0 was arbi t rary, the proof is complete. • 3 . 2 . T h e A $ f u n c t i o n a l s o n b a l l s o f H i l b e r t s p a c e s In this section, assume that H is a (real or complex) separable Hi lbert space of measurable functions / on a finite measure space (/,//). Assume that Hr\L°°(p) is dense in H. Wr i te || • || 205 Chapter III. Functionals on a set o f domains and on Dirichlet spaces for the norm in H and (•, •) for the inner product . Define: £ ( r ) = { / € f f : | | / | | < r } B = B(1) 8B = {feH: ll/H = 1} S(r,R) = {feH :r< ||/|| < R} S ( r ) = S ( r , l ) Note that S(r) is not weakly closed, its weak closure being al l of B. Given a funct ion <J> on [0,oo) or on C and a positive number a , define <&a(i) = <&(at). The fol lowing result is based on an insight at t r ibuted by Chang and Marsha l l [32] to Garnet t in the case of the Chang-Marsha l l inequality. T h e o r e m 3 . 5 . Suppose that $ is a finite increasing function on [0,oo) such that there exists an r > 0 with the property that A $ ( / ) < oo for every f £ B(r). Then A $ ( / ) < oo for every feH. Proof. F i x f e H . Then , since we have assumed that L°° D H is H, we may choose h £ def L°°(p)nH such that s = \\f-h\\ < r. Then , choose M so that \h\ < M < oo almost everywhere on 7, and choose N sufficiently large that rN/s > N + M. Then , A*(f) = J$(\f\)dp < J <f>(\f-h\+M)dp = [ $(\f-h\ + M)dp+ [ <S>{\f-h\ + M)dp J{\f-h\>N} J{\f-h\<N} < [ $(r\f-h\/s)dp+${N + M)p{\f-h\<N} J{\f-h\>N} < j$(r\f - h\/s)dp + $+{N + M)p(I) = A*(r(f-h)/s)+$+(N + M)ii(I). The second term is clearly f inite. A s to the first, it too is finite because of our assumption of the finiteness of A $ on B(r) and because of the fact that since | | r ( / — h)/s\\ = r we must have r(f-h)/s€ B(r). • 206 Chapter III. Functionals on a set of domains and on Dirichlet spaces The following result uses a similar method of proof. As a special case we will get the result of Cima and Matheson [35] on the weak continuity of A$ on 2J\{0} when <&(£) = e*2 for t e [0, oo). Theorem 3.6. Fix 0 < r < 1. Suppose that C —>• E is upper semicontinuous. Suppose moreover that there exists an increasing [0, oo) —> [0,oo) such that for every a > 1 we have <J>+(z) = o(\P(a:|,z|)) as \z\ —> oo (uniformly in argz). Finally assume that A$ is bounded on B(a) for every a < \/l — r2. Then A$ is weak upper semicontinuous on S(r). Remark 3.1. We may scale this result so as to replace the assumption 0 < r < 1 by the assumption 0 < r < R, and to assume that A$ is bounded on B(a) for every a < y/R2 — r2. Proof of Theorem 3.6. Let fn—*f weakly with /„, / € S(r). We must show that limsupA$(/n) < A$(/). Note that weak convergence implies that liminf > ll/H >r. (3.7) 71 —>-CO (The easiest way to see this is to note that | | / n | | > |(/n, /)|/||./||,' while ( /„ , / ) —> ||/||2-) Now, r < ll/H = sup ($,/). g€dB Using our density assumption, there exists a function g € L°°(p) D dB such that (g, /) > r. Put gn- fn - (fn,g)g- Then, (fn,g)g and gn are orthogonal, so that ||/n||2 = ||/„ - (fn,g)g\\2 + \\(fn,g)g\\2 = \\gn\\2 + \(fn,g)\2. Then, lim sup H l^l2 = limsup(||/n||2 - \(fn,g) n—too TI—foo <limsup(l-|(/ r i, f f)| 2) n—>-oo = 1-lim inf | (/„,</) | 2 7 1 — » C O = l-\(f,g)\2<l-r2, (3.8) 207 Chapter III. Functionals on a set of domains and on Dirichlet spaces since (fn,g) —>• {f,d) by of weak convergence. Weak convergence implies also that the numerical sequence {(/«,<?)} be bounded. Since g is almost everywhere bounded, there exists a finite number M such that For n sufficiently large we then have gn € B(j). Choose a > 1 such that ay < y/l — r2. I claim that A $ a is bounded on the fn corresponding to those gn which lie in B(j). Assume for now that this is proved. Then, since A $ a is finite on H by Theorem 3.5, and since gn £ B(j) for sufficiently large n, and finally since a > 1 so that $ + ( 2 ) = o(tya(\z\)), it follows from Theorem 3.2 that lim sup^^ A $ ( / n ) < A $ ( / ) . To prove the claim, note that we have \fn\<\gn\ + M almost everywhere, by choice of M. Since ay < y/l — r 2, we may choose (3 > a such that we still have 8y < s/l — r 2 . Choose T sufficiently large that 3T > a{T + M). Let 4 = { i e J : \gn{x)\ > T} and put Bn = I\An. Then, l(/n,5)fl(a:)| <M for every n € Z + and almost every x £ I. Now choose 7 such that lim sup \\gn\\2 < J2 < 1 — r2. < j y(a(\gn\ + M))dp Jl f V(a(\gn\ + M))dp+ f y(a(\gn\ + M))dp <Ay(f3gn) + p(I)V(a(T + M)). 208 Chapter III. Functionals on a set of domains and on Dirichlet spaces Now, the second term on the right hand side is finite and independent of n. The first term, on the other hand, is uniformly bounded on the set of those n for which gn £ B(7) as the inequali ty j/3 < \ / l — r 2 implies that A<p is bounded on B(jf3). • C o r o l l a r y 3 . 3 . Suppose that : [0, 00) —»• [0,oo) is an increasing function such that A $ is bounded on B(r) for some r > 0. Let $ + ( 2 ) = o(\I/(|,z|)) as t —> 00 (uniformly in a r g z ) . Then, A $ is norm upper semicontinuous on all of H if 3> is upper semicontinuous. Remark 3.2. O f course, norm upper semicontinuity does not by itself imply any boundedness results since unit balls of infinite dimensional Hi lber t spaces are not norm compact . Proof of Corollary. W i thou t loss of generality r = 1 so that A $ is bounded on B. Let / „ —> / in norm. There are two cases to be considered. Suppose first that / is the zero element of H. For n sufficiently large we wi l l have fn £ B. Then , since A $ is bounded by assumption on those fn for which this happens and since it is everywhere finite on H by Theorem 3.5, it follows that A# is bounded on the {fn} and so l imsup^^oo A $ ( / „ ) < A $ ( / ) by Theorem 3.2. (We have, of course, used the fact that if / „ —>• / in norm, then it also converges weakly.) On the other hand if | | / | | > 0, then let M = | | / | | 2 . We have | | / n | | 2 -> M. F i x e > 0 to be chosen later, supposing for now only that e < M. For n large enough we wil l have M - e < l l / l l 2 < M + e. Let r = M - e and R = M + e. Then for sufficiently large n we have fn € S(r,R). Note that R2 — r 2 —> 0 as e —»• 0 so we may choose s sufficiently smal l that R2 — r2 < 1. We then indeed have A $ bounded on B(a) for every a < y/R2 — r2 since A $ is bounded on B(l). App l y i ng Theorem 3.6 and Remark 3.1 we see that A $ is weak upper semicontinuous on S(r,R) so that l i m s u p ^ ^ A $ ( / n ) < A $ ( / ) since / „ S S(r,R) for sufficiently large n and certainly / £ S(r,R) by choice of r and R. • We now apply these results to the Chang-Marsha l l inequality. The weak cont inui ty result in the fol lowing corol lary is due to C i m a and Matheson [35]. 209 Chapter III. Functionals on a set of domains and on Dirichlet spaces C o r o l l a r y 3 .4 . Let = e*2 for t £ [0,oo) and consider the functional A $ on the Dirichlet space £>. Then, A $ is norm continuous on all ofT), and weak continuous on 03\{0}. Proof. Let H = T) and B = 03. App ly ing the Chang-Marsha l l inequali ty we see that the assumptions of Theorem 3.6 are satisfied wi th r = 0 and $ = \P. Since 5(0) = B \ { 0 } , this proves weak upper semicontinuity away f rom 0. A s for the norm upper semicontinuity, this follows f rom Coro l la ry 3.3 whose assumptions are verified wi th r = 1 because of the Chang-Marsha l l inequality. A s for weak and norm lower semicontinuity, this is a t r iv ia l consequence of L e m m a 3.1. • Remark 3.3. Note that A $ is not weak continuous at 0 € 05. We shall see a proof of this fact in §3.3.3. Note also that the proof of Coro l la ry 3.4 only really needs a weaker form of the Chang-Marsha l l inequal i ty—the boundedness of A $ on B(r) for every fixed r < 1. Th is weaker form is much easier to prove and its knowledge preceded that of that Chang-Marsha l l inequality, since in fact the estimate central to it was already contained in Beur l ing's thesis [23]. In part icular, by Coro l la ry 3.4 we see that A $ is finite everywhere on X> when <&(£) = e a ' 2 for t £ [0, oo) and a is arbi trary. The fol lowing result is at t r ibuted by Chang and Marsha l l [32] to Garne t t . It is an immediate consequence of Coro l la ry 3.4 and the fact that p d - z Q e 0 ^ 2 = 0 ( e ^ l 2 ' 2 ) as \z\ —>• oo whenever 3 > a and p is a polynomia l . C o r o l l a r y 3.5 ( G a r n e t t ) . Let $(*) = p(t)eat for p a polynomial, a any real number and t £ [0, oo). Then A $ is finite everywhere on X>. 3.3. Crit ically sharp inequalities and nonexistence of extremals We now discuss a the Chang-Marsha l l and Moser-Trudinger inequalit ies in an abstract sett ing. M u c h of the mater ial under this heading wi l l be taken f rom the author 's paper [88]. The methods here are entirely real variable based. 210 Chapter III. Functionals on a set of domains and on Dirichlet spaces 3 . 3 . 1 . T h e g e n e r a l r e s u l t s Let T be a collection of measurable functions on (I,p.). We shall throughout assume that 0 G T and that T is sequentially compact wi th respect to convergence in measure. Throughout §3.3 when we refer to concepts such as compactness, semicontinuity or cont inui ty wi th respect to convergence in measure we shal l mean sequential compactness, sequential semicontinuity or sequential continuity, respectively, al l w i th respect to convergence in measure. D e f i n i t i o n 3 . 1 . We say that an upper semicontinuous [0, oo) —> IK is c r i t i c a l for T, pro-v id ing: (i) A $ is upper semicontinuous on .F \ {0} wi th respect to convergence in measure, and (ii) A $ is not upper semicontinuous wi th respect to convergence in measure at 0 G T. Cond i t ion (ii) says that there is a sequence of fk G T converging to zero in measure, but with A$(/fc) converging to some number (possibly +oo) / which str ict ly greater than <&(0) = A$(0 ) . The fol lowing result then is of the same type as the work of Flores [51]. We write T o $ for the composit ion of the funct ions T and T h e o r e m 3.7 ( P r u s s [88]). Let $ be continuous and positive on [0, oo). Assume that $ is critical for T. Then there exists a positive, convex, increasing and infinitely differentiable function Y on [0, oo) with Y(y) < y for every y G [0, oo), l i m ^ o o ^1 = i a n d support bounded away from zero, such that A r 0 $ does not attain its supremum on T. Moreover, if $(0) = 0 then we may require that there be a sequence of fk G T converging to zero in measure such that l im sup^. A $ ( / ^ ) = supy e ^r Ar 0 $( / fc ) -A proof wi l l be given in §3.3.4. It is not known whether the assumption of continuity of $ can be weakened to upper semicontinuity. It is not hard to see that the continuity of a posit ive $ immediately implies the lower semicontinuity of A $ on al l of T by Fatou 's lemma (see L e m m a 3.1). 211 Chapter III. Functionals on a set of domains and on Dirichlet spaces We say that A$ is bounded on T if supye^ r A$(/) < oo. If $ is in addition critical for T then we say that the inequality supyg:rr A$(/) < oo is critically sharp. Theorem 3.7 then roughly says that if $ is continuous then even if a critically sharp inequality supy£^-A$(/) < oo attains its maximum, still we may perturb <3> in an asymptotically negligible way and lose the attainment of a maximum. Now recall that if $ is any positive measurable function on [0, oo) with A$ bounded on T, then, for every upper semicontinuous * with ty(t) — o($(£)) as £ oo, we have A$ upper semicontinuous with respect to convergence in measure on T, and in particular attaining its maximum there (Theorem 3.2). Theorem 3.7 shows that o(<&(£)) cannot be replaced by 0($>(t)) in that result, even under the assumption that 4> attains its maximum on T. Now, Theorem 3.2 says that a critically sharp inequality sup e^^ - A$(/) < oo cannot be improved by replacing $ by some \I> with <&(£) = o(ty(t)) as t -» oo since supye:r A<j(/) will then fail to be finite. We have the following partial converse to Theorem 3.2. As before, T is a collection of mea-surable functions on a finite measure space (I, p.), with 0 £ T and T compact with respect to convergence in measure. Theorem 3.8 (Pruss [88]). Let $ be continuous and positive, and suppose that A$ is con-tinuous on T with respect to convergence in measure. Then there exists a positive, convex and increasing V £ C°°[0,oo) with —>• oo as y —>• oo and Ar0$ bounded on T. Moreover, we may require that —>• oo as y —>• oo. A proof will be given in §3.3.4. As in the case of Theorem 3.7, it is not known whether the assumption of continuity can be weakened to upper semicontinuity. Corollary 3.6 (Pruss [88]). Let $ be continuous and positive with $(£) —> oo as t —» oo and suppose that A$ is continuous on T with respect to convergence in measure. Then there is a continuous and positive with $(£) = 0 ( ^ ( 7 ; ) ) as t —> 0 0 and with A<j continuous on T with respect to convergence in measure, and, in particular, bounded there. Moreover, if $ is 212 Chapter III. Functionals on a set of domains and on Dirichlet spaces convex (respectively, increasing, or convex increasing), then $ can be taken to also be convex (respectively, increasing, or convex increasing). Proof. First assume that we do not need ^ to be increasing or convex. Let where T is as in Theorem 3.8. Then it follows that = o(T($(t))) as t —> oo, so that by Theorem 3.2, it follows from the boundedness of Ar/0$ that A$ is continuous on J- with respect to convergence in measure. On the other hand, we also have = o(ty(t)) as t —)• co. Now, if we do want to be increasing and/or convex, then choose T as in the "moreover" of Theorem 3.8. Let Since ^jp- —>• oo as t —> oo, it follows from L'Hopital's Rule that t(y) = o(T(y)) and that y = o(T(y)), both as y —> oo. Furthermore, if V is infinitely differentiable, then so is f, and if T is convex then so is f. Then the desired result follows upon setting $ = T o $, and applying Theorem 3.2 as before in order to obtain the continuity of A$ on T with respect to convergence in measure. (Of course we also need to use the general fact that if F is increasing and convex while G is convex, then F o G is convex.) • We also have the following result which is complementary to Theorem 3.7 but which will turn out to be much easier to prove. The proof is again given in §3.3.4. The result is a modified version of a result of Pruss [88]. Theorem 3.9. Let $ 6 C°°[0,oo) be positive, convex and increasing with lim^co = co. Suppose that A$ is upper semicontinuous on .7-\{0} with respect to convergence in measure and that A$ is bounded on T. Then there exists a positive, convex, increasing and infinitely differentiable function \& on [0, oo) with ty(t) > <h(£) for every t 6 [0, oo) and with $(£) = \P(i) for all sufficiently large t, such that A$ does attain its supremum on T. 213 Chapter III. Functionals on a set of domains and on Dirichlet spaces 3.3.2. Application to the Moser-Trudinger inequality In the Moser-Trudinger inequality, T is the collection of real-valued absolutely continuous functions / on [0,oo) wi th / (0 ) = 0 and is the Moser-Trudinger inequality, as mentioned before. Car leson and Chang [30] then showed that the supremum is actual ly achieved at some / £ T, though it is not known what exactly this extremal / nor what the exact value of the supremum is. A s impl ic i t ly noted by Car leson and Chang [30, p. 117], T is compact wi th respect to uniform convergence on compact subsets of [0,oo), and in part icular wi th respect to convergence in measure. (Note also that T is the unit bal l of the Hi lbert space % of real valued absolutely continuous funct ions / on [0,oo) wi th / (0 ) — 0 and Note that the map / i-> f(x) is a bounded linear funct ional on 71, and compactness of T wi th respect to pointwise convergence, and in part icular convergence in measure, follows.) Fur thermore, $ is cr i t ical for T. Cond i t ion (i) was impl ic i t ly shown by Carleson and Chang [30, pp. 117-118], while condit ion (ii) is impl ic i t in the work of Moser [78]. We may see the val idi ty of condit ion (i) also from Theorem 3.6 as follows. For it is clear that rl is a Hi lber t space of functions on I = [0,oo). Let <P(£) = <&(£) = e ' 2 . The Moser inequali ty implies that A $ Q is bounded on T for all a < 1 (actually, this last fact is considerably simpler than the Moser inequal i ty) . Thus , the weak upper semicontinuity of A $ on ^ "\{0} follows from Theorem 3.6 wi th r — 0. F rom weak upper semicontinuity we easily obtain upper semicontinuity wi th respect to convergence in measure by using the weak compactness of J-. If I = [0, oo) and dp(x) — e x dx, then the inequali ty sup A $ ( / ) < oo 214 Chapter III. Functionals on a set of domains and on Dirichlet spaces On the other hand, A$ fails to be upper semicontinuous at 0 € T. To prove this we look at Moser's broken line functions, proceeding much like in [78] and the proof of case (c) of our Theorem 2.1. Let 3(x) = min(a;, 1) and put fn(x) — y/h~8(x/n). Then clearly /„ € T and roo rn roo rn / eftW-xdt= / ex2/n~xdt+ / en~xdt> / e~xdt+l. J0 J0 Jn Jo Now, the right hand side converges to 2 as n —> oo. On the other hand, it is easy to verify that /„ —> 0 in measure and A$(0) = 1 so that A$ indeed fails to be upper semicontinuous at 0 € T. Hence = e*2 is critical for T. The following result which was conjectured by McLeod and Peletier [77] then follows immediately from Theorem 3.7. Theorem 3.10. There exists a convex, increasing and smooth function T with 0 < T(y) < y for every y 6 [0,oo) and with lim^oo = 1, such that the supremum roo sup / r(e / 2W)e-^ (3.9) is not achieved over T. Of course it should be noted that (3.9) is finite. Theorem 3.10 shows that the existence of the extremal for Moser's inequality is in some way accidental, relying on non-asymptotic properties of the function e*2. 3.3.3. Application to the Chang-Mars hall inequality Now consider the setting of the Chang-Marshall inequality, with I = T and p being normalized Lebesgue measure. Again let <£(£) = e*2. We have already noted in Theorem 1.1 that weak convergence in 23 implies convergence in measure on T (and we have remarked that this was already known to Andreev and Matheson [5]), so that 23 is indeed compact with respect to convergence in measure on T. Cima and Matheson [35] have shown that A$ is weakly continuous on 23\{0}. This is also a direct consequence of the Chang-Marshall inequality and 3.6 with r = 0. Thus condition (i) of criticality is satisfied. 215 Chapter III. Functionals on a set of domains and on Dirichlet spaces O n the other hand, C i m a and Matheson [35] have shown that A $ fails to be weakly upper semicontinuous at 0 G 53. We give a concise proof of this fact. For , if A $ were upper semi-continuous on al l of 23, then by Theorem 3.8 there would be a \P = T o $ which grows str ict ly faster than $ and such that A<p is bounded. Bu t this contradicts Theorem 2.1. Thus condit ion (ii) of cr i t ical i ty is satisfied. (The same concise argument could have been used in the previous section to prove condit ion (ii) in the case of the Moser-Trudinger inequality, but for the sake of variety we preferred to give a more elementary argument there and a less elementary argument here.) Then , even though we do not know whether A $ achieves its max imum over 53, we do have the fol lowing result which follows f rom Theorems 3.7 and 3.9. • T h e o r e m 3 . 1 1 . There exist two C°°[0, oo) functions and \ P 2 such that for every t £ [0, oo) we have 0 < $ i ( i ) < e*2 < \P 2 (*) and A $ t is bounded on 53 for i = 1,2, but A $ ; does not achieve its supremum over 53 while A $ 2 does achieve its maximum over 53. One may take the to be convex and increasing with l i m / . - ^ e~ < 2 \P i ( i ) = 1 and \P 2 ( i ) = ^ for oil large t. Alec Matheson has k indly communicated to the author that he and Joseph C i m a had strongly suspected the t ru th of this result. Theorem 3.11 implies that one cannot use any asymptot ic arguments to prove or disprove the existence of an extremal in the Chang-Marsha l l inequali ty wi thout tak ing into account the exact behaviour of the funct ion e* 2 , not just asymptot ical ly, but also closer to the or ig in. Bu t this does not exclude the possibi l i ty of some abstract argument based on some global properties of e*2 (e.g., analyt ic i ty) . 3 .3 .4 . P r o o f s o f t h e r e s u l t s o n c r i t i c a l l y s h a r p i n e q u a l i t i e s If / and 4> are measurable on [0,oo), then wri te Then , the main step in the construct ion of for Theorem 1 is encapsulated in the following result. 216 Chapter III. Functionals on a set of domains and on Dirichlet spaces L e m m a 3 .2 . Let Q be a subset of L1[0,oo) containing the zero function, such that for each finite number T we have s u p g £ g \\g • 1[O,T]||L°° < ° ° - Assume that for every sequence gn of elements of Q, there exists a subsequence gnk which converges in measure to some g G L1[0,co) such that either g is almost everywhere null or else has \\g\\Li > l im sup^. \\gnk Hz,1 • Suppose further that \\ • \\Li fails to be upper semicontinuous at 0 G Q with respect to convergence in measure. Then, there exists a positive increasing function <p < 1 on [0, oo) such that lim^-^oo <j)(x) — 1, with || • not attaining its maximum on Q. Furthermore, (f> may be taken to be in C°°[0, co), with support bounded away from 0. Moreover if supgeg | |<? | |LI < oo then we may also require that there be a sequence gk 6 Q such that gk —>• 0 almost everywhere and l imsup f c ||<7A;||LI = suPa€ff l l5l l i i (0)-Assuming the lemma for now, we may proceed to prove Theorem 3.7. Proof of Theorem 3.7. W i t hou t loss of generality assume that $(0) = 0. For / € T and t G [0, oo), let mf(t) = p{x : $ ( | / ( x ) | ) > *}. Let Q = {mj : f G J7}. We shall apply L e m m a 3.2 to G. Let us verify its condit ions. Clear ly, every element of Q is pointwise bounded by p(I) < oo. Fur thermore, for TO/ G G we have poo I K I I L ' = / mf{t)dt = A*{f). Jo Then , using the lack of upper semicontinuity of A $ at zero, we may choose a sequence fk G T such that fk —> 0 in measure and l im supk A$(fk) > A$(0) = 0. Passing to a subsequence if necessary, we may assume that fk —> 0 almost everywhere. Then , l im supk $(|/fc|) = $(0) = 0 almost everywhere, by the cont inui ty of Hence, $(|/fc|) —> 0 almost everywhere, too, and hence also in measure. Thus , for every t > 0 we have mjk(t) —>• 0, and in part icular nxjk —V 0 in measure while l imsup f c | | T r i / f c ( t ) | | i , i = l im sup f c A$(fk) > 0, so that || • H^ i fails to be upper semicontinuous at zero in Q. Now, given any sequence TOjn of elements of Q, we may choose a subsequence mjn such that 217 Chapter III. Functionals on a set of domains and on Dirichlet spaces fnk converges in measure, using the compactness with respect to convergence in measure of T. Choosing a further subsequence if necessary, we may assume fnk converges almost everywhere. If the limit is almost everywhere zero then we are done. On the other hand, if fnk —> / where / does not vanish almost everywhere then we first of all have lim^ $(|/„ f c |) = $(|/|) by continuity of <&, and secondly, by the upper semicontinuity of A$ with respect to convergence in measure away from zero, we have lim supfc A$(|/„J) < A$(|/|). Then, since 3>(|/nJ) —> $(1/1) in measure, it follows that mj —> mj almost everywhere (in fact at all points of [0,oo) other than the at most countably many discontinuities of rrty), as can be easily verified. Also, |J7Ti/Hx,1 > limsupk H^ i . Hence, the conditions for the Lemma are satisfied. Choose <f> as in Lemma 3.2. Let fy r(y) = / <t>{x) dx. Jo It is easy to verify that = Ar0$(/)- Then the Theorem follows from the conclusions of Lemma 3.2. For example, the convexity of T follows from the fact that cf> is monotone increasing. • Lemma 3.3. Let & be a subset of L1[0',oo) containing the zero function, such that for each finite number T we have supgeg \\g • l ^ r ^ L 0 0 < oo. Assume that for every sequence gn of elements of Q, there exists a subsequence gnk which converges in measure to some g G oo) such that either g is almost everywhere null or else has ||ff||x,i > lim sup^ \\gnk WL1 • Suppose further that || • | | L i is uniformly bounded on all of Q and upper semicontinuous with respect to convergence in measure at 0 G Q. Then, there exists a positive and increasing function 4> > 1 on [0, oo) with lim^oo <f>(x) = oo and || • 1 £1(0) bounded on Q. Theorem 3.8 then follows from Lemma 3.3 in the same way as Theorem 3.7 had followed from Lemma 3.2. We now proceed to prove our two lemmata. Proof of Lemma 3.2. First suppose Q is not uniformly bounded in L1 norm. Let <f> be a positive 218 Chapter III. Functionals on a set of domains and on Dirichlet spaces increasing C°°[0, oo) function whose support is bounded away from zero and which has <p(x) — 1 for all x > 1. Then let gk be a sequence of elements of Q with ||fffc||£,i —> oo. Passing to a subsequence we can assume that for some g £ ^[0,00) we have gk g in measure. If g is almost everywhere null, then it is easy to see that the proof is complete since by the bounded convergence theorem (which is applicable because the {gk} are almost everywhere uniformly bounded on [0,1] by the hypotheses of the Lemma) we have \gk\ —> 0 so that Il5fc||£i(0) > \dk\ = Ikfelli i - Jo1 \9k\ a n d the r i g n t h a n d s i d e tends to 00, so that 00 as desired. Choosing a subsequence if necessary, then, we may assume that gk —> 0 almost everywhere and the Lemma follows. On the other hand, if g is not almost everywhere null then H^H i^ > limsupfc HS^ HL1 by our hypotheses. But, the right hand side is infinite, and this contradicts the fact that g £ L1. Now, assume that where the supremum is to be understood as taken over all sequences {<%} in Q tending to zero in measure. Since || • ||£,i fails to be upper semicontinuous at 0 £ Q with respect to convergence in measure, we have A > 0. Obviously, A < M. Replacing Q by {\g\ : g £ Q} if necessary, we may assume all functions in Q are positive. Choose 0 < a < 1 such that aM < A. For g £ Q, let M = sup ||</||£,i < 00. gee Let A= sup hm sup ||/A|J£,I {9k}cg k (3.10) so that (3.11) 219 Chapter III. Functionals on a set of domains and on Dirichlet spaces and pTg roo roo roo / 9 = 9- g=(l-a) g. (3.12) Jo Jo Jra JO Now define Gx = {9 • Tg > x}. Let I claim that f sup / geSxJo l imsupM^A. (3.13) x—too To show this, it suffices to prove that for any sequences xk —>• co and gk £ QXk such that JQ°° gk converges, we have \ i m k J0°° gk < A. Fix such sequences xk and gk. Passing to subsequences, if necessary, by our hypotheses we may assume that gk either converges to 0 in measure, or else it converges in measure to some nonzero g £ L^OjOo) with ||y||ii > lirnsup^ . ||<7fc||£,i. If it converges to 0 in measure then lim sup^ J0°° gk < A by definition of A. Otherwise note that since 9k £ Gxk, we have / ~ gk > a\\gk\\Li. Let hk{x) = gk(x) • l{x>Xky. We have hk -> 0 pointwise since xk —> 00. By Fatou's lemma then, roo roo lim inf / (gk - hk) > / g. k Jo Jo But roo rxk rr9k / {9k-hk)= / gk< gk = (1 - a)\\gk\\Li, Jo Jo Jo where we have used the fact that gk £ QXk together with (3.12). Thus, lim inf (1 - a)||flf*||Li > NL&i. k But since a < 1, this contradicts the facts that \\gW11 > limsup^ . ||<7fc||£i a R d that g does not almost everywhere vanish. Hence, the case where gk does not converge to zero in measure is impossible, and the claim is proved. 220 Chapter III. Functionals on a set of domains and on Dirichlet spaces Now define tb(x) = (-^-] (1 A inf A ~ / T » 9 \ . I c la im that tp(x) —>• 1 as x —> oo. To prove this, consider the funct ion A — at h{t) :i-a)f which is easily seen to be decreasing for t £ [0, M] since aM < A, and which satisfies h(X) = 1. B y (3.11) and (3.12) we then have *w=(ifi)(^«£(,.}*Gr'))-Bu t for g £ Qx we have J0°° g < Mx so that by the monotonic i ty of h on [0, M] we have il>{x) > j^{lAh(Mx)). Now by (3.13) we have l im infx^.oo h(Mx) > h(X) = 1 and hence l i m ^ o o ip(x) = 1 as desired. Note that tp is measurable as it is increasing. It is easy to verify that ||<7||£,i(,/,) < A for every g £ Q. For , given g £ Q w i th ||ff||£,i 7^ 0, we have roo rTg roo rrg roo roo roo / W>< / 9i>+ 9<^{rg) g+ flf < A - / g+ g = X, JO Jo J-rg Jo Jrg Jrg Jrg where we have used the monotonic i ty and choice of tp. The first inequali ty came from the facts that tp < 1 everywhere and that JT°° g > 0 if ||<7||£,i / 0. If we do not need <p to be C°°[0, 00) or to have support bounded away f rom zero, then just let 4> = tp. Otherwise, since tp is a increasing funct ion [0, 00) wi th l imit 1, we may easily choose a increasing C°°[0,oo) funct ion cp w i th support bounded away f rom 0 and wi th the properties that 0 < 4> < ip everywhere and that ep(x) —> 1 as x —> 00. We wi l l then necessarily st i l l have llffllno) < A f o r each-5 £ Q. We shal l now show that SUPII#IILI(0) = X. (3.14) 221 Chapter III. Functionals on a set of domains and on Dirichlet spaces To do this, fix e > 0. By definition of A, let gk be a sequence in Q with gk —> 0 in measure and II^IIL1 —* A- Choose T sufficiently large that 4>(x) > 1 — e for x > T. Since gk —^ 0 in measure and, by our hypotheses, the gk • l[o,x] a r e uniformly bounded in L°°, the bounded convergence theorem tells us that JQT gk —>• 0 as k —> oo. Since ||fffc||jr,i —>• A, we may choose K sufficiently large that gk > A — e for k > K. Then, for k > A' we have roo roo \\9k\\m*)>J 9k<t>>{l-e)J gk>{l-e)(\-e). Hence liminffc |JS'A;IIJG1 (0) > (1 — e){^ ~ e) f° r every e > 0, and so we see that indeed lim inf H^HLI^) > A and (3.14) follows from this and the already proved inequality ||#||LI(0) < A valid for every geG. The last sentence of the statement of the Lemma now follows upon taking a subsequence of the above gk which converges almost everywhere to zero. Thus, the Lemma is proved. • Proof of Lemma 3.3. As in the proof of Lemma 3.2, we may assume without loss of generality that all functions of Q are positive. Let roo M = sup / g. geQ Jo By assumption this will be finite. Let r o o Ux = sup / g. g€G Jx I claim that Ux —>• 0 as x —>• oo. For, fix any 0 < a < 1, and define ( roo roo ~> r; = i n f | r > 0 : y g < a J g\ . Let Qx = {g G Q : T® > x}. Exactly as in the proof of Lemma 3.2, we may show that if roo M" = sup / g, geGgJo 222 Chapter III. Functionals on a set of domains and on Dirichlet spaces then M" —>• 0 as x —>• oo. (For, in the present case A as defined by (3.10) wi l l be zero, by the assumption of upper semicontinuity wi th respect to convergence in measure to zero.) Now, fix e > 0 and choose 0 < a < 1 such that aM < e. Assume that x is sufficiently large that M° < e. Then , for such x and g £ Q, we have g < JQ°° g < M" < e providing g £ Gx. On the other hand, if g fi G% then < x, so that g < f™g = a g < aM < e. Hence, in either case f£° g < e, and so Ux —Y 0 as desired. Choose a sequence of finite posit ive numbers xk —> oo wi th the property that UXk < 2~k. Then set tp(x) = Card{A; £ Z + : xk < x}. Th is is a increasing posit ive funct ion on [0, oo), and we have ip(x) —> oo as x —> oo. Furthermore, if g £ Q then it is easy to verify that °° poo co \\9\\vw = / 3<YUx*^1' k=l J x * k=l by choice of xk- In fact, we also have || • ||£i(i+^,) bounded on Q since we had assumed that || • H^i is bounded on Q. Now, we may easily choose a increasing funct ion <f> £ C°°[0,oo) wi th the property that 1 < 4>(x) < l + ip(x) and <f>(x) —>• oo as x —> oo. The L e m m a then follows. • Proof of Theorem 3.9. If necessary replacing $ by $ — $ (0 ) , we may assume that $(0) = 0. If al l funct ions / in T satisfy A $ ( / ) = 0 then we are done. Otherwise, fix /o £ T such that A « ( / 0 ) > 0. Let M = s u p A $ ( / ) . Choose any e £ (0,p(I)). Choose A £ (0,oo) such that p{x:\~l < $( | /o (z ) | ) < A} > e . (Such a A exists since $ is everywhere finite while A$ ( /o ) > 0.) Now, choose a funct ion $ on [0,oo) and a positive number K which satisfy the following properties: 223 Chapter III. Functionals on a set of domains and on Dirichlet spaces (i) is convex, positive, increasing and C ° ° (ii) = Kt forte [0,A] (iii) $(£) = £ for all sufficiently large t (iv) > for all t (v) K > A r ^ l + M ) . To do this, first choose T sufficiently large that T _ 1 $(T) > Ae _ 1(l + M) and T > A. (This can be done since i _ 1$(i) —> oo as t —>• oo.) Let K = T _ 1 $(T). It easily follows that we may splice together a convex, positive, increasing C ° ° function \P(£) for £ G [0, oo) which equals i^i for t 6 [0,A], coincides with \P on [T + l,oo) and is at least as large as $ on (A,oo). (It is automatically at least as large as <& on [0, A] by choice of K and the convexity of <I>.) I claim that attains its maximum on T. For, A$ is upper semicontinuous on .F\{0} with respect to convergence in measure. To see this, note that A$ = A$ + A$_$. But $ — $ is a bounded function by condition (iii) of the choice of \P, so that A$_$ is continuous with respect to convergence in measure on all of T by the bounded convergence theorem and the continuity of $ — $. On the other hand, we had assumed the upper semicontinuity of A$ on ^\{0}. Let fn be a sequence such that lim Ay(fn) = supAy(g). (3.15) n-i-co gejr Passing to a subsequence, by compactness with respect to convergence in measure, we may assume that /„ —>• / in measure for some / G T. If / is not almost everywhere null, then X\mn^roo A\$(fn) < A$(/) by upper semicontinuity of A<p on J"\{0}, and hence A$ attains its maximum at / . Assume now that / vanishes almost everywhere. Then, lim A * ( / n ) = lim ( A * ( / n ) + A$_$(/„)).' n—too n—>oo 224 Chapter III. Functionals on a set of domains and on Dirichlet spaces Now, * — <& is a bounded continuous function and /„ —>• 0 in measure so that A$_$(/„) —>• \P(0) — $(0) as n —>• oo by the bounded convergence theorem. Since $(0) — (^0) = 0, we see that A<p_$(/n) —» 0. On the other hand, trivially lim sup A$(/n) < M. n—too In light of (3.15) we see that supA$(#)<M. (3.16) But now note that by conditions (ii) and (v) in the choice of \1/ and by the choice of e we see that •A*(/o) > £~\l + M)\\-1p{x : A"1 < $(|/„|) < A} > 1 + M, contradicting (3.16). Thus we see that the limit of the fn cannot be 0 and so A$ does attain its maximum on T. • 4. Properties of extremals of the A$ on Dirichlet spaces In this section we shall discuss properties of A$ functionals acting on the unit balls of Dirichlet spaces. 4.1. A variational equation Recall that if $ was a function from [0, oo) to R then A $ was an abbreviation for A $ ( | . Q . The following result due to the author is taken from the author's joint paper with Alec Matheson [75]. Theorem 4.1. Let u>: [0,oo) -> R . Write \P(i) = $(\/t)- Assume that \I> is differentiable on (0, oo), with \9'(t)\ < ceCt for some finite constants c and C, and every t > 0. Suppose f 6 2$ is an extremal function for A$. Write s*(f)= f | / | V ( | / | V 225 Chapter III. Functionals on a set of domains and on Dirichlet spaces Then if Sq,(f) vanishes, it follows that /$'(|/|2) vanishes almost everywhere on T. Assuming that S\$(f) does not vanish, we have f £ H2 and S*(f)zf' = V0{fV(\f\2)) onT, (4.1) where z stands for the identity function on T, and Vo is the orthogonal projection from L2(T) to H2(T)d^{f £ H2(T) : /(O) = 0}. Moreover, if we also have < 1, then 5$(/) must vanish. Remark 4-1- The functional A$ makes sense on 23 since the hypotheses guarantee that | \ P ( £ ) | < cC~l(eCt — 1) -(- |$(0)|, which, together with Corollary 3.5, guarantees the finiteness of A$(/) for every / £ 23 (indeed, for every / £ D). Similarly, the right hand side of (4.1) and the integral defining S\$ (/) make sense because of Corollary 3.5 and our assumption on the size of |*'|. Also, we note that the identity function on D always satisfies (4.1) even though it may not be extremal. (For example, it will not be extremal if $(£) = t2n and n is sufficiently large.) The interested reader may, of course, translate the conditions on $ into conditions on $, but they are perhaps more naturally stated for This will be even more true in the next theorem. If VP' (or, equivalently, $') does not vanish on (0,oo), and 5$(/) = 0, then the vanishing of /v[r'(|/|2) on T implies that / = 0, and (4.1) trivially continues to hold. The criterion (4.1) was shown by Andreev and Matheson (see [5, Cor. 3 and remarks following it]) in the special cases of the Chang-Marshall functions $a(i) = ea*2 and of the functions $(£) = t2n, under the auxiliary assumption that / ' £ H1. Our result above shows that this assumption will automatically be satisfied whenever / is extremal and does not vanish on (0,oo). Note that for any $ the functions / of the form f(z) = azn where n\a\2 = 1 (note that in particular this includes the identity function) provide solutions to (4.1). However, in general 226 Chapter III. Functionals on a set of domains and on Dirichlet spaces they may not be the only solutions. For instance, if = t2n where 2n > 4, then an extremal exists as noted before, and clearly the hypotheses of Theorem 4.1 are satisfied. But we have seen that the extremal in that case is not the identity function. Nor can it be of the form f(z) — azn with n|a|2 = 1 since for any such function we have A$(/) < A$(z), where z indicates the identity function. If one could prove that for all a < 1 sufficiently close to 1 all solutions / of (4.1) are of the form f(z) = az11 with n\a\2 = 1, then it would follow that sups6<g A$Q (g) < ea for a < 1 sufficiently close to 1, and so by a limiting argument we would conclude that A$j (g) < e = A$j (z), which would give an affirmative answer to Problem 3.1. Numerical solutions to (4.1) might lead to a better understanding of the extremal functions for A$. Open Problem 4.1. Investigate numerical algorithms for solving pseudodifferential equations on T of the form zf = v0{m\f\2)), where / is the boundary value of an analytic function and tp a sufficiently nice real-valued function. The following question is easily seen to be related to a conjecture that we shall give later (Conjecture IV. 1.2). Open Problem 4.2. If $(e*) is convex in t and $>'(x) > 0 for every x > 0,.then does it follow from (4.1) that / is automatically univalent? If so, then must it also be star-shaped? We now proceed to the proof of Theorem 4.1. We shall use the following trivial and very classical lemma to analyze the case where S$(/) = 0. We write HP(T) = {/ : / € HP(T)} for the antianalytic Hardy spaces. Lemma 4.1 (see Koosis [67, p. 87]). LetV G HP(T) orV <E #P(T) for some p, l<p<co, be real valued on T. Then V is almost everywhere constant. 227 Chapter III. Functionals on a set of domains and on Dirichlet spaces Proof of Lemma. Without loss of generality consider the case V € HP(T). By [67, p. 87] we can continue V analytically to all of C by setting V(z) = f(l/z) for \z\ > 1. Then V will be a bounded entire function, hence constant. • The careful reader will note that at several points in the following proof we would be able to assert that various functions lie in Lp for every p < oo, but we only write down that they lie in L2 or L1. We do this in order to make it clearer how to generalize the proof to yield more general results. Proof of Theorem 4.1. We proceed by a simple variational argument. Let / be extremal. Write / ( * ) = £ ~ = l « n * B -Suppose first that = 1. Write F(X) = | | / + Azn|||,, where we use zn as a short form for the function z H-+ zn. Since | | / | | D = 1 while ||zn||j) = y/n, it follows that, for |A| < l/y/n, we have F(X) > 0 and (/ + Xzn)/y/T{X) has unit Dirichlet norm. Suppose now that A is real. If / is extremal, then we must have dA*( ( /+Az") /V^vA) ) dX 0, (4.2) A=0 as long as this derivative exists. We shall prove this derivative exists, and compute its form. First note that F(X) = \ a-k\ + n\an + A|2 = I £ k\ak\2 J + n[(an + an)X + A2] \k=l (4.3) + n(2ARea„ + A2) = 1 + n(2ARean + A2). Now we have f(z) finite for almost every z G T. Furthermore, we may assume that \f(z) \ / 0 for almost every z. For if / vanishes on a set of positive measure then, being a function in D C H1, it would have to vanish identically and the Theorem would follow trivially. Moreover, if we restrict A to [—l/(2y/n), l/(2y/h~)] then F(X) will not vanish. Fixing z such that \f(z)\ < oo, we then 228 Chapter III. Functionals on a set of domains and on Dirichlet spaces have + A2r"|2/F(A)) absolutely continuous with respect to A on [—l/(2^/n), l/(2\/n)\. For, $ is differentiable except possibly at zero, the derivative is bounded in a neighbourhood of zero (since |$'(£)| < ceCt) and, if z is fixed, then \ f(z) + Az"j2 can vanish for at most finitely many (in fact, for at most two) values of A. Thus if — l/(2-v/n) < A < l/(2y/n), we will have F(t)(znf + znf + 2t\z\2n) - F'(t)\f + tzn\2 (4.4) A / dt 7 o V F(t) F(t)(2Re(znf) + 2t) - n(2Rean + 2t)\f + tzn\2 dt, (F(t))2 by (4.3) and the fact that \z\ = 1 on T. Of course, as long as 0 < \f(z)\ < oo (which happens for almost every z 6 T), if we let A —> 0 then this will converge to sdef d , * ( l / ( % ) Z ° ' 2 ) = 2 < R e ' 2 " ° / ) - "Re°»l/I2)*'(l/Ml2). (4.5) where we have used the fact that F(0) = \\f\% = 1. We wish to conclude that lim f Ax= [ 6, (4.6) and to this end we will dominate A\ by an Ll(T) function not depending on A. Now if -l/(2y/n) < A < l/(2y/n), then 1/4 < F(A) < 9/4, as can be easily verified. Thus, making simple estimates in (4.4) and using the assumption that |vP'(y)| < ceCy, we obtain, for - 1 / ( 2 ^ < A < l/(2y/n) and almost every z G T, |A A | < KeRVW\\f\ + 1 + (n + 1)(|/| + l) 2 ) , for some finite constant K depending only on c and C. But by Corollary 3.5, the right hand side of the last expression is integrable on T. Applying Lebesgue's dominated convergence theorem 229 Chapter III. Functionals on a set of domains and on Dirichlet spaces we find that (4.6) holds, so that (4.2) makes sense and we have dA$ ((f+\Z")/y/F(X) A=0 (4.7) / 2(Re(^/)-nRean|/| 2)tf'(|/| 2), by (4.5). But if / is extremal, then so is —if, since | / | = | — if \ and the Dirichlet norms are also the same. Applying (4.7) to —if then yields the same expression but with Im(zn/) and Ima„ in place of Re(znf) and Rean, respectively. Multiplying the new expression by i, adding to (4.7), and dividing everything by two, we find that for every positive integer n. Assume first that 5*(/) does not vanish. Then (4.8) would immediately yield (4.1) if we knew that zf had boundary values whose positive Fourier coefficients were {nan}. This would follow if we knew that / ' G H 1 , but unfortunately, we do not a priori know this, and so we must proceed more carefully. Let G(z) = f{z)^'(\f{z)\2) on T. By Corollary 3.5 and the hypothesis that |*'(y)| < cec*, we have G G X2(T). Hence V Q { G ) G H 2 ( T ) . But we see from (4.8) that the nth positive Fourier coefficient of G is S\a(f)nan, which is also the nth Taylor coefficient of Sn>(f)zf'. Now VQ(G) extends to a holomorphic function with Taylor coefficients equal to its positive Fourier coefficients, and the positive Fourier coefficients of VQ(G) must of course match those of G and these match the Taylor coefficients of 5#(/)z/'. It follows that S^(f)zf = Vo(G), which is precisely the equation (4.1). Furthermore, / ' G 7f2(T) since V 0 { G ) G H 2 ( T ) . Now if Sxs(f) vanishes, then by (4.8), the positive Fourier coefficients of the function /^'(|/|2) G L2(T) must vanish. But this implies that /\P'(|/|2) is the boundary value of an antianalytic function, call it h. By Corollary 3.5 and the inequality '^(y) < ceCy, h lies in H2(T). Therefore, l/l^'d/l 2) = h • f is also the boundary value of an antianalytic function from 771(T), since / is antianalytic and / G £> C H 2 . Applying Lemma 4.1, we see that l/P^'d/l2) is almost (4.8) 230 Chapter III. Functionals on a set of domains and on Dirichlet spaces everywhere constant on T . Bu t since 0 = Sq,(f) equals the integral of this constant about T , it follows that in fact | / | 2 # ' ( | / | 2 ) = 0 almost everywhere on T , so that / $ ' ( | / | 2 ) = 0 almost everywhere on T as desired. The only remaining th ing to do is to consider the case where | | / | | D < 1. In this case, for A sufficiently smal l we st i l l have / + Xzn G 55, so that we may use a simpler var iat ion. M u c h as before, but wi th the calculat ions being somewhat simpler, we find that the condit ion for every posit ive integer n. A s before, this implies that / $ ' ( | / | 2 ) is the boundary value of an ant ianalyt ic funct ion, say h, which as before wil l have to be in H2. Therefore, | / | 2 $ ' ( | / | 2 ) is the boundary value of /i • / G H1, so that its integral about T must equal h(0)f(0) = 0. Bu t its integral about T is just 5$ ( / ) . Thus , 5$ (/) vanishes and we may complete our argument in the same way as was done in the case of | | / | | o = 1, above, but using (4.9) in place of (4.8). • 4.2. Regularity of extremals Final ly , as corol lary to Theorem 4.1, we obtain a result on the regularity of extremal functions. Th is result is due to joint work of the author and A lec Matheson [75]. T h e o r e m 4 .2 ( M a t h e s o n a n d P r u s s [75]). Write $(£) = Fix an integer n > 0. Suppose that $ is n times differentiable on (0, oo). If n > 0 then assume further that \I/(n) is Lipschitz on bounded subintervals o / [0 ,oo) , and, if n = 0, then assume that $ is differentiable on (0,oo). Also assume that 0 < |$ ' ( t ) l < ceCi for some finite constants c and C, and for every t > 0. Finally assume that f G 55 is an extremal function for A $ . Then f is n times continuously differentiable on T . Furthermore, /(") is absolutely continuous on T, and in fact d A $ ( / + Az") dX A=0 = 0 makes sense, and applied to / and —if leads to the condit ion (4.9) /(n+i) e B M O A so that / ( n ) G A * . 231 Chapter III. Functionals on a set of domains and on Dirichlet spaces Remark 4-2. The Zygmund class A* is the set of functions g on T such that for every 0 6 [0, 2n) we have F(e<e+h^) + F{e^e~h)) - 2F(et9) = 0(h). Of course if A$ has no extremal functions, then the content of the Theorem as stated is null. Note, however, that as the proof will show, the conclusion of Theorem 4.2 holds not only for extremal functions / , but indeed for any functions / satisfying the conclusion of Theorem 4.1 with 5« ( / ) ^ 0. The following result follows immediately from Theorem 4.2. Corollary 4.1 (Matheson and Pruss [75]). Suppose $ : [0, oo) —>• K is such that t i-> 3>(\/?) is infinitely differentiate on (0, oo) and each of its derivatives is bounded near zero. Also assume that $'(£) ^ 0 on (0, oo), and that there is a finite constant C such that \<&'(t) \ = 0(eCt2) as t —>• oo. Then any extremal function for A$ must lie in C°°(D), i.e., for every non-negative k we must have /(fe) in the disc algebra. For the proof of Theorem 4.2, we now recall the following well-known result which is crucial to the proof of our regularity theorem, though in fact we only use the easy case p = 2 in our work. Lemma 4.2 (Privalov; see [41, Thm. 3.11]). Let f € Hp for a holomorphic function f on D and p > 1. Then f is absolutely continuous on T, continuous on D, and f has the boundary values f'{e10) =-ie-t9y±p-. We also need the following simple result, which we list here for ease of reference and to establish a convention which we will use in our proof of Theorem 4.2. Lemma 4.3. Let F be a Lipschitz function and G absolutely continuous. Then, F o G is absolutely continuous and (F o G)' = (F' o G)G' almost everywhere, with the convention that (F'(G(x))G'(x) = 0 whenever G'(x) = 0, whether F is differentiate at G(x) or not. Given this, we proceed to the proof of our regularity theorem. The main tool in the proof will be a qualitative analysis of condition (4.1). The careful reader will note that at some points 232 Chapter III. Functionals on a set of domains and on Dirichlet spaces in the proof we take some care to conclude that certain quantities are in L2, and only use the fact that other quantities are in L2, whereas it may be slightly simpler to replace all the L2 conditions by the condition np<co a n a 1 t° use the full M. Riesz theorem. This would allow us not to bother with keeping track of the degrees of the arguments of the polynomial pk+i-However, we choose to argue in L2 because the M. Riesz theorem is trivial for the L2 case and generalizes to L2M for arbitrary M C Z, so that it will be possible to adapt our proof to later obtain the more general Theorem 4.4. The following proof is due to the joint work of Matheson and Pruss [75]. Proof of Theorem 4-2. First note that for every k < n the function is bounded on the intervals [0, A] for every finite A. This follows by (n — k) integrations of the function \&(n) which is Lipschitz there. We may also assume that Sq(f) ^ 0. For if it does equal zero, then /$'(|/|2) vanishes on T. But $' can only vanish at zero, so we see that / = 0 in this case, and the Theorem is trivial then. We note that (4.1) can be written in the form Szf' = V0[p(fJ;*'(\f\2)], (4.10) where p is a polynomial in 2 + 1 variables and 5 = 5\p(/). The basic idea is to differentiate this expression repeatedly. Write for a function j on T. Note that Azn = nzn. If g is holomorphic on D and g' £ H1, then Ag = zg' by Lemma 4.2. Thus we can rewrite (4.10) as 5A/ = ^o[p(/,7;*'(|/|2))]. (4.11) 233 Chapter III. Functionals on a set of domains and on Dirichlet spaces The main thing to do now is to prove that our hypotheses in effect allow us to commute A with VQ the right number of times. We shall proceed iteratively. For the convenience of the reader, however, we first outline the idea of the proof in a less formal way in the case where $ G C°°[0,oo). In.that case we must show that / 6 C°°(T). By Theorem 4.1 and Lemma 4.2 we have Af = zf € H2 and / is absolutely continuous on T. Moreover, SAf = F1: where Fi =?(/,/; *'(|/|2)). Now, AF1=p2(z,z,fJ,f',7;V'(\f\2),y"(\f\2)), (4.12) for some polynomial p2. Moreover, the right hand side of (4.12) is only linear in / ' and / ' (i.e., / ' and / ' in it are never multiplied together nor are they raised to a power bigger than 1). The absolutely continuity of / implies that Fi =ip(f, /; W(\f\2)) is absolutely continuous on T since p is a polynomial. Together with an integration by parts and the fact that SAf = VQFI , this shows that the positive Fourier coefficients of AFi coincide with the positive Fourier coefficients of SA2f considered as a distribution on T. But by (4.12) since / ' G H2 it follows that AFi G L2(T) (here we have used the fact that the right side of (4.12) is only linear in / ' and J). Thus, S A 2 / G L2(T) and SA2f = P0F2, where F2 denotes the right hand side of (4.12). From the fact that A 2 / G L2(T), it follows (using Lemma 4.2) that / ' is absolutely continuous on T and that /" G H2. Thus F2 is absolutely continuous on T. Applying the above method one more time we conclude that AF2 is a polynomial in z, z, /, 7, /', 7, /", F, *'(l/l2), *"(l/l2), ^ "(l/l2), and that S A 3 / = VQAF2 G L2(T), 234 Chapter III. Functionals on a set of domains and on Dirichlet spaces so that / " ' G H2(T). I terating we conclude that £ H2(T) for al l natural k, which implies that / G C ° ° ( T ) as desired. We now return to the general case and proceed more rigorously. For the sake of our i terat ion, suppose that on T we have SAkf = V0\pk(z, z, f, / , / ' , / ' , / t * - 1 ) , / ( * - D ; (4.13) $'(| / | 2),$"(|/ | 2), . . . ,vi>( f c)(| / | 2))], with k < n, w i th pk a polynomial in 2(k + 1) + k variables, and wi th A3 f G H2 for every 0 < j < k. B y Theorem 4.1 and (4.11), we have this for k = 1. Us ing the fact that Ag = zg', we easily see that for any natural j we can write (in the sense of power series) for some set of universal constants a ^ / . Since A3 f G H2 for every 0 < j < k it follows that likewise G H2 whenever 0 < j < k. Thus by L e m m a 4.2 we have that is absolutely continuous on T for 0 < j < k — 1. Since sums and products of absolutely continuous functions are absolutely continuous, and since $ ( J ) for 0 < j < k is L ipschi tz on bounded subintervals of [0,oo) while / is absolutely continuous on T so that $ ^ ( | / | 2 ) must be absolutely continuous on T by L e m m a 4.3, it follows that (4.14) *'(l/|2),*"(l/|2),-,*(fc)(l/|2)) is absolutely continuous on T . B u t , in the presence of absolute continuity, we may use integra-t ion by parts to see that for every positive integer m, f2?r r2ir r e-imeAFk{eie) dO = m [ * e~imeFk(elB) dd Jo Jo r-2-K = Sm / e-imeAkf(eie) dB, Jo where we have applied (4.13) to obtain the last equality. ( 4- 1 5) 235 Chapter III. Functionals on a set of domains and on Dirichlet spaces Now it is easy to see from (4.14) and the definition of A that A F , = P k + 1 (z, z, f, /, /', T7, / W ; (4.16) * ' ( l / | 2 ) , * " ( l / | 2 ) , - , * ( / s + 1 ) ( l / | 2 ) ) for some polynomial pk+i in 2(k + 1) + (k + 1) variables, such that and f(k) are not raised to any power greater than one nor are they multiplied together on the right hand side of (4.16) (i.e., fixing all arguments other than the (2k + l)st and (2A; + 2)nd in pk+\ we obtain something of degree one), and where we have used the convention of Lemma 4.3 at points z of T where ^i(k+1)(\f(z)\2) is undefined. This convention may be necessary if k = n and n > 0, since then we only know that ^>(fc+1) exists almost everywhere, and the set of points z of T at which ty(k+1)(\f\2(z)) is undefined may well have positive measure, since after all | / | is free to be constant (as long as this constant is not zero) on a subset of T with positive measure, even if / itself is not everywhere constant in D. Since we know that / is absolutely continuous on T, it must be bounded there. As noted at the beginning of the proof, the hypotheses of the Theorem imply that is bounded everywhere on [0, II/HTO] for j < n. This is also obviously true for j = n + 1 if n — 0 by our condition on the size of j^ Ef/1. In the case where j = k = n + 1 and n > 0, we note that the derivative of the function \p( n) which is Lipschitz on [0, H / H 2 * , ] is bounded, by the Lipschitz constant wherever it exists. Moreover, for almost every z € T such that |/(z)|2 falls into the exceptional set where is not denned, we have, following the convention of Lemma 4.3, some factor equal to zero multiplying the ^ " H l / M D m t ne right hand side of (4.16). Finally, since is in H2, from Lemma 4.2 we conclude that all arguments of pk+\ m (4-16) are bounded, except possibly for and But these latter two arguments are never multiplied together, nor are they ever raised to any power, so we see that since they lie in L2(T), it follows that in fact the right hand side of (4.16) must lie in L2(T). Thus, V0(AFk) € H2(T), as well, by the L2 case of the theorem of M. Riesz (Theorem 1.3.4). Now write g(z) = SAkf(z) = £~=1 amzm for z <G D. We have zg'(z) = £~=1 mamzm in D. In this notation, (4.15) tells us that for m positive, the mth Fourier coefficient of AFt on T is 236 Chapter III. Functionals on a set of domains and on Dirichlet spaces mam, and since Vo(AFk) £ H2(T), it follows as in the proof of Theorem 4.1 that we must have zg' = Vo(AFk). Bu t zg' = SAk+1f, so that we obtain S A * + 1 / = Vo\pk+i (z, z, /, /, /', T 7,/<*), JW; (4.17) ¥'(l/l3),*ff(l/lV-.*(*+1)(l/l2))]. with A k + 1 f = S~lzg' £ H2 since 5 ^ 0 . Bu t this was precisely what was needed for the iterat ion to continue. Thus i terat ing, we see that Akf £ H2 for each k < n + 1. A s argued before, it follows that f(n+i) g JJ2^ a n ( j s o by L e m m a 4.2 we have absolutely continuous on T , and of course then / must be n t imes differentiable, as desired. F ina l ly , since / ( n ) must be bounded, being absolutely continuous, and on the bounded interval [0, H/H^ ] we have $ ( n + 1 ) bounded wherever defined, it follows f rom (4.16) that AFn must lie in L°°(T) (where as before we use the convention of L e m m a 4.3) so that by (4.17) it follows that A n + 1 / £ B M O A , and so 6 B M O A , by the argument which we have used twice before in order to pass from estimates on the A3 f to ones on the f(J\ B u t , B M O A is contained in the Bloch space, and a derivative g' of a holomorphic funct ion g is in the B loch space if and only if g £ A * (see, e.g., [100, vol . I, p. 163]) so that / ( " ) £ A * . • 4.3. The strict analytic radial increase property ( S A R I P ) Definition 4.1. We say that $ : C -» E has the strict analytic radial increase property (SARIP) if the fol lowing condit ions are satisfied: (i) $ is Bore l measurable (ii) $ is lower semicontinuous at 0 (iii) whenever / £ 2$ has / (0 ) = 0 and / ^ 0 then we have j * ( A / ) > I $(/) for every A > 1. 237 Chapter III. Functionals on a set of domains and on Dirichlet spaces Definition 4.2. We say that <£: C —> R has the smooth strict analytic radial increase property (SSARIP) if it has the SARIP and the following additional conditions are satisfied: (i) ^ G C 1 ^ 2 ) (ii) if / G 33 has /(0) = 0 and / ^ 0 then we have I (/,(Vd>)(/))>0, (4.18) • JT providing | (/, (V$)(/)) | G LX{T). On the left hand side of (4.18), we use (•, •) for the Euclidean inner product on R 2 and we make the identification C = R 2. Note that non-trivial positive linear combinations of functions having SARIP (respectively, SSARIP) also have SARIP (respectively, SSARIP). The properties SARIP and SSARIP will make useful assumptions later in the study of the extremals of the A$. Remark 4-3. Let <h have the SARIP (respectively, SSARIP). Let I be a linear functional from R 2 to R. Then $ + L has the SARIP (respectively, SSARIP). To see this, note that f (* + L)(f)= f <&(/) + I L(f)= [ <&(/) + (>, J T J T J T J T for all / G 33 with /(0) = 0 since L o f is harmonic for / holomorphic so that JjL(f) = L(/(0)) = L(0) = 0. The following two examples are easy to verify. Example 4-1- If A H * <b(\z) is strictly increasing on [0, oo) for every z £ C\{0} then # has the SARIP. In particular, if &(z) = (p{\z\) for <f> strictly increasing on [0,oo), then $ has SARIP. Example 4-2. Let 0 be a function on R which is strictly increasing on [0, oo) and strictly de-creasing on (-00 ,0] then <&(z) — <h(Rez) has SARIP. 238 Chapter III. Functionals on a set of domains and on Dirichlet spaces Finally we have the following example. Example 4-3. Let cp be a convex function on R which is strictly convex at 0. Then $(z) = cp(Re z) has SARIP. To see this, by strict convexity at 0 choose a k 6 R such that d>(0) + kx < 4>(x) for all i ^ O . Let ip(x) = <f>(x) — kx — <p(0). This is a convex function which is everywhere strictly positive except at 0 where it vanishes. I claim that ip is strictly increasing on [0, oo) and strictly decreasing on (—oo,0]. It suffices to prove the strict increase on [0,oo) since the other strict monotonicity property follows by applying the same argument to <£(—(•)) and ip{—{•))• To obtain a contradiction, suppose that there exist 0 < X\ < x2 < oo such that ip(xi) > ip(x2). By convexity, the line joining (xi, ip{x{)) with (x2, f(x2)) will be below the graph of / except over the interval [xi,^]- In particular, this line will be below (0,ip(0)). But since ip(xi) > ip(x2), it follows that this line is above the point (0,ip(x2)). Hence, ip(x2) < ip(0) = 0, which is a contradiction to the strict positivity of ip away from 0. Hence, ^ (z) = ^(Rez) has SARIP by Example 4.2. But $ — 3>(0) differs from \P only by a linear functional so that by Remark 4.3 it follows that $ - $(0) has SARIP. It follows that $ has SARIP. We may modify the preceding example as follows. Example 4-4- Let cp £ C 1 ( R ) be convex and strictly convex at 0. Then &(z) = cp(Rez) has SSARIP. To see this, note first that by the previous example it has SARIP. Then, by adding a linear function if necessary, we may assume that </>'(0) = 0. Let / G 23 be such that / ^ 0. Then, Now, because (p'(0) = 0 and cp is convex, it follows that cp'(x) > 0 for x > 0 and cp'(x) < 0 for x < 0. Hence, (Re f)cp'(Re /) is everywhere non-negative on T and vanishes only where cp'(Ref) vanishes. Thus, (4.19) is non-negative. To obtain a contradiction, assume that it vanishes. Then, cp'(Ref) vanishes almost everywhere on T. Now, strict convexity of cp at 0 implies that either cp'(x) > 0 for all x > 0 or cp'(x) < 0 for all x < 0 or both. Moreover, / does not almost everywhere vanish on T and / has mean zero over T so that neither / + nor / ~ can almost everywhere vanish on T . Hence, cp'(Ref) cannot almost everywhere vanish on (4.19) 239 Chapter III. Functionals on a set of domains and on Dirichlet spaces T because <f>' is non-zero on at least one of the two half lines (—oo,0) and (0,oo). Part of the usefulness of SARIP lies in the following proposition. P r o p o s i t i o n 4 . 1 . Fix a £ [0,oo). Let $ be any real-valued SARIP function on C such that A$ achieves a maximum over 23a at f £ 23a. Then ||/||©Q = 1. Proof. By SARIP the function A i-> A$(A • Id) is strictly increasing on (0,1], where Id is the identity function on D. Taking the limit as A —>• 0 and using the lower semicontinuity of <E> at 0, we see that in fact A H-> A$(A • Id) is strictly increasing on [0,1]. Hence, A$(0) < A$(Id). Hence, our extremal / cannot be the zero function. I f 0 < H/ltaa < 1 t h e n let A = | | / | | ^ > 1. We have Xf £ 23a. By SARIP we have A « ( A / ) > 4 . 4 . S o m e e x t e n s i o n s In this section we consider the more general A$ functionals for $ a real function on C. The results of §4.1 and §4.2 will essentially be special cases of the work of the present section. Let $ : R 2 —> IK. When convenient, we shall make the identification C = R 2. In the foregoing, (V<&)(/) will indicate V<& evaluated at / , where V$ is the gradient. We denote the euclidean inner product of x, y £ R 2 by (x, y). Let Vo and z be as in Theorem 4.1. The following result generalizes Theorem 4.1. T h e o r e m 4.3. Let $ : K 2 —> R. Assume that A$ is defined on 23, and that $ has first partials <&j everywhere on R2, satisfying \&,j(z)\ < cec^2, for every z £ R 2, where j — 1, 2, and c and C are any finite constants. Suppose f £ 23 is an extremal function for A$. Write contradicting the assumption that A$ achieves a maximum at / . • 240 Chapter III. Functionals on a set of domains and on Dirichlet spaces Assuming that Q $ ( / ) does not vanish, we have f £ H2 and Q*(f)zf' = P0[(y*)(f)], onT. (4.20) Moreover, if we also have < 1 then Q $ ( / ) must vanish. Remark 4-4- The proof of the last sentence of the Theorem is an easy to just i fy differentiation of the funct ion r A $ ( r / ) at r = 1. Note also that if $ satisfies S S A R I P then Q $ ( / ) cannot be equal to zero. For , by S S A R I P if Q$(f) = 0 then / = 0, while 0 cannot be extremal by S A R I P and Propos i t ion 4.1. One proves the rest of the Theorem by using exact ly the same variat ional methods as in the proof of Theorem 4.1. It should also be noted that at the point at which we had previously asserted that if / is extremal then so is —if, one must modify the remark to note that if / is extremal for A$(.) then —if is extremal for A$(,(.)). The careful reader wi l l note that Theorem 4.3 is not a complete generalization of Theorem 4.1. F i rs t of a l l , Theorem 4.1 had a discussion of what happens when S<p(/) vanishes. Secondly, Theorem 4.1, when translated into the language of Theorem 4.3, d id not require the first partials of $ to exist at zero. We may explain this as follows. The proof of Theorem 4.1 which was adapted to yield Theo-rem 4.3 sti l l goes through if we only assume that first of a l l , $ j exists on R2\E for some set E such that \{el6 : f(el$) £ E}\ = 0, and that secondly, $ is Lipschi tz on compact subsets of some e-neighbourhood of the image of / . In Theorem 4.3 we in effect took E = {0} and noted that if | / | £ E on a set of posit ive measure then / = 0 and the results are t r iv ia l then. Hence, if we wish we could loosen the condit ion of the differentiabil i ty of $ at zero. The fol lowing result is a generalization of Theorem 4.2, which, as we had noted, was due to joint work of Matheson and Pruss [75]. T h e o r e m 4 .4 ([75]). Fix n > 0. Let $ : C —>• R . Assume that A $ is defined on *B, and that $ has first partials $ j everywhere on R 2, satisfying |$j(,z)| < ceG^ for every z £ K 2 , where 241 Chapter III. Functionals on a set of domains and on Dirichlet spaces 1 < j < 2 and c andC are any finite constants. Assume that 4> satisfies SSARIP. Suppose that all the (pure and mixed) partials of $ of order < n exist. If n > 0 then assume further that all the (pure and mixed) nth order partials are Lipschitz on compact subsets of R 2 . Suppose f G 25 is an extremal function for A $ . Then f is n times continuously differentiable on T . Furthermore, the nth derivative of f on T is absolutely continuous and lies in A * ( T ) . Remark 4-5. It is in fact possible to obtain analogues of Theorems 4.3 and 4.4 for 25 a in place of 25, for a > 1. The methods are essentially the same, except that the variat ional equation (4.20) has OO Xy/(n)*n n=l in place of zf. Indeed, this much wi l l even work for ct G (0,1), and so we can f ind an analogue of Theorem 4.3 for al l a > 0. O f course, for a < 1, the square exponential growth condit ion on $>tj(z) wi l l have to be replaced by a polynomial condi t ion, wi th the degree of the polynomial depending on a and determined by use of Theorem 3.3 and the requirements of the proof. O n the other hand, for a > 1 the square exponential growth condit ion can be dropped, and replaced by boundedness on compacta . The proof of the modif ied Theorem 4.4 for 2 5 a , a > 1, is much the same as that of the ordinary Theorems 4.2 and 4.4. The only th ing to note is that on the left hand sides of various equations instead of having Akf we wi l l have oo A f c _ 1 ^ naf(n)zn. 71=1 However, this does not affect the conclusions very much, since if this quant i ty lies in Z. 2 (T) then a fortiori so does Akf. Ac tua l ly , we can do better: we can conclude that G L2(T), where [a\ is the smallest integer greater than or equal to a. We can also conclude that / is n + [a\ — 1 t imes continuously differentiable. Since these various extensions are not of great interest to us at present, we leave the details to the reader. 242 Chapter III. Functionals on a set of domains and on Dirichlet spaces 5. Symmetric decreasing rearrangement and Dirichlet norms Let / £ Da for a £ [0,oo). Then / has finite nontangential boundary values almost everywhere on T since o a C o 0 C h2. Let f® be the funct ion on T given by the symmetr ic decreasing rear-rangement of n.t . l im / (see §1.6.2). Since / £ L2(T), we must likewise have / ® £ L2(T). Moreover , / ® has mean zero on T since / does and since /® and / are equimeasurable. (Given equimeasurabi l i ty, use Remark 1.2.2. Equimeasurabi l i ty follows from Remark 1.6.7.) Let / ® : D -> E be the Poisson extension of f® : T -> E v ia the Poisson integral (Theorem 1.3.3); we wi l l then have / ® £ h2(TS>). T h e o r e m 5 . 1 . Let f E.Da for 0 < a < 2. Let f® be as above. Then, l l / @ l k < l l / l k - (5-1) Suppose moreover that 0 < a < 2 and that equality holds in (5.1). Then f is a rotation of f®, i.e., there exists w £ T such that f(z) = f®(zw) for all z £ D. The proof wi l l be given later. Remark 5.1. For a = 0, we always have equali ty in (5.1), since both sides of (5.1) are then equal to the L2(T) norm, and symmetr ic decreasing rearrangement preserves L2 norms. Remark 5.2. If ce = 2 then it is well known that there exists / £ da such that / is not a rotation of / ® but | | /®| |o Q = | | / | | n Q . To see this, note that for any function g £ C ^ D ) D U 2 we have where g'(eie) = j§g(el6). (This is easy to see by wr i t ing the right hand side of (5.2) in terms of the Fourier series expansion of g'.) Now, let fi and / 2 be functions in CX(T) wi th the following propert ies: (i) fi and / 2 are symmetr ic decreasing (ii) fi is constant on Aid={el6 : |0| < 7r /2}, while / 2 is constant on A 2 = f { e ^ : |0 —7r| < 3VT /4 } (iii) fi + fi has mean zero. 243 Chapter III. Functionals on a set of domains and on Dirichlet spaces Now, define f(eie) = fi{eie) + f2(e^9+^^^). Then , f®=f1 + f2. Th is is easiest seen in light of F igure 5.1 by compar ing sizes of level sets of / and of f\ + f2. It is moreover clear that / is not a rotat ion of / ® . Let g2(et6) - f2(ei(e+(>*/4V). Then g2 is constant on e~iw/4A2, while / i is constant on Ai, so that the supports of g'2 and /{ are disjoint. Likewise, f2 is constant on A2) so that the supports of f2 and f[ are disjoint. Hence, r2ir r2ir r2TV / \f'(ei8)\2d6= \fl(eie)\2d0+ \9'2(ei6)\2d9 Jo Jo Jo = r\fi(el9)\2d9+ r\f'2(ew)\2 d6 Jo Jo = r\U@)'(e^)\2d6. Jo From (5.2) we conclude that | | /®||o 2 = | | / | |o 2 a s desired. Now, given an analyt ic funct ion F £ S a for some finite a £ [0, 2], let f = KeF. Then , / £ 0 a , and ||/ | |?) a = | |F | | iD a (Remark 1.2). Let / ® be the symmetr ic decreasing rearrangement of / as before; then / ® £ Da and there exists a unique funct ion G £ D „ such that R e G = / ® ; we wil l have | |G| | io a = | | /®| |o Q - (We put G = Vf® in the notat ion of §1.3.3.) We then define F® = G and can reformulate Theorem 5.1 as follows. C o r o l l a r y 5 . 1 . Let F £ ® a for 0 < a < 2. Let F® be as above. Then, l l ^ ® l k < l l ^ l k -Suppose moreover that 0 < a < 2 and that equality holds in (5.1). Then F is a rotation of F®, i.e., there exists w £ T such that F(z) = F®(zw) for all z £ ED. The funct ion F® is in fact univalent on D and the image F®[D] is Steiner symmetr ic. Th is follows f rom the fol lowing Propos i t ion . P r o p o s i t i o n 5 . 1 . Let f £ Hr(D) be such that Re / is symmetric decreasing on T . Assume f is not constant. Then f is univalent and the image /[D] is Steiner symmetric about the real axis. 244 Chapter III. Functionals on a set of domains and on Dirichlet spaces -71 -371/4 -71/2 -7l/4 0 7T/4 7T/2 37l/4 7C -71 -371/4 -7T/2 -7l/4 0 7l/4 7C/2 37T/4 71 -71 -371/4 -7C/2 -7l/4 0 7C/4 7T/2 37T/4 71 Note: appropriate constants should be added to the functions to ensure that/ has mean zero. Figure 5.1: The functions fi, f2, f and / ® 245 Chapter III. Functionals on a set of domains and on Dirichlet spaces The proof will be given later. Corollary 5.2. Let cp be a function on R such that $(z) = (^Re z) has SARIP. Fix 0 < a < 2. Suppose that A$ attains its maximum over Q3a at f £ 2$a. Then, there exists a w £ T suc/i i/iai z H-7- Re/(zw) symmetric decreasing on T (i.e., Re/ is the rotation of a symmetric decreasing function). Moreover, the image /[D] is Steiner symmetric and f is univalent. Proof of Corollary 5.2. Suppose that Re / is not the rotation of a symmetric decreasing function in the sense specified in the statement of the Corollary. Then, ||/®||:oQ < ||/||5)Q < 1 in light of Corollary 5.1. But A$(/®) = A$(/) by Proposition 1.2.1 since T has finite measure. Hence, A$ attains a maximum at /®. But by Proposition 4.1 we then obtain a contradiction since Il/lb. < I-Hence Re / is the rotation of a symmetric decreasing function, i.e., g(z) = Re f(zw) is symmetric decreasing. Moreover, by Proposition 4.1 we have / ^ 0 so that the rest of the conclusions of the theorem follow by applying Proposition 5.1 to g. • Proof of Proposition 5.1. First, fix r £ (0,1). Let fr(z) = f(zr). Put u = Re/ r and v = Im gr. By Corollary 1.6.3, the function u is symmetrically strictly decreasing on T. Now, let Dr = /r[D]. We shall prove that Dr is Steiner symmetric about the real axis and that fr is univalent on D. It will then follow that / is univalent on D(r). Taking r —> 1— it will follow that / is univalent on D. Moreover, /[D] = Uo<r<i Dr, and the union of Steiner symmetric domains is Steiner symmetric, so that we will thus be done. Note that by another limiting argument it suffices to consider only those values of r for which / has no zeroes on T(r). The symmetry of u implies that u(z) = u{z) for all z £ D. It follows from Remark 1.3.2 that v(z) = —v(z) since v is the conjugate function of u. In particular, it follows that u(l) = v(—1) = 0 and that Dr is symmetric under reflection in the real axis. I first claim that it is false that for every 9 £ (0,7r) we have v(el8) < 0. For, suppose that indeed v(elS) < 0 for each 9 £ (0,7r). 246 Chapter III. Functionals on a set of domains and on Dirichlet spaces We have assumed that fr has no zeroes on T. Hence, since u is symmetric strictly decreasing while v(eie) < 0 for 0 £ (0,VT) SO that v(eie) > 0 for 9 £ ( -TT ,0) since v(z) = -v(z), it follows that fr(et6) winds once around 0 in the negative direction as 9 goes from 0 to 2n. But this is impossible if fr is to be analytic. Hence we see that v(et9) cannot be non-positive for every 0 G ( O , 7 r ) . I now claim that v(et9) > 0 everywhere on (0, n). For suppose that on the contrary there exists 9o £ (0,TT) such that v(el9°) < 0. Since we have already seen that v(et9) cannot be non-positive for every 9, it follows that it must be strictly positive at some 9 — 9i £ ( 0 , 7 r ) , and by continuity it follows then that there must be a 9 £ (0,7r) for which v(e%9) = 0. We shall prove that this leads to a contradiction. For, let x = u(et9). We have x £ Dr. Let y = sup{y £ K : x + iy £ Dr}. If y = —oo then the vertical line at abscissa x never meets Dr and so x must either be the maximum or the minimum of u(et(p) so that 6 £ {0, n} by the strictness of the symmetric decreasing character of u, and so we have a contradiction as 9 £ (0 ,7r). Since Dr is an open bounded domain symmetric under reflection in the real axis, it follows that 0 < y < oo. Moreover, x + iy £ Dr = / r [D] so that there is a sequence zn £ D such that fr(zn) —>• x + iy. Passing to a subsequence if necessary, we may assume that zn converges to some point z of D. By continuity of fr on D we have fr{z) = x + iy. First suppose that z £ D. Then, since fr is non-constant it follows that fr(z) fi dDr since fr is an open mapping, and we have a contradiction since x + iy £ dDr. Suppose now that z £ dD. Then x + iy = u(z) + iv(z). Write z = eiv for 9 £ ( - 7 T , TT]. We have x = u{eiv). But x — u(el9). Since u is symmetrically strictly decreasing it follows that (p — ±9. But v(e'9) = 0 and v(e~'i9) = 0 likewise since v(Z) = —v(Z). Hence, v(e%v) = 0. Hence y = 0. But this is an immediate contradiction since we have already seen that y > 0. Thus, indeed v(et9) > 0 on (0 ,7r), and v(ei9) < 0 on (—7r,0). Since u is symmetrically strictly decreasing it follows that u + iv is one-to-one on T. By Darboux's theorem it follows that fr = u + iv is univalent in D. We must now show that Dr is Steiner symmetric about the real axis. We have already noted that Dr is reflection symmetric about the real axis. Fix x + iy £ Dr 247 Chapter III. Functionals on a set of domains and on Dirichlet spaces with y > 0 and i £ K , We must show that the line segment joining x + iy with x — iy lies in Dr. Let yi = sup{«/' : x + iy' £ Dr}. As before, we have x + iyi £ dDr and x + iyi = fr(zi) for some z\ £ T. Moreover y £ (0, j/i). To obtain a contradiction, suppose that there exists y' £ (—y,y) such that x + iy' £ Dr. Let y2 = sup{y' < y : x + iy' £ Dr}. We then have y2 £ (—y, y) and x + iy2 G dDr. As before, it follows that x + iy2 = fr(z2) for some z2 £ T. Then, it follows that u(z2) — u(zi) = x. Hence, z\ = z2 or = ~z2. In either case it follows that |u( i^)| = |u(^2)| because of the reflection antisymmetry of v. Thus, |yi| = \y2\. But this contradicts the facts that y £ (0, j/i) and y2 £ (-y,y). Hence we see that Dr is indeed Steiner symmetric. • We now proceed to the proof of Theorem 5.1. Proof of Theorem 5.1. Assume that 0 < a < 2, since the case a = 0 is trivial, and the case a = 2 can be obtained from the case a £ (0, 2) by taking the limit as a —> 2— and noting that t h e n \\f\Uc, l l / l k f o r e v e r y fixed / e D 2 -We use ideas from a proof of the Stein-Weiss interpolation theorem [22, Chapter 5]. For / £ Do, define ^(/,i)= inf ll/ollio+tll^Hi, Jo+h=J where the infimum is taken over all decompositions f = fo + fi where /,• £ D8- for i £ {0, 2}. We compute an explicit formula for £ | (/,£)• To do this, note that elementary considerations show that for any fixed w > 0, t > 0 and a £ K the quantity (a — x)2 + twx2 attains its minimum over x £ K precisely at a x = 1 + tw Let j? = {cn : n £ Z+} U {sn : n £ Z+}. 248 Chapter III. Functionals on a set of domains and on Dirichlet spaces For t G let n(t) = m if 6 = cm or t = sm. Then, J > « ( t ) 6 2 ( / ) . Letting iy = n2(£), a = t(f) and a; = fi(/2), it follows that the infimum in the definition of &2(f,t) is actually attained and ^(/,*) = ll/-/2 |li 0+*||/2 | |g 2, where for every t 6 we have «<A) = t ( / ) l + in2(6)' so that (We use the short forms n"(C) = (n(E))p and C( / ) = (e(/))p.) Thus, = v^n 2(E)€ 2(/) fe(i + '»2(e)" (5.3) Suppose now that <f> is a smooth non-negative function on (0, oo). Using Fubini's theorem and then making the change of variable 1 + tn2(t) = u we obtain £( / , dt = ^) « 2 ( / ) (u - l)u~ V (^y) n"2(«) (5.4) Now, let <^ (£) = cgt~1~9 for a positive constant c#. Then, it is easy to see that the right hand side of (5.4) will be of the form YJc^2em\f)=ce\\!\\i2e, where /oo (tt - l)tt_1(tt - I)'1'8 du. 249 Chapter III. Functionals on a set of domains and on Dirichlet spaces It is easy to see that for 0 < 9 < 1 this integral is finite. Then , choose eg so that C$ = 1, and so we wi l l have roo / 4(f,t)4>(t)dt=\\f\\la, (5.5) Jo where a = 28. I now c la im that for 0 < t < oo we have £%(f®, t) < fi%(f, t) wi th equality if and only if / is a rotat ion of f®. Once we prove this c la im, we wi l l be done by (5.5). Rewr i te (5.3) as t2f ^2(/>o = £( e 2 (/)-r - + tn?(t) = 2 | | / | | 2 r 2 ^ - ^ T * 2 / LHJ) Z . 1 + i n 2 ( e ) -teA w O f course | | / | | L 2 ( T ) = | | / ® | U a ( T ) . Let t2f It tea w for 0 < t < oo. To prove the c la im we now need only show that Jt(f) < 7 t ( /® ) with equality if and only if / is a rotat ion of f®. Define CO Kt(eie) = Y —^ cos n9. n=l Then , 7*(/) = 2</,/<t*/), where (•, •) is the inner product in L2(T). Bu t it is not difficult to verify that R , i B ) = acosha(n-d) _ 1 t { ' 4s inh7ra 2 ' for 0 < 9 < 2n, where a = t~ll2. Thus , Kt is symmetr ical ly str ict ly decreasing. Thus , the inequali ty 7o( / ) < 7o( /®) follows from Beckner 's rearrangement theorem (Theorem 1.6.4). • 250 Chapter III. Functionals on a set of domains and on Dirichlet spaces 6. Baernstein's sub-Steiner rearrangement We now define Baernstein 's sub-Steiner rearrangement [12]. The terminology is slightly mis-leading in that it is actual ly not known whether this is a rearrangement in the sense of Defini-t ion 1.2.2. Let D be an Hp domain containing the origin for some p > 1. Let / : D —> D be a uniformizer wi th / ( 0 ) = 0 (see §1.5.5). Then / £ HP{D). Let f® be the unique Hp funct ion such that f® (0) = 0 and Re / ® is the symmetr ic decreasing rearrangement of Re / . Here we use the M . Riesz theorem (Theorem 1.3.4) which asserts that the conjugate function of an LP(T) func-t ion lies in LP(T), for p > 1, while of course R e / ® £ LP(T) since / £ LP(T). In light of Propos i t ion 5.1, the funct ion / ® is univalent and its image is Steiner symmetr ic about the real axis. Let £> B = / ® [ D ] . We cal l DB the B a e r n s t e i n s u b - S t e i n e r r e a r r a n g e m e n t o f D. T h e o r e m 6 . 1 . Let D be a Greenian FLP domain containing the origin for some p > 1. Let cp be any Borel measurable function on D such that Y$(D) is defined for 3>(z) = ch(Rez). Then, r$(DB) = r*(D). Proof. We have / ® ( 0 ) = 0. Moreover, f® is a uniformizer (in fact, it is a Riemann map since DB is s imply connected) and has Steiner symmetr ic image. Hence, by Coro l la ry 1.1, the domain is an Hp domain since / ® £ Hp. In part icular, it is a Nevanl inna domain. The image of f® cannot be al l of C (otherwise the inverse funct ion ( / ® ) _ 1 is a bounded non-constant entire funct ion), and it is easy to see if that z £ dDB then the Steiner symmetry guarantees that Theorem 1.5.3 is appl icable (just take 6(r) = (7r/2) sgn Im z if Im z ^ 0, and 6(r) = 7 r / 2 ± (7r/2) sgnRe z if I m z = 0; of course z ^ 0 since 0 £ DB), and so DB is Greenian. Hence, Theorem 1.5.7 is applicable and r $ ( D B ) = / * ( / © ) = / <KRe/®) = [ <KRe/) = I *(/) = U{D), JT JT JT JT providing that T$(D) and T$(DB) are f inite. (The first and last equality followed f rom Theo-rem 1.5.7. The th i rd equali ty followed f rom the definit ion of / ® and Proposi t ion 1.2.1.) In case 251 Chapter III. Functionals on a set of domains and on Dirichlet spaces one of r$(73) and T$(Z) B) is infinite, then they both must be infinite and of the same sign, because we can approximate our <h by bounded functions. • Let B now indicate Steiner symmetr izat ion about the real axis. Then , we know that T$(D) < T$(DB) whenever &(z) = <f>(Rez) for a convex funct ion <f> (Theorem 1.6.6). Since T$(D) = r$ (D B ) and DB is Steiner symmetr ic , it would seem that DB is some "smal ler" version of the Steiner symmetr izat ion. In fact, we have the fol lowing result, which provides an answer to a question of Baernstein [12] and justif ies the name s u b - S t e i n e r as applied to the (-) B operat ion. The main result of the present section is as follows. T h e o r e m 6 .2 . Let D be an Hp domain containing the origin for some p > 1. Then DBCDB. (6.1) More generally, if f is any holomorphic function whose image is contained in D, then /® [D] C DB. Since A r e a ( J 9 B ) = A r e a ( D ) , we obtain the fol lowing Corol lary. C o r o l l a r y 6 . 1 . Let D be an Hp domain for some p > 1 with 0 G D. Then, A r e a ( D B ) < A r e a ( D ) . Recal l that B is the collection of al l domains of area at most ir which contain the origin. C o r o l l a r y 6 .2 . Let <f> be a measurable function on E such that A $ attains a maximum over 03 where $ ( z ) = 4>(Rez). Then either r$(D) < $(0) for all D G B or there is a univalent function f G 03 with Steiner symmetric image such that T$ attains a maximum over B at / [D] . Note that unfortunately do not have any uniqueness result, even under some assumption such as S A R I P . It would be nice, for instance, to be able to say that all domains at which T$ attains its max imum are of the form /[D] where / is as in the Corol lary. « 252 Chapter III. Functionals on a set of domains and on Dirichlet spaces It would also be nice if we knew that if A$ attains a maximum over 5$ at / 6 2$ then T$ attains its maximum at /[D]. Note that it is easy to see that the condition T$(D) < #(0) for all D £ B cannot happen if, for instance, <f> has a minimum at 0 and is not constant. (Just choose D a domain of area n which has the property that it reaches far enough so that a part of its boundary with strictly positive harmonic measure lies in {z : <&(z) > $(0) } so that T$(D), being a weighted average of $ with the weight given by harmonic measure, must be strictly larger than $(0).) Proof of Corollary 6.2. Let / i € 03 be such that A$ attains its maximum at f\. Assume first that f\ ^ 0. Let / = ff. Then / is univalent with Steiner symmetric image. Moreover, / £ 23 (Corollary 5.1) so that /[D] £ B. I claim that T$ attains a maximum over B at /[D]. To see this, let D £ B. Then, r * (£> B ) = r*(£>). Let h be the Riemann map from D onto DB with h(0) = 0. By Theorem 1.2 we have r$ (D B ) = A$(/i). But AreaDB < AreaD < TT so that h £ 23 as h is univalent and has image area at most ir. Thus, A*(h) < A $ (/) . But / is univalent also, so that A*(/)<r*(/[o>]). Putting all the preceding displayed inequalities together we see that r*(D)<r*(/[D>]), and we conclude that indeed F$ attains a maximum at /[D]. Suppose now that f\ = 0. Then, A$ is bounded above by $ (0 ) . The same argument as above then shows that T$ is bounded above by $ (0 ) . • 253 Chapter III. Functionals on a set of domains and on Dirichlet spaces To prove Theorem 6.2 we now define the Nevanlinna counting function. Given an analytic function F on D and r £ (0,1), let n(r,w;F) be the number of solutions z £ D(r) of the equation w = F(z), counting multiplicities. The Nevanlinna counting function then is: NF(w) = / rc(r, w; F) r~1dr. Jo Remark 6.1. The function NF vanishes at w £ C if and only if w fi F[D]. We recall the following very useful theorem of Baernstein. Use B to denote Steiner symmetriza-tion about the real axis. Write Np for (NF)B. The function /® is defined as before. Theorem 6.3 (Baernstein [8]). Let F £ H1^). Then, pY fY J ^Nf{x + iy) dy< J NF® (x + iy) dy, (6.2) for every Y £ (0, oo], with equality for Y — oo. Assume this Theorem. The following Lemma is also useful in conjunction with it. Lemma 6.1. Let F £ HP(J}) for some p > 1. Then, /oo NF(x + iy) dy < oo, •oo for every x £ E . Proof of Lemma. Note that /oo poo NF{x + iy)dy= / Nf{x + iy)dy. •oo J—oo This follows by the fact that (for fixed x) B is a measure preserving rearrangement when restricted to x + iE (with respect to one-dimensional Lebesgue measure) and we can apply Proposition 1.2.1 (ii)- Hence since Theorem 6.3 guarantees equality for Y — oo in (6.2), we need only prove the lemma for F® in place of F. Let G = F®. Then G £ Hp since F £ Hp. As in the proof of Theorem 6.1, we have G a univalent function onto a Steiner symmetric Greenian domain. Define <p(2r) = max(0, (Rez) - x). 254 Chapter III. Functionals on a set of domains and on Dirichlet spaces This is a subharmonic function (Theorem 1.4.3) and we may write LHM(2 ,$;D) = $(z)+ / g(z, w; D) dp,$(w). JD Now, by Theorem 1.5.11 we then have /oo g{z,x + iy;D)dy, (6.3) • C O for some strictly positive constant c (we used here the fact that g(z, -;D) vanishes outside D to extend the range of integration from {y : x + iy G D} to all of E). By Theorems III.1.2 and 1.5.4 we have LHM(0,$;£>) = r*(£>) = A*(G) = f max(0, (ReG) - x). JT The right hand side is finite since G G HP(B) for some p > 1 so that G G LP(T) (all we need is G G H1 here in fact). Since $(0) is finite, it follows from (6.3) that /oo g(z, x + iy; D) dy < oo. (6.4) -oo But now G is univalent. Thus, n(r, w;G) = l { r > | G - i ( ^ ) | } , and so * c ( w ) = logJG=W Hence NQ(W) = g(w,0;D) by Theorem 1.5.8. Since g(w,0;D) = g(0,w;D), we are done by (6.4). • Proof of Theorem 6.2. We need only prove the "Moreover", since the rest follows from the special case where / is a uniformizer of D sending 0 to 0. Fix XQ G E. Let LQ = {xo+iy : y G E}. Put L — LQ n DB and M = LQ D D B . Write Ai for one-dimensional Lebesgue measure. We must prove that L C M. Because both DB and DB are Steiner symmetric, it suffices to prove that Ai(L) < Ai(M). 255 Chapter III. Functionals on a set of domains and on Dirichlet spaces If Ai(M) is infinite, then we are done. Hence, suppose that Yi= f|Ai(M) is finite. Since Nj vanishes outside D it follows that Nf can be nonzero on a set of measure at most 2Yi, and hence NB vanishes on L\{xo + iy : \y\ > Yi} by definition of the Steiner rearrangement. But we have equality in (6.2) for Y = oo so that it follows that On the other hand by (6.2) with Y — Y\. Since Nj® > 0 and by Lemma 6.1 all our integrals are finite, it follows that Now, L is precisely the set of points of LQ at which Nj® is non-zero. It follows that AI(JL) < 2YX = Ai(M) as desired. • 256 Chapter I V Radial rearrangement Overview In this chapter, "doma in " shall mean "Greenian doma in " . Our interest in this chapter is a conjecture of Matheson and Pruss [75] to the effect that there exists a way of replacing a domain D by a simply connected and star-shaped domain D of not bigger area such that the T$ functionals are increased by the replacement, where $ is a continuous funct ion on [0, oo) such that t 1-4 <&(e*) is increasing and convex. A positive answer to this conjecture would allow for a t ighter connection between the A $ and T$ functionals. Unfortunately, we do not manage to obtain an answer to the conjecture, although we do get some part ia l results and some interesting evidence. We state the relevant conjecture and discuss it in §1.1. In §1.2 we discuss some consequences of our conjectures. Then , in §1.3, we state a conjecture (Conjecture 1.4) as to how we think D should be defined v ia circular symmetr izat ion and Marcus ' radial rearrangement [70], and we discuss a few cases in which the conjecture holds. We state a weaker conjecture concerning harmonic measures (Conjecture 1.6). We also give a few counterexamples contradict ing some possible extensions of our various conjectures. It is unfortunate that al l our main conjectures remain open. However, in §2 we do give some part ia l results. For instance, we show that the harmonic measure functionals wr are increased by our conjectured choice of D providing the original D is simply connected (Theorem 2.3). The proof uses simple connect iv i ty in an essential way. 257 Chapter IV. Radial rearrangement In §3 we discuss our problems as transferred to the cylinder. This makes the constructions and conjectures a little more intuitive. We also discuss two-sided lengthwise Steiner symmetrization, and obtain a partial result (Theorem 3.1) analogous to our Conjecture 1.6. Finally, we discuss a "cutting" operation which one might conjecture to increase the wr functionals for circularly symmetric domains. In §4 we discuss a formulation of our Conjecture 1.4 in terms of Green's functions, and discuss why our Conjecture 1.6 on harmonic measures is weaker than Conjecture 1.4, and how Marcus' result [70] on the increase of the inner radius under radial rearrangement is connected with our Conjecture 1.4. Then, in §5 we generalize Marcus' above-mentioned result, proving our Conjecture 1.4 in the special case of t ^ ^$(e*) being concave. In §6 we discuss a one-sided Steiner rearrangement due to Haliste [56] and state her result on the effect of this rearrangement on harmonic measures (Theorem 6.1). Then, in §7, we discuss a Brownian motion formulation of our conjectures. Having done so, we state a rather natural one-dimensional discrete version of Conjecture 1.6 which we shall prove in §9. We note that while this one-dimensional discrete version holds not only for a simple random walk, but also for a random walk with probability p of going to the right and 1 — p of going to the left, this is not the case with Conjecture 1.6. Indeed, in §7.1 we prove that the analogue of the cylindrical Brownian motion version of Conjecture 1.6 fails when the lengthwise component of the Brownian motion is replaced by a uniform motion to the right. This seems to provide some evidence against Conjecture 1.6, although the particular counterexample domain produced here does have Conjecture 1.6 satisfied because of simple connectivity and Theorem 2.3. Finally, in §7.2 we prove that a somewhat natural conjecture about Brownian motion exit times, analogous to Conjecture 1.4, is false. However, this too does not really provide evidence against Conjecture 1.4, but is mainly a consequence of the fact that radial rearrangement does not preserve areas. Most of the material in sections 1-7 is taken from the author's paper [85]. 258 Chapter IV. Radial rearrangement In §8 we prove extensions of the Beur l ing shove theorem on the harmonic measure of slit discs [23, pp. 58-62]. In part icular , we shall prove Conjecture 1.6 in the case of a circularly symmetr ic and starshaped domain slit along the negative real axis. Our extensions of the Beur l ing shove theorem in the 2-dimensional case go further than the extensions of Essen and Hal iste [45]; however, their methods are also valid in higher dimensions, while confine ourselves to domains in R 2 . Par ts of §8 are taken from the author 's paper [84]. Then , in §9 we give the discrete one-dimensional version of Conjecture 1.6 and prove it. We also give a few other results. The discrete one-dimensional result is actual ly completely elementary, as are its proofs (broken up between sections 9.2 and 9.3). The proofs are based on an explicit formula (Theorem 9.6) for the probabi l i ty of a random walk surviv ing various dangers on its route so as to exit a bl ind alley. O u r results in §9 are simi lar to a discrete result of Essen [43], and in §9.4 we discuss the connection wi th a continuous result of Essen [42]. A l l the mater ial in §9, excepting Theorem 9.2, is taken f rom the author 's paper [87]. F ina l ly , in §10 we prove a part ia l result concerning a horizontal-convexity analogue of a weaker version of the Matheson-Pruss Conjecture 1.1. Th is result provides some evidence for that conjecture. Our proof uses the var iat ional equation for extremals of A $ given in §111.4.4. It also uses Baernstein 's sub-Steiner rearrangement f rom §111.5, though the proof could also be done wi th standard Steiner rearrangement and Steiner analogues of the results of Baernstein [7]. On first reading, the reader may wish to omit sections 5, 6 and 9.3, the details of the con-struct ions in sections 7.1 and 7.2, as well as some of the details in the proof of L e m m a 10.1 of §10.1. 259 Chapter IV. Radial rearrangement 1. Conjectures and counterexamples 1.1. The primary conjectures Throughout this chapter, let T be the collection of all functions # on [0, oo) such that t >->• <J>(e*) is convex increasing and $ is continuous at 0. By Theorem 1.4.4, T is precisely the set of functions $ such that z i—^ $(|.z|) is subharmonic on C. Consider the following conjecture. Conjecture 1.1 (Matheson and Pruss [75]). For any domain U containing the origin and of finite area there exists a star-shaped domain U with AreaC/ < AreaC/ such that r$(C/) > r$(E/) for every $ G T. A weaker conjecture but still very much of interest would be as follows. Conjecture 1.2. For any domain U containing the origin and of finite area and any fixed $ G T there exists a star-shaped domain U with AreaC/ < AreaC/ and r$(C7) > T$(C/). The difference between Conjectures 1.1 and 1.2 is that the latter allows the domain U to depend on the choice of the particular function $ G T. Recall that if U is star-shaped then it is simply connected (this is easy to verify directly since U is then contractible). Unfortunately, Conjecture 1.2 and a fortiori Conjecture 1.1 are still open. 1.2. Consequences of a positive answer to Conjecture 1.2 Let us formulate an even weaker version of Conjecture 1.2. It will be weaker because star-shaped domains are automatically simply connected. Conjecture 1.3. For any domain U containing the origin and of finite area and any $ G J-there exists a simply connected U with AreaC/ < AreaC/ and T$(C/) > r$(C/). 260 Chapter IV. Radial rearrangement This conjecture is still open, too. Let B, as before, be the set of all domains of area at most 7r which contain 0. P r o p o s i t i o n 1.1. Suppose that Conjecture 1.3 is valid for some $ £ T such that $(£) = o(et2) as t —)• oo. Then T$ attains its supremum over B. Moreover, there exists a simply connected extremal domain in B at which T$ is maximized. Proof. By Theorem 3.4, there exists / € 23 such that A$(/) > A$(g) for all g £ 23. Let D = /[D]. Let [/ be an arbitrary domain in B. Then by Theorem III.1.2 and Conjecture 1.3 we have U { u ) <u(u) = A9(g), where g is a Riemann map from D onto U with g(0) = 0. But Areaf/ < 7r so g £ 23 and hence A*(5) < A$(/). By Theorem III.1.2 we then have A$(/) < r*(£>), and so T^(U) < r*(D) for all U € B. Hence T$ attains its maximum over B. Moreover, if it attains this maximum at D, it likewise attains it at D and hence there exists an extremal simply connected domain. • Sakai [95] had conjectured that T$p attains a maximum over B where &p(t) = tp for 0 < p < 00. Hence, an affirmative answer to Conjecture 1.3 implies an answer to Sakai's conjecture. P r o p o s i t i o n 1.2. Let $p(i) = tp. Then T^p(D) < T$(D) for every D £ B providing p £ [0, 2]. If Conjecture 1.3 holds for <J>p then this is also true for p £ (2,4]. Proof. The case of p £ [0,2] is the well-known Alexander-Taylor-Ullman inequality [3] (see Kobayashi [66] for another proof). (More precisely, the Alexander-Taylor-Ullman inequality is the case p — 2, and, as Sakai [95] notes, the case p < 2 follows from Holder's inequality.) The case p £ (2, 4] follows from Conjecture 1.3 and the fact that the inequality is valid for simply connected domains D. To see the validity for simply connected domains, it suffices to use The-orem III.1.2 and the fact that A$ (/) < A$ (Id) for p £ [0, 4], where Id is the identity function and / is any function in 23. This latter inequality has been proved by Matheson [74]. Professor 261 Chapter IV. Radial rearrangement Sakai has kindly informed the author that the desired inequality in the simply connected case was also obtained by Professors N. Suita and S. Kobayashi. • The inequality in the above proposition was conjectured for p 6 (2,4] by Sakai [95]. Hence, an affirmative answer to Conjecture 1.3 would imply an affirmative answer to yet another conjecture of Sakai. As can be seen, it would also simplify the proof of the Alexander-Taylor-Ullman inequality, since the inequality A$p(/) < A$(Id) for / 6 23 is quite easy to prove for p G [0,2]. Indeed, it suffices to prove the latter inequality for p = 2 since the general case then follows by Holder's inequality. But for p = 2 the inequality is essentially trivial since then A$2(/) = T,n=i \f(n)\2 while H/lli, = E^°=i n\f(n)|2 < 1, which easily shows that A$2 attains Indeed, it follows from the Chang-Marshall inequality for univalent functions. Since the proof of the Chang-Marshall inequality given in [72] is much simpler (and it also simplifies considerably in the univalent case) than the proof of Essen's inequality [44], we see that an affirmative answer to Conjecture 1.2 would imply a simpler proof of Essen's inequality. 1.3. Radial rearrangement A tool which one would think is very natural for attacking Conjecture 1.1 is Marcus' radial rearrangement [70]. The author is grateful to Professor Albert Baernstein II for having sug-gested the use of Marcus' radial rearrangement for this purpose. Let U be a set in the plane containing a neighbourhood of the origin and choose e > 0 such that D(e) C U, where D(r) indicates an open disc of radius r about 0. Define its maximum over 23 precisely at the functions of the form c • Id where |c| = 1. Finally, we note that if Conjecture 1.2 holds, then the Essen inequality sup T$(D) < oo DeB for = e*2 follows from the Chang-Marshall inequality sup A$(/) < oo. 262 Chapter IV. Radial rearrangement Figure 1.1: A multiply connected domain for Example 1.1, together with its radial rearrange-ment. Then the set U* = {re* : J\~Ldp < Re{9;U)^ is called the (Marcus) radial rearrangement of U. Observe that we are using a logarithmic metric in the definitions. Note that D(e) C [/* and that if U is open then 6 H> RS(9;U) is lower semicontinuous (cf. Proposition 1.6.1) and hence £7* is open. It is easy to verify that £7* is independent of the choice of £ and that Area(£/*) < Area(£7) with equality if and only if £7* and U coincide almost everywhere with respect to Lebesgue area measure. Note that U* is always star-shaped and that £/* = U if and only if U is star-shaped. Finally, one may observe that is a rearrangement in the sense of Definition 1.2.2 on the rj-pseudotopology of all measurable subsets of C containing a neighbourhood of the origin. Marcus [70] has shown that if decreases neither inner radii (see inequality (4.2), below) nor certain capacities. Example 1.1. One might naturally conjecture that [7* has the desired property that r$(<7*) > r$(C7). However, this is not the case. Let U be any domain with the following properties for some positive r and 6, where £ is as before: 263 Chapter IV. Radial rearrangement 264 Chapter IV. Radial rearrangement (i) The harmonic measure of {\z\ = r} n d(D(r) f l U) at zero in U does not vanish. (ii) RE(9; U) < p~ldp for every 9. Such a domain can easily be exhibi ted; see F igure 1.1 (left) for a mult iply-connected example, and F igure 1.2 for a simply-connected example. G iven such a domain , (ii) implies that we wi l l have £ / * C D( r ' ) where r' = r — 5 (see F igure 1.1, r ight). Choose x £ (r', r ) . Define $(£) = max(0, t — x). Then $ is convex, hence in J7, and it is easy to verify that r$([/) > 0 by (i). O n the other hand, as J/* C D(r') we must have <&(|z|) vanishing on the closure of U* so that r$([/*) = 0, and thus radial rearrangement by itself does not give a solut ion to Conjecture 1.1. Now, given a domain U, let U® be its circular symmetr izat ion (see §1.6.1). We have Area(f /®) = Area ( t f ) . A n d TQ{U®) > T9(U) for every $ € T by Coro l la ry 1.6.1. Note that if U is c i rcular ly symmetr ic then so is U* though the converse does not in general hold. It is easy to see that no domain satisfying (i) and (ii) of Example 1.1 can be circularly symmetr ic . C o n j e c t u r e 1.4. Conf ined to the class of circular ly symmetr ic domains, radial rearrangement does not decrease any of the funct ionals T$ for $ G f . A n affirmative answer to this question would give an affirmative answer to Conjecture 1.1 since we could then let U = (U®)*. Wh i le Conjecture 1.4 is in general open, we can prove it for a very simple class of domains. Example 1.2. Consider the closed polar rectangle H = {reie :rl<r<r2,\9-Tr\< 90}, where 0 < r i < r 2 < 1 and #o < Let U = D\H. 265 Chapter IV. Radial rearrangement Then U is circularly symmetric. It is easy to verify that U* = T»\H', where H' = jre8'" : ^ < r < 1,\9 - vr| < 6»0 j Letting A = r^ "1, we see that we may also write (1.1) U* = J}\\H, with A > 1, and where XH — {Xz : z £ H}. Fix r < 1. We then have w(O,(0E/)n{|z| < r};U) = u[0,(dH)n{\z\ < r } ; D \ f f ) > w(0, (0 f f ) n {|z| < r}; D(A - 1 ) \ i7 ) > u(0,(dH) n {|z| < A- 1r};D(A- 1)\/Z") = w(0, (SA#)n{>| < r};D\Ar7), where we have used the monotonicity of u(z,A;V) with respect to A and V (Remark 1.5.3), together with invariance under scaling. Now, let v(r;V)=u(z,{\z\>r};V). (1.2) It is not difficult to see that if $ £ T and h is the least harmonic majorant of z i-4 $(|z|) in a domain V, then /•oo n(0) = $ ( 0 ) + / &(r)v(r;V)dr. Jo From (1.1) it follows that v(r;U) < v(r; *7*) and thus it follows that T$(U) < r$([/*) as desired. In the example above, we had a stronger conclusion than simply increase of the T$ for $ £ T. This is contained in the following conjecture. 266 Chapter IV. Radial rearrangement C o n j e c t u r e 1.5. (Known to be false!) If U is circular ly symmetr ic then v(r;U*) > v(r;U) for every r > 0, where v(-; •) is defined by (1.2). To see that this is false, let £7 = D \ ( ( - l , - i ] n 0 O ( i ) ) . Ev ident ly U is circular ly symmetr ic and [7 * = D\(—1, — | ] . Set r = ^. Clear ly, t ; ( i ; t / ) = a ; ( 0 , 5 D ( i ) ; D ( i ) \ ( - i , - I ] ) . Define g(z) = u(z,dB^y,B^)\(-l~l}), and / i ( z ) = w (z,{\z\ > i}f l9[ /*) . Then , it is easy to see that h(z) < 1 for every z G T ( ^ ) \ { —|} and that h{z) = 0 for every z G (— | , — \~\. On the other hand g{z) = 1 for each z G <9-C(|) and again g(z) = 0 for z £ (— | , — | ] . Hence the max imum principle applied in the domain D ( | ) \ ( —| , — |] implies that h(z) < g(z) for every z in this domain . In part icular v{\\U) = g(0) > h(0) = w ( | ; t / * ) . O f course, one might say that this is not really a counterexample to Conjecture 1.5 since U is not connected and domains are usually taken to be connected. Bu t we can make U connected! F i x a smal l e > 0. Let Ue = D \ ( ( - 1 , - g ] U {\eiB : £ < 9 < 2ir - e}) . The U£ are connected and circular ly symmetr ic for every positive e. Bu t as e —>• 0, we have v(^; UE) - » u ( | ; U), and certainly U* = D\(—1, - 5 ] = so that for sufficiently smal l £ we have w( | ; Ue) > u ( | ; £ /* ) = u(^ ; £ / * ) and we t ru ly have a counterexample. We now present the fol lowing bona fide Conjecture. C o n j e c t u r e 1.6. Let U be a circular ly symmetr ic domain, and let Ur — D ( r ) n t / . Let wr(U) — u(0,dB(r)ndUr;Ur). Then wr{U) < wr{U*). 267 Chapter IV. Radial rearrangement Note that by Theorem 1.6.2 (which is due to Baernstein [7]), an analogue of this for circular symmetr izat ion holds, i.e., wr(U) < wr(U®) for any domain U containing the origin. Since wr(U) = vr(Ur), Conjecture 1.6 is weaker than the false Conjecture 1.5. It is also weaker than Conjecture 1.4, as wi l l be seen in §4. However, Conjecture 1.6 has the one advantage that, as we shal l see below in Theorem 2.3, it is actual ly known to be true when the domain U is s imply connected. It is also known to be true in the case where U is of the form D \7 where / is a finite union of closed intervals on the negative real axis. Th is fact is known as Beurl ing's shove theorem and is proved in his thesis [23, pp. 58-62]; an account of this may also be found in [79, §IV.5.4], and a more general result also valid in higher dimensions was obtained by Essen and Hal iste [45]. We note here that in fact Beur l ing 's method of proof readily also gives the fol lowing result which is a ful l answer to Conjecture 1.4 in the case of the slit disc mentioned above. A proof of a generalization wi l l be given in §8. T h e o r e m 1 .1 . Let I be a finite union of closed intervals in [—1,0). Then r*(D\/) <r*((B\/)*) for any $ 6 T. One possible approach to proving Conjecture 1.6 given the solution in the s imply connected case would be to at tempt to reduce a general c ircular ly symmetr ic domain to a simply connected one by using a shoving argument like Beur l ing 's , this t ime moving holes which are circularly symmetr ic about the negative real axis instead of intervals. We shall make this idea precise in Prob lem 3.1 of §3. However, Beur l ing 's proof does not seem to go through direct ly for this case; the reader cognizant of Beur l ing 's proof wi l l recognize that the diff iculty is that the various monotonic i ty properties of the Green's functions used in the proof are no long applicable when the holes are not contained in the negative real axis. F ina l l y we remark that if the ful l Conjecture 1.6 were known to be true, then the proof of Essen's inequal i ty [44] would be greatly simpl i f ied. 268 Chapter IV. Radial rearrangement 2. Some positive results The fol lowing result, which wi l l be proved in Remark 8.5 of §8, generalizes Theorem 1.1, above. T h e o r e m 2 . 1 . Let D be a star-shaped and circularly symmetric domain. Let I be a finite union of closed intervals in (—oo,0). Let U = D\I. Then F$(U) < T$(U*) for all <& 6 T. The fol lowing result is a special case of Conjecture 1.4. T h e o r e m 2 .2 . Let $ 6 T be such that t H-> Jj5>(e*) is concave on (—oo, logi?). Then r*(u) < r«j>([/*) for any domain U C D(R). A proof of this as a consequence of a variant of a special case of a result of A l v i no , Lions and Trombet t i [4] wi l l be outl ined in §5, below. However, Theorem 2.2 does not assume circular symmetry of U, and as seen in Examp le 1.1, a method that does not use circular symmetry (or something like it) cannot in general be expected to yield a ful l positive answer to Conjecture 1.4. We now state the posit ive result for s imply connected and circular ly symmetr ic domains which constitutes the main part of the evidence for Conjecture 1.6. T h e o r e m 2 . 3 . Let U be a simply connected circularly symmetric domain. Then wr(U) < wr(U*), with definitions as in Conjecture 1.6. Th is shall be seen to follow f rom the fol lowing more general result, the proof of which takes only a l i t t le more effort. G iven z (E C , wri te [0, z) — {Az : A 6 [0,1)}. T h e o r e m 2 .4 . Let U be a simply connected domain which is symmetric with respect to reflec-tion about the real axis and satisfies [—p, r) C U C D(r) for some p > 0. Let C be a symmetric arc in T ( r ) f l dU, centred on the point r. Let C ' = { z e C : [ 0 , z ) c f / } . 269 Chapter IV. Radial rearrangement Then co(-p,C;U)<u;(-p,C,;U'k). It is easy to see that C C dU* DC. Theorem 2.3 then follows by apply ing Theorem 2.4 to the Ur w i th C = dD{r) n dUr and p = 0. It is easy to see that Theorem 2.4 need no longer hold if p < 0. To see this, consider the circular ly symmetr ic domains U£ = B\{rel8 : \ < r < \,s < 9 < 2ir - e} for e > 0, note that U* = D\{reie :\<r <l,e <9 <2ir -e) and f ix p € (—1, — T h e n , it is easy to see that u{—p, dD D dU£; Ue) is bounded away from zero as e -> 0 while u(-p, dBn dU*; U*) -> 0 as e -> 0 so that Theorem 2.4 wi l l not hold if s is sufficiently smal l . F ina l ly , note that C wi l l be connected in Theorem 2.4 if U is s imply connected. Now we proceed to the proof of Theorem 2.4. F i r s t however we need some background results so that we can use a special case of a quite general result of Baernstein [11]. The reader interested in the many different kinds of rearrangements which al l yield analogous results is referred to [11]. Let / be the interval (—7r, 7r], and let F be a posit ive L ipschi tz function on I x K . Recal l that the Steiner rearrangement about the real axis SB of a set S C J x R was defined by Sa=U[{x}x{y:\y\<±\{t:(x,t)eS}\}}, x€l where \{t : (x,t) 6 S}\ indicates the Lebesgue measure of {x : (x,t) 6 5 } . G iven a function F on / x K we may define FB as in §1.2. Then we have the fol lowing result, where both integrals are taken wi th respect to Lebesgue area measure. For $(£) = t2, these integrals are known as D i r i c h l e t i n t e g r a l s . T h e o r e m 2 .5 ( B a e r n s t e i n [11, C o r . 3 ] ) . Let F be Lipschitz and positive on I x R and as-sume that for every y we have F(x, y) —> 0 as y —> ± o o . Then FB is also Lipschitz and for any 270 Chapter IV. Radial rearrangement convex increasing function <& we have JlxR JlxR f $ ( | V F B | ) < / *(|VF|) As a corollary, we obtain the following modified version of a result of Marcus [70, Thm. 1]. For a positive function / which is Lipschitz on D and satisfies /(0) = 0, and for (x,y) G I X R, we define F(x,y) = f(e~\y\+%x). It is easy to verify that this is Lipschitz on I X R. Then, identifying R 2 with C, let log z be the branch of the logarithm with Im log z E I, and set f+(z) = FB(—ilog(z)) for z G D. Then, f+ will be radially increasing on D. Note that f+ = —(—/)* as can be easily verified. Theorem 2.6. Let f be Lipschitz and positive on D and assume that /(0) = 0. Then f+ is Lipschitz on compact subsets o/D\{0), satisfies /*(0) = O(z) as z —> 0, and has This follows immediately from Theorem 2.5 with = t2 and from the well-known conformal invariance of Dirichlet integrals, where we use the conformal map —ilogz from D\(—1,0] onto the upper half of I x R, and then note that the Dirichlet integral for FB over the lower half of I X R is the same as that over the upper half of it. The only subtlety is with proving the Lipschitz character of f+. From Theorem 2.5 we find that is Lipschitz on compact subsets of D\[—1,0]. Rotational symmetry in the definition of f+ (i.e., applying the above to def fv(z) = f(elvz) and noting that (/¥,)^ .(z) = f+(etipz)) shows that in fact it must be Lipschitz on compact subsets of all of D\{0}. Now /(0) = 0 and so I claim that for every r > 0 we have sup f(z) > sup U(z). z£B(r) z£D(r) Thus fif(z) = O(z) as z —> 0 since f(z) = O(z) as z —> 0. To prove the claim, note that for any A > 0. To prove this it suffices to show that equality holds when we intersect both sides with a ray starting from the origin, and to do this one needs to note that the logarithmic {z G D : f(z) < A } * = {z G D : U(z) < A}, (2.1) 271 Chapter IV. Radial rearrangement metric defining radial rearrangement precisely corresponds to the composition with — ilogz in the definition of f+. We now proceed to the proof of our Theorem 2.4. Proof of Theorem 2.4- Without loss of generality set r = 1. As usual, by an internal exhaustion like the one in [7, Proof of Thm. 7] we may assume that all our domains have smooth bound-aries. We shall assume for now that p = 0 and at the end of the proof we discuss the minor modifications necessary to take care of the case p > 0. We now use the method of Haliste [56]. Let V be any subdomain of D such that [0,1) C V and V is symmetric under reflection in the real axis. Let Ti1 be a symmetric arc about 1 in dTS> D dV. Let / be the (unique) holomorphic map of V onto the disc D with /(0) = 0 and /'(0) > 0. Then f(E) is a symmetric arc of <9D centred about 1, and its harmonic measure at 0 in D equals its normalized Lebesgue measure. By conformal invariance, this normalized harmonic measure also equals u(0, E;V). Now, as in [56] (but for convenience with reversed boundary values so that Theorem 2.6 would work), let u = UE,V be the solution of the following mixed Dirichlet-Neumann problem on D\[0,1): * e [ 0 , l ) , (2.2a) z e dJ}\f(E), (2.2b) z e f(E), (2.2c) z€D\[0,l), (2.2d) where 4^ denotes a normal derivative. Let <f>(E,V) = [[ \VuEy\\ J J D\[0,1) where all such integrals are understood to be taken with respect to Lebesgue area measure. Then, gluing two copies of D together along the arc f(E) to form a Riemann surface, and applying the Dirichlet and maximum principles on it, we easily see that (p(E, V) must be strictly decreasing with respect to the length of the arc f{E). But since the length of this arc is proportional to u(0, E;V), there must be a strictly decreasing function tp : [0, oo) —• [0, oo) dn u(z) = 0, u(z) = 1, u(z) = 0, Au(z) = 0, 272 Chapter IV. Radial rearrangement such that iP(u(0,E;V)) = cp(E,V). (2.3) Haliste [56, equation (3.6)] gives an explicit expression for ip in terms of elliptic integrals. Now, by conformal invariance and the fact that / sends 0 to 0, E onto f(E) and [0,1) onto [0,1) (this last fact holding due to the reflection symmetry of V), it follows that we may instead consider the function SE,V = f / _ 1 o UE,V and we will have <j>(E,V)= ff |Vs £ ) V - | 2 , (2.4) JJ V\[0,1) and more moreover the function s = SE,V will be the solution to the following mixed Dirichlet-Neumann problem on V\[0,1): s(z) = 0, ze [o,i), (2.5a) s(z) = 1, z e dv\E, (2.5b) zeE, (2.5c) As(z) = 0, ze v\[o,i). (2.5d) The proof of Theorem 2.4 is now quite simple. Take s — sc,u with the above definition and set s(z) = 1 for z £ D\C7. If U has a nice boundary then s has no problem with satisfying the conditions of Theorem 2.6. Hence, fl I W < If | V S | 2 . (2.6) J J O J J D Now, it is easy to verify that s and s+ are identically 1 in D\U and D\U*, respectively; for s+ this follows from (2.1) with A = 1, together with the fact that {z G D : s(z) < 1} = U. Thus the integrands in (2.6) are supported on U and £/* respectively, so that / / r / * | V ( s c t / ) * 1 2 - / / y l V s ^ l 2 ' (2J) since we have a nice boundary which thus has Lebesgue measure zero. Furthermore, if U is symmetric about the real axis and simply connected then so is [/"*, and s+ is identically 1 on dU*\C since for z € C we have s*(z) = max[0z) s = 1. Clearly, too, is identically 0 273 Chapter IV. Radial rearrangement on [0,1). Hence s+ — (sc,u)+ satisfies the two Dirichlet boundary conditions which would be imposed on sc",(C/*) t>v (2.5a) and (2.5b), though in general it will fail to satisfy the Neumann condition (2.5c) and the harmonicity condition (2.5d). Then, it follows by the Dirichlet principle with free boundary values that which combined with (2.4) and (2.7) yields HC,u+)<<fi(c, u). By (2.3) it follows that u(0,C';[/*) > w(0,C;l/), since ip is strictly decreasing. This completes the proof in the case of p — 0. If p > 0 then we proceed just as above, the main difference being that the map / , instead of taking 0 to 0, is now required to take — p to 0; note that the condition /'(0) > 0 is equivalent to the condition f'(—p) > 0 since / is to be univalent. Then, instead of considering the solution s = spy to (2.5a)-(2.5d) we now consider the solution s = S - P < E , V to the mixed Dirichlet-Neumann problem (2.5a')-(2.5d') obtained from (2.5a)-(2.5d) by replacing [0,1) in (2.5a) and (2.5d) by [—p, 1). The rest of the proof goes through. For, we still have ib(u(-p,E;V))= ff \Vs_PiEtV\2, JJ y\[o,i) with exactly the same function ip as before. Moreover, if s = S - P I E , U then s+ satisfies the two Dirichlet boundary conditions that would be imposed on s_pEt(u*) by (2.5a') and (2.5b), where (2.5a') is (2.5a) with [0,1) replaced by [—p, 1). The reader waiting to see where the assumption p > 0 is used may be pleased to note that it is used precisely in the assertion that s+ satisfies (2.5a') for the rearranged domain V = U*. • 274 Chapter IV. Radial rearrangement 3. Transferring the problems to the cylinder and the question of two-sided lengthwise Steiner symmetrization W i t h no loss of generality we may assume that our domain U is contained in the unit disc (use def scaling and approximat ion to handle the general case). Let / : O \ {0 ) —> V _ = (—oo,0) X T be the conformal isomorphism f(z) = (log\z\,eiai**). (3.1) Note that V - is a semi-infinite cyl inder. We shall sometimes have a certain tendency to im-pl ic i t ly consider the set (—oo, 0] x T as the closure V - of V - , and to extend things defined on V ~ automat ica l ly to V " . Now, for W C V " and u £ T , let X(u;W) = \{x < 0:(x,u) fi W}\ and set W* = {(x, u) G V ~ : x < —X(u; W)}. It is easy to verify that A r e a ( V ~ \ I ¥ ) = A r e a ( V ~ \ F y * ) and W\{0}]* = f[U*\{0}]. We say that a domain W C V - is c i r c u l a r l y s y m m e t r i c if {x} X T is the cartesian product of {x} w i th a circular interval centred about 1 G T , for each x £ (—oo, 0]. C lear ly W is a circularly symmetr ic subset of V - if and only if f - 1 ^ ] is a circular ly symmetr ic subset of the plane. Let Q be the collection of convex increasing functions $ on (—oo, 0]. Then , it is easy to see that we may reformulate Conjecture 1.4 and Conjecture 1.6 to be conjectures concerning least harmonic majorants of (x,t) ^ \P(£) for \P( i ) , harmonic measures and circular ly symmetr ic domains in V - . Th is is done expl ic i t ly in the language of Brownian motion in §7. Now, we present a conjecture which is more general than Conjecture 1.6. A n analogous con-jecture more general that Conjecture 1.4 could perhaps also be formulated. Let V = R X T be a doubly infinite cyl inder. We may of course easily define what it means for a domain on V 275 Chapter IV. Radial rearrangement to be circularly symmetric. We now say that a domain W C V is flip-symmetric if for every [x, t) £ V we have (x, t) £ W if and only if (-x, t) £ W. Now, given t £ T, let L(t-W) = \{x:(x,t)eW}\, where | • | indicates Lebesgue measure on R. Define the (two-sided) lengthwise Steiner symmetrization of W C V to be W^d={(x,t) : \x\ < L(t,W)/2}. Clearly Area(PF~) — Area(VF). We may analogously define flip symmetry and lengthwise Steiner symmetrization for domains which are subsets of C = R X R, except that in the above definitions we replace t £ T by y £ R. Given W a subset of V or C, let 7 be T or R, respectively. Given M > 0, we define1 nM(W) = LO ((0, 0), ({-M, M} x Y) n W; ((-M, M) x Y) n W) , where of course (—M, M) is an open interval with boundary {—M, M}. It would certainly be hopeless to suppose that &M(W) < QM(W~) in general since Example 1.1 can be adapted to this situation. Hence, if W C V then we will impose circular symmetry on W while if W C C then we will usually impose Steiner symmetry about the real axis. From Theorem 1.6.7 it follows that QM(U) < SIM{UB) since nM(u) = n'{u') + n'{-u'), where U' = U n ((-M, M) x R) and n'{D) = w(0, {Re z = M} n D; D). But Steiner symmetry is still insufficient, since we may easily come up with examples where £IM(W) > 0 while {-M, M} X Y is disjoint from W~ so that QM(W~) = 0. For example, let W = (-M/3,4M/3) X Y and note that W~ = (-5M/6,5M/6) X Y. Hence we also require flip-symmetry, which in the case of Y = R, is by definition the same as reflection symmetry about the real axis. Of course harmonic measures can be easily defined on the manifold V . 276 Chapter IV. Radial rearrangement Conjecture 3.1. Let W C V be both flip-symmetric and circularly symmetric. Then « A f ( W ) > nM(w) for every M > 0. To see that this is more general than Conjecture 1.6, note that given a domain W CV with W = f[U\{0}] for U C D with 0 eU (where / is our conformal map from O\{0} to V - ) , there is an R > 0 such that (—oo, —R] X T C W since U is open at 0. Then, let Wr = ( ( - r , r ) X T ) U { ( ± ( r + £ ) , * ) : x G [0, R), (x - R, t) e W}, where the ± means that for each admissible x and t, we throw both ((r + x),t) and (—(r + x), t) into Wr. Then, it is easy to see that Qn+r(Wr) —> w\{U), and all the rearrangements are nicely preserved by the correspondences so that Conjecture 3.1 indeed implies Conjecture 1.6. Theorem 3.1. Let W C C be a simply connected domain which is symmetric under reflection in the real and imaginary axes. Moreover, assume that the interval (—M, M ) lies in W. Let Ci be an interval of ({M} x E) n W centred on M, and let C = C\ U —C\. Let C = {(±M, y) : ( - M , M) x {y} C W}. Assume that C is an interval. Then w(0, C; ( ( - M , M) X E ) n W) < w(0, C; ({-M, M) x E) D W~). If W is Steiner symmetric about the real axis then C will automatically be an interval. Moreover, we see that J2M(W) < QM(W~) under the conditions of the Theorem with C = ({M} x E) Pi W. In particular we have the following result. Corollary 3.1. IfW is both flip-symmetric [about the imaginary axis) and Steiner symmetric about the real axis then nM{w) < nM{w=) for every M > 0. 277 Chapter IV. Radial rearrangement This is a special case of Conjecture 3.1. For, given a domain W', by approximation we may assume that W is bounded, and then by scaling we may assume that it is a subset of R X (—n, n). But such a subset can be mapped conformally into V - via (x,y) i->- (x,eiy), and this map preserves all the relevant symmetries and symmetrizations so that Conjecture 3.1 would indeed imply the inequality QM{W) < QM(W~). Outline of proof of Theorem 3.1. The proof is done in a manner very similar to the proof of Theorem 2.4. We assume FF is a subset of (—M, M) x R. We map W onto D by a holomorphic map g in such a way that 0 goes to 0 while the interval ( — M, M) goes to the interval (—1,1). Then, g[C] consists of two circular intervals of equal length, centred on 1 and —1 respectively. The length of each of these intervals then is proportional to the harmonic measure w(0,C; W). Moreover, this length is strictly inversely monotonely related to the Dirichlet integral of the solution s — sc,w of the boundary value problem s(z) = i, z e ( - i , i ) , s(z) = 0, z 6 d®\g[C], —s(z) = 0, ze g[C], A s ( z ) = 0, z e B>\(-1,1), in D. And as in the proof of Theorem 2.4, this problem can be pulled back by g~l to a problem on W. We may now complete the proof as in Theorem 2.4, except that we can no longer use Theorem 2.5 or 2.6. Let u = UQ,W — sc,w 0 <7_1- By rescaling, we may assume that M = w. Define u = 0 on ((—7r, TT) x R)\W. Thus extended, u on (—n, IT) X R will satisfy any regularity conditions that we may later need, providing W is nice enough. We could just apply a theorem on Steiner symmetrization (about the imaginary axis) on (—7r,7r) X R to u, but at least in [11] we do not seem to find an explicit statement of the kind of theorem we need. Instead, [11] gives a (l,2)-cap symmetrization theorem which we can use as follows. The set (—7r,7r) X R can of course be mapped conformally via (£, y) i—>• ey+lt onto T x R. This lets us transfer u to a function on T X R. We then apply (l,2)-cap symmetrization (see [11]) to the transferred 278 Chapter IV. Radial rearrangement function and we pull back to get a function on (—TT, TT) X R. This process will not increase Dirichlet integrals [11, Cor. 3]. The function we obtain from u by this process will turn out to satisfy the Dirichlet boundary conditions (but perhaps not the Neumann or harmonicity conditions) that would be imposed on the function uc,^w~y The proof is then completed just as in the case of Theorem 2.4 by the Dirichlet principle with free boundary values. • Corollary 3.1 was also independently obtained by Baernstein [13] via much the same adaptation of the method of proof of Theorem 2.4. Conjecture 3.2. Let $ be an even convex function on K. Let U be a flip-symmetric subset of, respectively, V or C, and assume it to be, respectively, circularly symmetric or Steiner symmetric about the real axis. Then if T$ indicates, respectively, the value at (0,1) € V or the value at 0 € C of the least harmonic majorant of (x,t) i—>• $(x) on U, then T$(U~) > T$(U). Actually, while the Conjecture 3.2 is still open, in the special case of U C C in §10 we will obtain a certain partial replacement. Now, given a domain W C V~, and given a half-open interval (a, 6] C (—oo, 0], we may define W(atb] = lnt({(x + b - a,t) : x < a, (x,t) G W} U.{(x,t) :b<x, (x,t) G W}), where Int X denotes the union of all open subsets of X. The domain M^(a,6] m a y be considered to be W with the ring (a, b] X T cut out. Intuitively, then, one would expect that the probability that a Brownian motion starting at a point at infinity (to the left of {0} X T) arrives at {0} X T without touching the complement W would increase if we replace W with VF(a,6]i since W(^a^ is in some way "shorter." Of course, in reality, one cannot talk about a finite shortening of the length of a set which has infinite length. In the most interesting cases we will have (a, b] X T C W. We may pull the transformation i')(a,b] to the punctured disc to get U(r'R] = Int Q jsei8 : 0 < s < r, se10 G £/j U {se* : r < s, seie G U^j , 279 Chapter IV. Radial rearrangement for U C D and 0 < r < R < 1. We wi l l have f[U^\{0}] = / [ C / \ { 0 } ] ( l o g r i l o g f i ] , where / is our conformal map of D\ {0} onto V " defined by (3.1). Note also that A r e a f / ( r - R l < AreaU if r < R. The above-mentioned intui t ion concerning Brownian mot ion then suggests the fol lowing prob-lem. O p e n P r o b l e m 3 . 1 . D o we have wi (U) < Wi([/('•.«])? More generally, do we have T$(U) < r ^ C / t ^ l ) for every <& £ TI A s it stands, the answer is negative, and a counterexample is provided by the domain on the left side of F igure 1.1 (cf. Examp le 1.1). However, the Prob lem is st i l l open under the addit ional assumption that U be circular ly symmetr ic , and this is the case we consider in the discussion below. We should also remark here that the fact that the second part of the Prob lem is more general than the first is a consequence of Theorem 4.1 in §4, below. Note that (U^r,R^)* C [7 * wi th equali ty if and only if the annulus {sel8 : s £ (r, R]} is contained in U possibly modulo a set whose intersection wi th any ray through the origin has null Lebesgue length measure. Note also that if a domain U is finitely connected and circular ly symmetr ic then a finite number of appl icat ions of (•)(r'-Rl wi th (r, R] a max ima l interval chosen so that the annulus {setB : s £ (r, R]} is contained in U wi l l reduce U to a simply connected and circular ly symmetr ic domain U' w i th ( £ / ' ) * = £ / * . A l imi t ing argument could then be employed in the case of an infinitely connected domain U to reduce it, too, by an infinite number of appl icat ions of (•)(r'-Rl to a simply connected domain U' wi th (U')* = U*. Then , an affirmative answer to the first part of the Prob lem would imply that wi(U') > W\(U). Bu t by Theorem 2.3 we would then have wi(U*) = wi((£/')*) > wi(U') since 11' is simply connected. Hence, an affirmative answer to the first part of the Prob lem would imply an affirmative answer to Conjecture 1.6. It may also be worth not ing Prob lem 3.1 can be considered as concerned about a generalization of the method of proof of Beur l ing 's shove theorem. A s further mot ivat ion, note that since an affirmative answer to the second part of the Problem 280 Chapter IV. Radial rearrangement would give us a way to increase the functionals T$ by replacing a domain with a simply con-nected one without increasing the area of the domain, it would yield an affirmative answer to Conjecture 1.3. 4. Formulation in terms of Green's functions Now we would like to note a well-known result which was alluded to before. The following result is an immediate consequence of Proposition 1.6.2. Theorem 4.1. For any pair of domains U and V the following are equivalent: (a) For every $ £ T we have T$(V) > T$(U). (b) For every r > 0 we have /•27T r2lX / g(rei8,O;V)d0> / g[rel\ 0; U) dB. Jo Jo It might well eventually turn out that to prove Conjecture 1.4 it would easier to prove condi-tion (b) of Theorem 4.1 with V = U*. We now note that Conjecture 1.4 implies Conjecture 1.6 because harmonic measures on the boundary of a domain correspond to normal derivatives of Green's functions, so that (at least for a sufficiently nice domain) 2 l W r { U ) = - T Wre"0;Ur) M = ^ ( r _ p ) _ r / ^ ^ a 0 ; ^ ^ ( 4 1 ) r J0 ^r p-tr- JQ as noted in [10]. Now if Conjecture 1.6 holds then by Theorem 4.1 we have an inequality between the right hand side of (4.1) and the same right hand side but with Ur replaced by (£/ r)*, so that the desired inequality wr(U) < wr(U*) then follows from the easy observation that ( C / r ) * C (U*)r and t o r ( ( [ / r ) * ) < wr(U*). (Of course, if our domains are not sufficiently nice then we have to use an approximation argument, but this is easy.) Assuming U C D(r), Conjecture 1.6 is an inequality between g(-,0;U) and p(-,0;!7*) near T(r). Let us consider such inequalities near 0. Given a domain U containing the origin, its 281 Chapter IV. Radial rearrangement Green's function can be written in the form g(0, z;U) = log + log p + o(l) as z —> 0, where p = p(U) is a constant known as the inner radius of U about 0. Then, inequalities between g(-,0; U) and g(-, 0; [/*) near zero correspond to inequalities between p(U) and p(U*). Marcus [70, Thm. 3] had shown that for any domain U (not necessarily simply connected or circularly symmetric) we have p(U) < p(U*). (4.2) (See the following section for a reference to another proof of this fact.) Hence, in the simply connected circularly symmetric case we know that we have the correct inequality between g(-, 0; U) and g(-, 0; U*) near <9D(r) and near 0; for general domains we only know that we have it near 0. 5. The case where t i—>• d^Jfe ^ is concave Now, return to the cylinder V considered in §3. Given a real-valued function / on V, consider the function defined on V by f**{x,t) = sup / f(w,t)dw, E J E where the supremum is taken over all measurable sets JSCR with Lebesgue measure 2\x\. The function /** may be called the (lengthwise) Baernstein *-function of / . See Baernstein [11] where many similar objects are considered. We recall the following theorem which is a modified version of a very special case of a theorem of Alvino, Lions and Trombetti [4] (see also [11, Thm. 7] for another way to prove such results). Say that a function / on R is s y m m e t r i c d e c r e a s i n g if f(x) = f(—x) for every i 6 i and / is decreasing on [0, oo). T h e o r e m 5 . 1 . Let V C V and let A(-,i) be a symmetric decreasing function on R for every t G T . Let u be the solution of -Au = A 282 Chapter IV. Radial rearrangement in V, with boundary value 0 on dV. Let v be the solution of the analogous problem on V~. Extend u and v to vanish identically outside V and V~, respectively. Then everywhere on V . Note that the second inequal i ty in (5.1) is a t r iv ia l consequence of the definit ion of (•)**. Now, suppose $ is a convex increasing funct ion in C2[0,R) wi th $ ' concave and $ " (0 ) finite. Assume that V C (-R, R) X T is sufficiently regular. Let h be the least harmonic majorant of (x,t) i—y # (z) in V. Then , A n d of course h — $ vanishes on dV. Let H be the least harmonic majorant of (x,t) i-> *&(x) on V~. Then by Theorem 5.1 we have u**(x,t) < / v(s,t)ds, J-\x\ where u = h — \J/ and v = H — $ . It easily follows f rom this and the definit ion of u** that for every i 6 T , Thus , if we apply an approximat ion argument, we wi l l obtain the fol lowing result. C o r o l l a r y 5 . 1 . Fix t € T . Let \P be a convex increasing function on [0,R) such that $' is concave and $ " (0 ) is finite. Suppose V C V . Let h be the least harmonic majorant of (x,t) H-> $ ( | x | ) on V, and let H be the least harmonic majorant of (x,t) H->- $( |X | ) on V~. (5.1) -A{h-'V)(x,t) = # " ( z ) . sup u(x, t) < u(0, t). x Then h(0,t) < H(0,t) for every t G T . A t this point we note that to prove Theorem 2.2, one may use a l imi t ing argument together wi th Coro l la ry 5.1, fol lowing the method that was used in §3 to show that Conjecture 3.1 283 Chapter IV. Radial rearrangement implies Conjecture 1.6. Note that Theorem 2.2 generalizes the inequality (4.2), above, which was due to Marcus [70]. To see this, assume without loss of generality that U C D be sufficiently regular and let $>(t) = logi on [0,1], so that the least harmonic majorant of $ on U equals g(-, 0; U) — log p|. Also note that Theorem 2.2 does not need any circular symmetry condition on U, so it is difficult to imagine using this method to prove results like Conjecture 1.4 in light of the fact that Conjecture 1.4 need not hold if U fails to be circularly symmetric. 6. Haliste's one-sided lengthwise Steiner rearrangement We take the occasion to discuss a situation analogous to Theorems 2.4 and 3.1. As before let U be a domain in the plane. Now, for M > 0, define WM(U) = w(0, {Rez <M}DU, {Rez = M} n U). Given a domain U, we define a domain U^M\ which we shall term the one-sided lengthwise Steiner rearrangement at abscissa M, by the equality = {(x, y) : M - \{t < M : (t, y) e U}\ < x < M}. This definition also makes sense for M < 0, and was first studied by Haliste [56]. As in the case of radial and circular rearrangement, for any real M the set t/(M' is open if U is open. Furthermore, Area(C/(M)) = Area(c7n {Rez < M}). Now, it would be hopeless to desire that in general WM{U^M^) > WM(U) since C/W need not contain 0 even if U does. However, we can assert that WM(U^M^) > WM(U) provided U is Steiner symmetric. Theorem 6.1 (Haliste [56, Thm. 4.2]). Let U be a simply connected domain in the plane which is reflection symmetric about the real axis and contains the interval [0, M). Then WM(U)<WM{U(M)). (6.1) In particular, (6.1) will hold whenever U is Steiner symmetric. Moreover, by Theorem 1.6.7, we have WM(U) < WM(UB), SO that we also have the following inequality. 284 Chapter IV. Radial rearrangement Corollary 6.1. Let U be any domain containing the origin. Then for any positive M we have WM(U)<WM((UB)W). Haliste's proof [56] of Theorem 6.1 basically follows along much the same lines as that of Theorem 2.4 or Theorem 3.1. 7. Brownian motion, simple discrete analogues and exit times We may now use Brownian motion to transfer Conjectures 1.4 and 1.6 to the cylinder V -considered in §3. Let BT be a Brownian motion on V - . Let TW = inf {t > 0 : BT £ W}. We use PZ(-) and EZ[-] to indicate probabilities and expectations, respectively, when Bo is conditioned to be z. Moreover, we use P _ 0 0(-) and E~°°[-] to indicate the limit of PZ(-) and EZ[-], respectively, as z -> (—oo, u) (in all cases which we consider, this limit will not depend on u). Throughout, W will indicate an open subset of V~ which contains some semi-infinite cylinder (-oo, —R] x T (for some R > 0), so that U {0} will be an open subset of D, where / is given by (3.1). Write TT: V~ —>• (—oo,0) for the projection onto the first coordinate. Recalling the well-known connection between Brownian motion, harmonic measure and harmonic majorants, the following two conjectures are equivalent to Conjecture 1.4 and Conjecture 1.6, respectively. Let Q be the collection of all increasing convex functions on [—oo,0]. Conjecture 1.4'. Let W be circularly symmetric and let $ G Q. Then E~°°[^(K(BTW))] < E~°°[$(TT(BTW<))]. Conjecture 1.6'. Let W be circularly symmetric. Then P-^(TT(BTW) = 0) < P-°°(n(BTW4) = 0). 285 Chapter IV. Radial rearrangement By Theorem 2.3, Conjecture 1.6' will be true if V~\W is connected. We now give a simple one-dimensional discrete model for these problems. Let R{, i £ Z Q , be a simple random walk on ZQ = {0, —1, —2,...} with P(i? J +i = Ri ± 1 | Ri) = \ for all i. Let {sn}n=_00 be a sequence of numbers in [0,1] with So = 0. Assume that sn = 1 for n sufficiently close to — oo. Consider the following process. Let Xi, i 6 Z Q , be a sequence of i.i.d. random variables uniformly distributed on [0,1]. We will kill the random walk at time TS = inf{i > 0 : X{ > sRi}. Thus, the random walk Ri at step % has a probability of surviving equal to SR{. This rs is quite analogous to our previous r\y, since the likelihood of our Brownian motion terminating in {x} X T is heuristically expected to depend inversely on the size of the set ({x} X T) fl W, so that the size of this set is analogous to sx. What is the discrete analogue of (•)*• then? Well, if W is circularly symmetric, then W* is W with the slices ({x} x T) fl W reordered so their sizes decrease as x increases. Hence, let {snt}n=_00 be the decreasing rearrangement of the sn. We will have 0 = SQ* < < < Let P~°°(-) and E~°°[-] indicate probabilities and expectations, respectively, after having taken the limit of the starting point of the random walk tending to —oo. The obvious analogue of Conjecture 1.6 is as follows. Theorem 7 .1. p - 0 0 ^ = o) < p-°° ( i? T ^ = 0). This will be a consequence of Theorem 9.1 in §9, below. To get this out of Theorem 9.1, we note that En[$>(RTs)] and Pn(RTs = 0), as well as their analogues with s* in place of s, will both necessarily be constant for n < N where N < 0 is chosen so that sn = 1 whenever n < N, so that nothing will change if we make things finite by introducing a reflecting boundary condition at n, or even the somewhat different boundary condition used in §9. The natural analogue of Conjecture 1.4 is still open. 286 Chapter IV. Radial rearrangement 7.1. Uniform motion to the right: a counterexample Now, in §9, Theorem 7.1 was in fact proved not just for random walks with equal probabilities of going forward and backward, but also for walks which have a probability p of going forward and 1 — p of going backward. (For p < ^ we have to change the statement of Theorem 7.1 somewhat, though, since otherwise both of the limits in it are zero. However, if instead of taking the limit we simply start the random walk at some point x such that sxi = 1 for every x' < x then Theorem 7.1 will continue to hold and will make more sense for p < \.) One might hope that such a generalization to non-equal forward/reverse movement probabilities would also be possible in the two dimensional case of Conjecture 1.6'. But, this is not true in the natural adaptation of Conjecture 1.6' to the case p — 1. We now show this impossibility of extending Conjecture 1.6' to the case n = 1. Of course, p = 1 means that the random walk constantly moves to the right. Hence, in our analogue we shall replace the process Bt in Conjecture 1.6' by a process whose first coordinate moves uniformly to the right while whose second coordinate is a Brownian motion St on the circle T. For convenience, we index our process with time on (—oo, 0] so that the process is t >->• (£,£*) on V - . Let W C V~ be circularly symmetric and such that (—00, —T] X T C W for some finite T. Then, the analogue of p = 1 in Conjecture 1.6' is that P((t,St) e w,vt e (-00,0)) < P{{t,st) e wM,vt e (-00,0)). Write F(w) = P((t,st) ew,vte (-00,0)). To prove the existence of a counterexample, first let H be the semicircle {elS : \6\ < TT/2}, let H ° — T\H be its complement, and then define WT,S = V"\[([-T - e, 0) x {-1}) U ({-e} x H C ) ] . (See Figure 7.1.) Then, w ? i e = v - \ ( [ - r - e , o ) x { - i } ) . (See Figure 7.2.) 287 Chapter IV. Radial rearrangement Legend: Black lines indicate the complement of WTi A=(-T-s-l) 5 K - 6 - 1 ) C=(-s-0 £>=(-e ,0 £ = ( - 1 , 0 ) F=0,0) Figure 7.1: The domain WT,S-288 A Legend: Black lines indicate the complement ot WTe. A=(-T-s-l) £ = ( - 1 , 0 ) ^=(1,0) Figure 7.2: The domain W££ For a domain W, let H(W) = P((t,St) eW,Vte ( -oo ,0 ) and S0€ H). I c la im that one can choose e > 0 and T € (0, oo) such that Assume this c la im for now. Let WS = V " \ [ ( [ - r - e, 0) x {!}) U ([-e, - £ + x H% for 0 < (5 < e, and note that = V " \ ( [ - r - e, 0) x {1} U [-*, 0) x #c). It is easy to verify that F(WS) F(W T , e ) , 289 Chapter IV. Radial rearrangement as S —> 0 for fixed e > 0, while F(Ws*)-+H(WT,e), again as 8 —> 0. Choosing S sufficiently small, may easily ensure that F(W*) < F(W$) because of (7.1). It remains to prove (7.1). To do this, let a(T,e) = F(WT>s), and 3{T) = H{W+0). Of course, Wj>0 = WT,O- It 1 S e a s Y to see that B(T + e) = H(W*E)- Also easy to verify is the fact that as e I 0 then a(T,e) t H(WT,o) = H(W£Q) = 3{T). We need an estimate of the rate of convergence in this limit. To obtain this estimate, note that we may write a(T,e) - F(WT,£) = H(WT,o)P(St G T\{-l},Vt G (-6,0) | 5_£ G H). For, we may use the Markov property of Brownian motion together with the fact that H(WT,O) can be seen to be equal to the probability of our process arriving from (—oo,0) to {—e} X H without having touched d(Wx,s), because of translation invariance and the relation (d(WT,e)) n ((-oo, -e) x T) = {{t - e, w) : t < 0, {t, w) G d{WT,o)}-Now, 1 - P(St G T\{-1}, Vi G (-e, 0) | 5_£ G H) = P{St = 1 for some t G (-£, 0) | 5_e G H) < c i e - C 2 / £ , for some finite positive constants c\ and c2 depending only on the rate of the Brownian motion St. To obtain the last inequality, we have noted that there is a positive distance between H and the point —1, and so we can get a bound of the form c\e~C2l£ for the probability of a Brownian 290 Chapter IV. Radial rearrangement motion crossing this distance wi th in a t ime s. Hence, (3(T)-a(T,e) = H(WTo)(l - P(St G T \ { - l } , V t G (-6,0) | S-e G H)) (7.3) < H(WTt0)cie-c^E < C l e - C 2 / £ , for every e > 0. To obtain a contradict ion, suppose now that (7.1) is always false. Then , for every e > 0 and each finite posit ive T we have a(T,e) < f3(T + e) as (3(T + e) = H(W^e). Thus , by (7.3) we must have (3(T) - j3(T + e) < j3(T)-a(T,e) < cxe-^le. Since /? is monotone decreasing, i t follows that \P(T) -/3(T')\ < C l e - C 2 / I T - T ' l . Div id ing both sides by T — T' and tak ing the l imit as T' - 4 T we see that /3 is differentiable everywhere on (0, oo) and that (5' = 0. Hence, (3 is constant. Bu t , limT->o+ P(T) = \H\ = \ while limj^ oo /3(T) = 0, so that /3 is not constant. Hence we have a contradict ion, and so (7.1) must be true for some choice of e and T. Note that the counterexample is of the form (7.2). It is easy to see that the complement of a domain defined by (7.2) is connected, so that such a domain corresponds to a simply connected domain on the disc. B y Theorem 2.3, then, it follows that Conjecture 1.6' does hold for our present counterexamples to the p = 1 analogue of Conjecture 1.6'. Hence, these part icular counterexamples cannot be used to disprove Conjecture 1.6'. 7.2. Exit times of Brownian motion We now should mention something about exit t imes of Brownian mot ion, given their connection wi th the T$ as discussed by Burkholder [28]. A s before, let TW = i n f { i > 0 : Bt fi W}. It makes l i t t le sense to ask whether we have P - ° ° ( T W > A) < P - ° ° ( T W ^ > A), for each A > 0, because if (—oo, T] X T C W for some T > —oo then both sides of the above inequali ty wi l l equal 1 for al l f inite A. We may, however, obtain such a result in the sett ing of 291 Chapter IV. Radial rearrangement §3 and §5 for the symmetrization — where the domains W are subsets of the full cylinder V, and the Brownian motions start somewhere on {0} X T. Indeed, if z £ {0} X T and W is an arbitrary domain on V (not necessarily circularly symmetric) then PZ(TW > \ ) < P Z ( T W , > A), for all A > 0. The proof is almost identical to that of Haliste [56, Thm. 8.1] (see also [25]). At this point we demonstrate why the cylindrical setting is more natural in terms of Brownian motion. For, suppose now that Bt is Brownian motion in the plane. Let TU = inf{£ > 0 : Bt £ U} as before. We may ask whether it is true that P°(ru > A) < P°(T[/* > A) (7.4) for all A > 0 if U is a planar domain. We might wish to restrict this question to circularly symmetric U. But, as it turns out, this inequality is false, even for circularly symmetric U. (The technical reason why a proof along the lines of [56, Proof of Thm. 8.1] cannot be constructed is the lack of an appropriate convolution-rearrangement inequality caused by the fact that the rearrangement U »-» U* is not measure preserving.) We now prove that (7.4) is not true in general. Example 7.1. Fix a small positive <5, a large positive R and an even larger positive L. Let Us,L,R = (> = \ A < U {x + iy : 0 < x < L, \y\ < 6} U {z : \z - L\ < R}. The domain US,R,L can be visualized as a balloon tied with a string to the origin, with the radius of the balloon being R, and the string being of width 26 and length L - R. I claim that for appropriate choices of 6, R, L and for every A > A0 for some A0 = Xo(S,R,L) < oo, the inequality (7.4) is false for U — U$,L,R- T O prove this, first let h(6, L, R) = sup{| Im z\ : z e U*LR}. It is easy to verify that for fixed 5 and R we have h(S, L, R) S as L —> oo. From now on, then, for any fixed S and R we will be implicitly choosing a specific L = L(8,R) such that h(5,L,R)<25. 292 Chapter IV. Radial rearrangement We may write B T = ( B ^ \ B ^ ) , where the B \ ^ are independent Brownian motions on K . Define TJJ(\,H) = P° I sup I B ^ K H ) , \o<t<\ J for nonnegative A and positive H. If we put B ^ in place of B ^ in the above definition this will of course change nothing. Note that PX{ sup \B[1)\ <H) < ij>(\,H) (7.5) 0<<<A for any x £ K. One way to see this is to note that, as is well known, [56, Proof of Thm. 8.1] implies that for any domain D we have PZ(TD>\)<PR*Z(TDB>\), (7.6) for any real A. Let D = {z £ C : | Imz| < H}. Then, Pz{rD > A) = PIMZ( sup | B t ( 2 ) | < H), o<t<\ and D3 = D. Setting z = ix and noting that and B ^ behave in exactly the same way, we obtain (7.5) from (7.6). Inequality (7.5) combined with the Markov property of Brownian motion implies that ^(S + T,H)<i>(S,H)iP(T,H). Let cj>(\) = ip(X, 1). By scaling of Brownian motion we have tp(X, H) — cj>(X/\fH). As cj>(S+T) < d>(S)<f>(T), the function log<?!> is a negative subadditive function on (0,oo). By [62, Thm. 6.6.1], the limit lim^oo A - 1 log<j>(X) exists and equals inf^e(0oo) A - 1 log<f>(\). Let j be this limit. Then, since <f>(X) < 1 for t > 0, it follows that 7 < 0, and so <fi(X) behaves like e7A for large t. We now estimate P°(T~USLR > A) and P°(TTJ* > A). Since h(6,L,R) < 26, we have ' ' U6,L,R US,L,R C K x (-26,26). Thus, P°(TU* > A) < P ° ( r R x ( _ 2 5 , 2 5 ) > A) = V(A,2£) = <j>(\/V28). (7.7) 81L,R 293 Chapter IV. Radial rearrangement O n the other hand, let e = e(8, L, R) be the probabi l i ty that the random walk Bt start ing at 0 hits the interval [L — R/2, L + R/2] before hi t t ing dUs,L,R- C lear ly e > 0. Then we may apply the strong M a r k o v property to conclude that ^ ( ^ , » > A ) > £ i e [ i J n f i + s / ! ] P « ( ^ > A ) . Let QR,X be the square (x — KR, X + KR) X (—KR, KR), where K — — 1) ~ 0.4114. It is easy to see that for x € [L — R/2, L + R/2], the square QR)X is contained in US,L,R- Thus , f ^ ^ , , , > A ) > £ i e | i J / „ f i + R / 2 ] / » ( r « J i , , > A ) . (7.8) Bu t , by t ranslat ion invariance and the spl i t t ing of Bt into the independent coordinates B^ and B^2\ we have P X ( T Q R I X > \ ) = P ° ( T Q R I 0 > \ ) = P° [ sup | 5 J 1 } | <KR)P°\ sup | £ f ( 2 ) | < KR] \ ° < < < A / \ °<<< A / (7.9) = (TP(\,KR))2 = (cf>(X/V^R))2. Now, choose R and 5 such that 2/VKR < l/y/26. Then I c la im that .. (<f>(\/VnR))2 hm v y v 1 >! = oo. (7.10) A^^= 4>{X/V26) If this c la im is just then we are finished by (7.7), (7.8) and (7.9). Bu t to prove the claim it suffices to take a logar i thm of the rat io wi th in the l imit on the left hand side of (7.10), and to apply the choice of R and 8 as well as the fact that t~l log <?!>(£) —» 7 < 0. 8. The Beurling shove theorem and extensions Let D be a domain in the plane, containing the interval (—1,0]. Let / be a finite union of closed intervals in [—1,0). Define W(I;D) = io{0,dD;D\I). 294 Chapter IV. Radial rearrangement Let / * " be a single interval of the form [—1, —a] such that the logari thmic length of / * " equals that of 7. More precisely, the logar i thmic length of 7 is Thus, J - = [ - 1 , - e - " * 7 ) ] . Then Beur l ing in his thesis has shown that W(I;D) > W ( / * - ; D ) . Th is inequal i ty is known as Beur l ing 's shove theorem [23, pp. 5 8 - 6 2 ] 2 ; the name (suggested by Professor Baernstein) comes from the fact that can be visualized as I wi th al l of its intervals "shoved" over to the right. A generalization of Beur l ing 's shove theorem to higher dimensions and to some more general domains D has been formulated by Essen and Hal iste [45]. Remark 8.1. If D is a star-shaped domain wi th —1 € dD and ( — 1,0] C D, then for any finite union I of closed subintervals of [—1,0) we have where (•)* is M a r c u s ' radial rearrangement. Thus , Beur l ing 's shove theorem says precisely that w i ( 0 \ J ) < W ! ( ( D \ 7 ) * ) . In fact Beur l ing 's shove theorem implies Conjecture 1.6 for the domain D = D \ 7 (we already alluded to this in §1.3) . To see this, we must prove that for all r > 0. For r > 1 this is t r iv ia l , since both sides of the above inequali ty vanish. The case of r = 1 has already been done and is a direct consequence of the shove theorem. It remains to consider the case of r < 1. G iven a subset J of [—1, 0), wri te Jr = J f l [—r, 0). I c la im that the 2Unfortunately the thesis [23] is not widely available, but an account of the theorem and of its proof can also be found in [79, §IV.5.4]. (£>\/)* = D\I (8.1) wr(B\I) < wr{B\I^) 295 Chapter IV. Radial rearrangement following chain is valid: = w(0,<9O(r);D W U ) (8.2a) = u;(0,<9O(r);O (8.2b) = w(0,5D;D\r" (8.2c) < LL>(0,dtyB\(r (8.2d) = w(0,<9O(r);D (8.2e) < w(0,<9O(r);D W \ ( / ^ ) r ) (8.2f) = w(0,<9O(r);D ( r ) \ / - ) (8-2g) = wr(B\I^). (8.2h) We now justify each step here. First, (8.2a) and (8.2h) follow directly from the definitions of wr. Equalities (8.2b) and (8.2g) are trivial since D(r)\7 = B(r)\Jr and D(r)\J*- = B(r)\(I*-)r. Also, (8.2c) and (8.2e) follow from the scaling invariance of harmonic measure. Inequality (8.2d) comes from Beurling's shove theorem. Only inequality (8.2f) remains to be verified. It follows from the monotonicity of harmonic measure with respect to increase of the domain as soon as we prove the inclusion (/-) r C r ^ - 1 / ^ ] . (8.3) Now, to prove (8.3), since both sides of (8.3) are possibly empty intervals of the form [—r, —a], it suffices to prove that P((nr) < pirir-'lrD- (8-4) 296 Chapter IV. Radial rearrangement Without loss of generality assume that (/*~)r 7^ 0- Now, the logarithmic length p is invariant under dilation and under the (•)*" operation so that pirir-Xr) = pdr-ilrD = p ^ h ) = p(Ir) f° , ,dx f° . .dx , = j_ 1iAx) = J l/ru(-i,-r]W^ + logr = p(Ir U (-1, r]) + log r = p((/r U (-1, r])^) + log r > + log r /"° / ,dx , = / l/-(a;)-r-r-r-logr J-i \x\ where we have used the fact that [—1, — r] C I*~ which follows from the assumption that ( / - ) r ^ 0-The following result is implicit in Beurling's proof [23] and in the work of Essen and Haliste [45]. Proposition 8.1 (Beurling [23]). Let D be a domain in the plane containing the interval (—1,0]. Suppose that D satisfies both of the following two conditions: (i) for any fixed x £ ( — 1,0), the Green's function g(-,x;D) is increasing on the interval (—l,x) and decreasing on the interval (x,0) (ii) the domain D is star-shaped. Then, for every finite union I of closed subintervals of [—1, 0) we have W(T,D)>W(I^;D). (8.5) A proof of a more general result (Theorem 8.2) will be given later. Given this, to prove Beurling's shove theorem for D = D it suffices to verify (i), which can easily be done since g{-, -;D) = Q has an explicit expression given by (1.8.5), so that (i) can be verified by elementary methods. We now use Proposition 8.1 to show that the inequality (8.5) also holds for two quite different classes of star-shaped domains D. Under assumption (a), the 297 Chapter IV. Radial rearrangement following result generalizes the one-dimensional case of Essen-Haliste's [45; Thm. 1]. Under assumption (b), we shall use Theorem 1.8.2 in our proof. T h e o r e m 8 . 1 . Let D be a domain in the plane with (—1,0] C D. Assume that D satisfies at least one of the following auxiliary conditions: (a) D is simply connected and reflection symmetric about the real axis (b) D contains the disc D(—|; \). Then condition (i) of Proposition 8.1 holds. Hence if D is also star-shaped, then for any finite union I of closed subintervals of [—1,0) we have W(I;D) > W(I*~;D). A proof will be given after the remarks below. The above theorem was given in Pruss [84]. Remark 8.2. Note that if D is star-shaped then it is automatically simply connected so that (a) in that case only requires reflection symmetry about the real axis. It is not hard to trace through the proofs to conclude that if D is star-shaped and satisfies at least one of the conditions (a) and (b) of Theorem 8.1, then equality holds in (8.5) if and only if I = I*-. A discrete analogue of the Beurling shove theorem for D = O was given in §11.10. It is not known whether an analogue of Theorem 8.1 under either of its auxiliary conditions can also be proved in the discrete setting of that section. A generalization of the shove theorem to a case where D = D but I is allowed to range over all closed subsets of [—1,1]\{0} has been conjectured by Segawa [96, Remark on p. 183]. It can be seen without undue difficulty that the star-shapedness of D does not by itself imply condition (i) of Proposition 8.1. (A counterexample may be constructed by letting De = C\{—reie : r > 1}, fixing 0<ce<l</5<7<oo and noting that if e > 0 is sufficiently small then g(/3,~f;De) < g(a,j;De).) However, the author does not know of any counterexample to the inequality W(I\D) > W(I^;D) for D star-shaped, and so the question of whether the star-shapedness of D is by itself sufficient to guarantee (8.5) appears to be open. 298 Chapter IV. Radial rearrangement Proof of Theorem 8.1. First assume that condition (a) holds. Let L = D n (—oo, 0] 3 (—1,0]. By simple connectivity and reflection symmetry, the set L is an interval. Let / be the Riemann map of D onto D with /(0) = 0 and /'(0) > 0. By reflection symmetry it is easy to see that / is real on L and in fact that it is a strictly increasing bijection of L onto the interval (—1,0]. But, g(r,R; D) = g(f{r), /(i?);D), and so by the monotonicity of / and by the fact that D satisfies (i) of Proposition 8.1, it follows that D must also satisfy it. Now assume instead that condition (b) holds. Given any domain D in the plane and a point z £ D, define D'z = z + {—z + D)', where {—z + D)' is defined as in §1.8 via equation (1.8.1). Then, using condition (b), it is easy to verify on geometric grounds that for any R £ (—1,0) we have (—1,0) C D'R. But by Theorem 8.2 we have g(-,R;D) radially decreasing away from R on D'R, hence a fortiori also on (—1,0), so that condition (i) of Proposition 8.1 follows immediately. • We now state a very general proposition which together with Theorem 8.1 gives a number of results of the type of the shove theorem. T h e o r e m 8 . 2 . Let D be a domain in the plane containing the interval (—1,0]. Suppose that D satisfies both of the following two conditions: (i) for any fixed x. £ ( — 1,0), the Green's function g(-,x;D) is increasing on the interval (—1, x) and decreasing on the interval (x, 0) (ii) the domain D is star-shaped. Let $ be a real function on D. For ( £ dD let * ( C ) = lim sup $ ( 2 : ) as z tends to £ from within D. Make the following assumptions: (a) $ is subharmonic on D and continuous on (—1,0); moreover, & > on dD (b) the solution h of the Dirichlet problem on D with boundary value $|9jD on dD has h(x) monotone increasing on (—1,0) 299 Chapter IV. Radial rearrangement (c) $ is monotone decreasing and continuous on (—1,0). Then, r*(D\i) < r*(D\in, where T$(C/) is the value at 0 of the solution of the Dirichlet problem on U with boundary value $\dU on dU. Remark 8.3. Theorem 8.1 gives us sufficient conditions for (i) and (ii) to hold, so that in combination with it we get a generalized shove theorem. Remark 8.4- Proposition 8.1 follows from Theorem 8.2. To see this, let $ be the function which is identically 1 on dD and identically 0 in D. Clearly $ satisfies (a)-(c) (in (b) we will of course have h = 1). Moreover, it is clear that in this case Y^(D\I) = W(I;D) and T9(D\I~) = W{I~;D). Remark 8.5. Theorem 8.2 implies Theorem 2.1. To see this, note that D satisfies conditions (i) and (ii) of Theorem 8.2 by Theorem 8.1, since any circularly symmetric set is reflection symmetric with respect to the real axis. Since D ^ C (as D is Greenian since we had assumed that all domains are Greenian in this chapter) and as D is circularly symmetric, we will have def L= sup{L' < 0 : L' fi D} not equal to —oo. Rescaling the problem if necessary we may assume that L — —1, so that —1 £ dD. Replacing 7 by 7 fl [—1,0), we may assume that 7 is a subset of [—1,0). Let <3> £ T. This $ satisfies (a)-(c). Indeed, (a) and (c) follow from the assumption that $ £ T. Now, to verify condition (b) note that h(z) = LHM(z, $; TJ) by Theorem 1.5.4 and so condition (b) follows by Theorem 1.6.5. Then the conclusion of Theorem 2.1 follows from Theorems 8.1 and 8.2 together with equality (8.1), since D is star-shaped and —1 £ dD. Theorem 8.2 and Theorem 8.1 together yield the following generalization of Beurling's shove theorem which is interesting even in the case D — D. We call a set A a s m o o t h a r c if it is the homeomorphic image of either an interval or the circle. T h e o r e m 8 .3 . Let D be a star-shaped C1 domain which is reflection symmetric about the real axis and contains the interval (—1,0]. Let A C D be a smooth arc which is also reflection 300 Chapter IV. Radial rearrangement symmetric about the real axis and which intersects the positive real axis. Then, U(0,A;D\I)<L,(0,A;D\I-)-Proof. Conditions (i) and (ii) of Theorem 8.2 hold by Theorem 8.1 (i). Define $ to vanish on D and be equal to the indicator function 1A of A on dD. Conditions (a) and (c) of Theorem 8.2 then are trivial. It suffices to verify condition (b). We have \P = 0 so that h(z) = u>(z, A; D). Let / be a Riemann map of D onto the left half plane H = {z £ C : Re z < 0} with /(0) = —1 and /'(0) > 0. Then, / maps A into interval of the imaginary axis which is symmetric about the positive real axis. Moreover, / is monotone increasing on (—oo, 0]HD and maps (—oo, 0]nZ) onto (—oo,0]. (The preceding two sentences both use the reflection symmetry of D.) By conformal invariance of harmonic measure we have h{z) = u{f{z),f(A);H). To verify condition (b) of Theorem 8.2, because of the monotonicity of / on (—oo, 0] fl D it suffices to prove that u{w,B;H) is increasing for w £ (-co, 0) and B an interval of iK symmetric about the point 0. We now do this. Let —oo < w' < w < 0. Let A = w'/w > 1. Then since both the domain H and harmonic measure are invariant under dilations we have u(w, B; H) = u(Xw, XB; H) = u(w', XB; H) > OJ(W',B;H), as desired. The last inequality followed from the fact that XB D B and from the monotonicity of harmonic measure with respect to the set being measured. • It remains to prove Theorem 8.2. The careful reader will notice that the proof bears some resemblance to that of Theorem II.10.1. Indeed, both proofs are based on the ideas of Beurling's original proof of his shove theorem (see [23, pp. 58-62] or [79, §IV.5.4]). 301 Chapter IV. Radial rearrangement I 7 i ^1 ^ • • • • • • JC0=-1 * i 0 Figure 8.1: The decomposition of I into 7i and 72 in a case where n = 3. Proof of Theorem 8.2. Let n be the number of disjoint intervals in the set (—1, 0)\7. If n = 1 then I must be of the form [—1, —a] and I*~ = I so that we are done. Proceed by induction. Suppose that n > 1 and that our Proposition has already been proved for n — 1. Let zo = inf[-l,0)\/. Then XQ either equals —1 or is the right endpoint of one of the disjoint intervals making up I. Let x\ = inf{a; > xo • x fi I}. Then x\ is the left endpoint of an interval of 7, x\ > XQ and (a:o,a;i) is one of the disjoint intervals making up the set (—1,0)\7. Let A = XQ/X\ > 0. Put h = 7 n[-l , z 0 ] and (See Figure 8.1.) Then I = h U 72. Put and 72 = /n[a;i,0). J l = J i Jo = A72. Finally, set J = Jx U J 2 . Then, because xi is the left endpoint of an interval of 72 and XQ is either —1 or the right endpoint of an interval of 7i it follows that (—1, 0)\J consists of precisely 302 Chapter IV. Radial rearrangement n — 1 disjoint intervals, since dilation of T 2 by the factor A will merge the left-most interval of I2 with the right-most interval of I\. The induction hypothesis then implies that r * ( £ > \ j ) < r* . (D\ j* - ) . Moreover, p(I) = p(h) + p(h) = p(Ji) + p(J2) = P(J) because of the dilation invariance of the logarithmic measure p. Hence, J* - = J*~ so that r*(D\j) < r » ( D \ / ^ ) . We will thus be done as soon as we can prove that r « ( D \ 7 ) < r « ( D \ J ) . (8.6) (Note the analogy between inequality (8.6) and inequality (II.10.3) which played a similar role in the proof of Theorem II.10.1.) Let / be the solution of the Dirichlet problem in D\I with boundary value $ on d(D\I). I claim that / is subharmonic on TJ if we define f(x) = $(x) for x S 7. To see this, let be as in the statement of our proposition and additionally let *B(z) — <&(z) for z 6 TJ\T. Let g be the solution of the Dirichlet problem on TJ\T with boundary value ^ on d(D\I). Then, by the subharmonic function inequality [39, inequality (1.VIII.8.4)] which is essentially a refined version of Theorem 1.4.2 we have g > $ everywhere on D. Now to show that / is subharmonic on TJ, it suffices to verify the subharmonicity at every point x of T n TJ since / is harmonic everywhere else. Such a point x is regular (with respect to the domain TJ\T) by Theorem 1.5.3 since I is a collection of intervals and lies in the complement of TJ. Hence, / is continuous at x. Moreover, we have / > g on TJ since $ > $ everywhere. But g(x) = f(x) = <&(x) by regularity of x and continuity of $ and ^ at x. We have already seen that g > $ on TJ. Hence / > $ on TJ. Then, if r > 0 is sufficiently small, we have f(X) = $(X) < \ II *<^2 [[ f> 303 Chapter IV. Radial rearrangement where we have used the subharmonicity of $ on D. Hence / is indeed subharmonic at x, and thus everywhere on D. Note that D is regular by Theorem 1.5.3 since D is star-shaped so that for each point of its boundary the ray pointed directly away from the origin must lie in the complement of D. I claim that h is the least harmonic majorant of / on D, where h is as in condition (b) of our Proposition. To see this, note first that h indeed is harmonic on D. We must prove that h > / there. Let <p = h — f. Then, <f> is the solution of the Dirichlet problem with boundary value 0 on dD and h — f on I D D. But on I D D, f agrees with $. But h is larger than $ on dD, and so by the maximum principle must everywhere be larger than $ and hence must be larger than /on / f lDso that <j> indeed is positive on d(D\I) and hence is positive on all of D by the maximum principle. Hence h is a harmonic majorant of / . Moreover, since (f> vanishes on dD it follows that we must indeed have h equal to the least harmonic majorant of / . Then, by the Riesz decomposition (Theorem 1.5.11) write f(z) = h(z)- g(z,w;D)dpf. JD But / is harmonic on D\I so that the support of pj must lie within I (Remark 1.5.9). Hence, f(z) = h(z)- g(z,w;D)dpf- g(z, w; D) dpj. Jii Jh Let /*(*) = / 9{z,u>;D) dpf, Jlk for k = 1,2. Then fk is a harmonic function on D\Ik (this follows from the harmonicity of g(-,w; D) away from w). Define F(z) = h(z)-f1(z)-f2(X-1z) on D. Then, F is harmonic on D\J as can be easily seen (use star-shapedness to note that if z G D then \~xz G D as well). Let / be the solution of the Dirichlet problem on D\J with boundary value $ on d(D\J). I claim that F < f. Suppose that this is true. Then, we will 304 Chapter IV. Radial rearrangement have r « ( i > \ / ) = /(o) = / i ( o ) - / i ( o ) - / 2 ( o ) = h(o) - MO) - MX-'O) = F(O) < f(o) = r * ( D \ j ) , and the proof of (8.6) will be complete. Hence we must verify that F < / . By the maximum principle since both F and / are harmonic in D\J it suffices to show that if zn —> C S 9(D\I) with zn G then limsup(F(zn) - / > „ ) ) < 0. n Note that F(zn) — f(zn) = f2(zn) — f2(X~1zn). Suppose first that z G dD. Since we may neglect subsets of the boundary with null harmonic measure, we may assume that z fi D fl J since 3D DD fl J can contain at most one point by the star-shapedness of D. Then, recall that D was a regular domain and that / — h was the solution of a Dirichlet problem in D\l with boundary value 0 on dD. The boundary value function for this Dirichlet problem is continuous on dD\D fl I (indeed, it is identically zero there) so that it follows that (f — h)(zn) —>• 0. Hence, fi(zn) + f2(zn) —> 0. Since both / i and f2 are positive, it follows that limsup/2(z„) = 0. Hence, limsup(F(zn) - f(zn)) = limsup(/2(zn) - f2{X~1zn)) < 0, n n as desired. Now consider z G J. Set both / equal to $ on J and / equal to $ on I. Regularity of D\I and D\J (easy to check via Theorem 1.5.3) shows that F, f and / are continuous in D since $ is continuous on I n D. Hence, what we must verify is that F(z) < f(z). Now, f(z) = ${z). Thus, we must prove that F(z) < $(z). Consider first the case where z G J\ = I\. Then, F(z) = h(z)-f1(z)-f2(X-1z) < h{z) - h{z) - f2(z) = * ( » , 305 Chapter IV. Radial rearrangement where the inequality followed from the fact that g(X~1z,w; D) > g(z,w;D) for w £ I2 and z £ I\ since then z < \~lz < w and we may apply assumption (i). Consider now the case where z G J2- Then, A _ 1 z G I2 and F(z) = h(z)-f1(z)-f2(\-1z) </ i (A- 1 z)-/ 1 (z)-/ 2 (A- 1 z) <h{\~lz)-h^z)-f2{\-lz) = / ( A - 1 * ) = Hx~lz) < * ( * ) , as desired. To justify the above inequalities, note that the first inequality followed from as-sumption (b) on the monotonicity of h. The second inequality followed from the inequality g(z, w; D) > g(\~xz, w; D) valid for z G J2 and w G h by assumption (i) as then w < z < \~lz. The final inequality followed from assumption (c). Hence we have seen that indeed F < $ on J and our proof is complete. • 9. A discrete one-dimensional analogue 9.1. Statement of results Fix p G [0,1]. Let {rp : i G Z^} be a random walk on {1, 2,..., N + 1}, with rg = 1, ^ K + i = r? + 1 I r?) = p, P(rpi+1 = n — 1 I rp = n) = 1 — p, if n > 1, and P(rf + 1 = l | r f = l) = l - p . Thus, we have a simple random walk on a "blind alley," with the boundary condition that at the "wall" (i.e., at 1) when we try to go to the left then we stay put. The open end of the blind alley is at iV + 1. 306 Chapter IV. Radial rearrangement Let si, s2, • • •, SN G [0,1] be given. Every time the random walk rp is at a point n G {1, 2,..., N}, let there be a new danger (independent of anything that had happened until that time, and in particular independent of the outcomes of any previous visits to the point n) and let the probability of surviving it be sn. Let PN(si,... ,sjv) be the probability that the random walk has survived all the time up to its arrival at the point N + 1. More precisely, let Xo, Xi,... be random variables which are independent and identically uniformly distributed on [0,1]. Let TN = inf{t > 0 : rf = N+ 1}. Of course P(TN < oo) = 1 if p > 0. Then we have PN(su ..., sN) = P ^ f| {Xi < srf}j. Note that PN(su...,sN) = SiS2...sN, (9.1) PN(su...,sN) = 0 and P£(l, , . . , l ) = l, for every p > 0. Since we have a random walk in a blind alley, and there are dangers, we may term the model a random walk in a dangerous blind alley. Theorem 9.1. Let S i , . . . , S N G [0,1], and let ,.. .,sN be si,..., S N rewritten in decreasing order. Then for p G [0,1] we have pN(s1,:..,sN) < PN(SI ,...,sN), with equality if and only if at least one of the following conditions holds: (a) (s!,...,sN) = (sf,...,sN) (b) Sk = 0 for some k G {1,..., A^ } (c) P = 1 307 Chapter IV. Radial rearrangement ( d ) p = 0. Th is result is analogous to an inequali ty of Essen [43, T h m . 2] concerning rearrangement in a certain second order difference equat ion. His difference equation is very similar to that which 1/2 must be solved to compute P^J ( s i , . . . ,syv)> but there are sti l l some essential differences. We wi l l say more regarding the work of Essen in §9.3, below, where we shall state the actual 112 difference equations whose solution gives P^ , and in §9.4 where shall discuss the connection wi th Essen's analogous continuous case [42, T h m . 5.2]. It is quite possible that Essen's methods [43] could be adapted to prove Theorem 9.1, at least in the case p = \ , even though his results do not appear to apply directly. However, we prefer to use different tactics (keeping the same overall strategy) which, in an elementary way, exploit the l ineari ty properties of a funct ion appearing in the expl ici t formula for PN. Our proof wi l l be given in §9.2. F ina l ly , it should be noted that it does not seem that the methods of Baernstein [11] can be used to prove results like Theorem 9.1. The heuristics behind Theorem 9.1 say that if we consider the random walk only unti l such t ime as it hits the point N +1, then it wi l l spend more t ime further away from this point than it does nearer to it, so we wi l l improve safety if we reorder the dangers so the more dangerous ones are near iV + 1 where the random walk spends less t ime. The author has not found a way of making this intui t ion into a rigorous proof. One might hope to find a probabi l ist ic proof along these lines, but no such proof appears to be available right now, and it does not appear at al l easy to produce such a proof. If p = ^ then Theorem 9.1 may be thought of as a discrete one-dimensional analogue of Conjecture 1.6. The author is grateful to Professor A lber t Baernstein II for suggesting that the author also consider the case p ^ | . Theorem 9.1 says that PN(si,..., SN) increases if we rewrite the sn in decreasing order. One might hope that on the other hand if we rewrite them in increasing order, then PN(si,..., s^y) decreases. However, the hope would be in vain, as can be seen from the fol lowing theorem. 308 Chapter IV. Radial rearrangement Theorem 9.2. Fix p £ (0,1). Then, for all sufficiently small e > 0 we have 1,1). We now give a certain simple result which is not completely trivial. Theorem 9.3. Let s\,..., SN £ [0,1] be given. Fix j £ {1,..., iV}. Then for p £ [0,1] we have with the obvious conventions if j is 1 or N. Moreover, equality holds if and only if at least one of the following conditions holds: (a) there is a k £ {1,..., j — 1, j + 1,..., ./V} with sk = 0 (b) sk = 1 for every k £ {1,..., j} i (c) p = 1 and Sj = 1 Intuitively Theorem 9.3 says that if we make a dangerous road shorter by removing a segment then the road becomes safer for a random walk. We will give a proof of Theorem 9.3 in §9.2 as a by-product of our proof of Theorem 9.1. This result is an analogue to the first part of Problem 3.1, in that the passage from s\,..., SJV to si,..., S j - i , Sj+i, • • •, SJV is analogous to the cutting operation in that Problem. Now, let si, S2, • • • £ [0,1] be an infinite sequence. Define the random walk rf on Z + with the same transition probabilities as the previous walk on {1,..., N + 1}. Let Ls be the first time that the random walk fails to survive a step. More precisely, we define P N ( S 1 : . . . , S N ) < P ^ ^ S i , . . . , Sj-i,Sj+i, . . . , SN), (9.2) (d) p = 0. Ls = i n f > 0 : X{ > sr,}• The proof of the following result is given in [87] where it is done via the methods of §11.9.3. We shall not give the proof in this thesis. 309 Chapter IV. Radial rearrangement Theorem 9.4. Let si, S 2 , • • • 6 [0,1] and let sJ, sJ, . . . be the decreasing rearrangement of the Si. Letp e [0, §]. Then P(LS > n) < P(Lsy > n), for every n > 0. It is not known whether the condition p € [0, |] can be relaxed topg [0,1], although it is easy to see that Theorem 9.4 does hold for p = 1. Open Problem 9.1. Does Theorem 9.4 also hold for p 6 (|, 1)? Now, fix p € [0,1]. Let $ be a real valued function on Z+ satisfying the "convexity" (one might also use the term "subharmonicity") condition $(n) < ( l-p)$(n-l)+p$(n+l) , (9.3) for n € Z + , where <&(0) ==$(1). The condition (9.3) is equivalent to positing that <I>(r;) is a submartingale. It is easy to inductively see (starting with the fact that $(1) = 3>(0) so that $(1) > $(0)) that (9.3) implies that $ is increasing. Open Problem 9.2. Does it follow from (9.3) that E[$(rL,)] < £ [ $ ( r x a Y ) ] ? If p — | then this is a one-dimensional discrete analogue of Conjecture 1.4. Now we just wish to note that some sort of convexity condition like (9.3) on $ in addition to the increasing character of $ is necessary if p £ (0,1). For, if we do not have this condition, then we may adapt our counterexample to Conjecture 1.5. Set si = ^, s2 = 0, S3 = \ and s4 = s5 = • • • = 0, and let $(re) be 0 for n < 1 and 1 otherwise; a simple computation then shows that then the answer to Problem 9.2 would be negative. Note also that if we let syv+i = SN+2 = • • • = 0 and set $(n) = max(n - N,0) then E[$(rLs)] = PN(s\,..., SN) SO that Theorem 9.1 is a special case of Problem 9.2. 310 Chapter IV. Radial rearrangement Finally, the following result should surprise no one, but we state it for completeness. If we increase the probability of going towards our goal then certainly the probability of arriving at it should increase. T h e o r e m 9 . 5 . Let 0 < p < r < 1 and let si,..., SJV G [0,1]. Then P N ( S 1 , . . . , S N ) < P N ( S ! , . . . , S N ) , with equality if and only if one of the following conditions holds: (a) Sk = 0 for some k £ { 1 , . . . , TV} (b) si = • • • = SN — 1 and p > 0. We now outline a proof of Theorem 9.5, leaving the details as an exercise to the reader. Consider a more general case of a random walk defined as above, but instead of having a constant probability p of going to the right and 1 — p of going to the left, allow this probability to vary with position, so that the probability of moving to the right from B E { 1 , . . . , N} is tn £ [0,1] and the probability of moving to the left is 1 — tn. As before, moving to the left from 1 results in standing still. Just as before, we can define the probability of the random walk getting from 1 to + 1 without having fallen into any of the dangers. I claim that this probability will increase if any one of the tn is increased; clearly this would a more general result than Theorem 9.5 (though of course we would have to ensure that appropriate conditions of equality hold, the verification of which we leave as an exercise for the reader). To prove the claim, fix n. Assume n > 1; the case n = 1 is handled similarly. We want to see the dependence on tn. So, let A be the probability that a random walk (with movement probabilities defined by the tj) starting from n — 1 will eventually arrive at n without having fallen into any of the dangers. Let B be the probability that such a random walk starting from n + l eventually arrives at n without having fallen into any of the dangers and without having first arrived at iV + 1. Let C be the probability that such a random walk random when started from n + l eventually arrives at N without having fallen into any of the dangers and without 311 Chapter IV. Radial rearrangement having first arrived at n. Finally, let P be the probability that a random walk starting at n eventually arrives at N + 1 without having fallen into any of the dangers. The probability of a random walk from 1 arriving safely at N + 1 is proportional to P, so we need only compute how P depends on tn. Also, A, B and C are independent of tn and satisfy the equation From this point on it is an elementary exercise to verify that P increases with tn, and to determine the conditions under which the increase fails to be strict. 9.2. Various useful identities, formulae and some proofs In this section we shall prove Theorems 9.1, 9.2 and 9.3, assuming an explicit formula (Theo-rem 9.6, below) for PN(si,.. . ,SJV). The proof of this formula will be given in §9.3. First we note a simple probabilistic identity which will later prove to be of use. Suppose p £ (0,1), N > 2 and si — 1. Then it does not matter how long the random walk spends at the point 1, since it will survive to eventually leave 1 and go to 2. Whenever it subsequently goes left from 2, it will survive until its eventual return to 2. Hence, we may form a certain correspondence between random walks on {1, 2,..., N} and those on {2,..., A^ } in such a way as to prove that P = sn{l - tn)AP + sntn(BP + C). PN(l,s2,...,sN) = P\ * _ l ( S 2 , - - - , S j v ) - (9.4) It is trivial to also verify that this continues to hold if p £ {0,1}. Now, for positive n, let ihN^n(ai,..., a^ v) be the sum of all terms of the form (9.5) with 1 < ii < h + 1< i2 < i2 + K • • • < in < in + 1 < N. 312 Chapter IV. Radial rearrangement Explicitly we have i>N,n(ai, • • •, GAT) iV-2n+l N-2n+3 N-l il — l J2=ii+2 i n = t n _ i + 2 with the convention that empty sums are equal to zero. Clearly Vw.n is a function of N variables, is linear in each variable if the others are fixed, and vanishes identically for 2n > N. Let L f J VN(a1,...,aN) = 1+ ^ (-l)n^iv,n(ai,...,a7v), n=l for N £ ZQ , where [x\ denotes the greatest integer not exceeding x. Note that = 1 for i v e { 0 , i } . Now, I claim that #jv+i(ai,. ..,aN+i) = *Ar(a2, a3, • ..,aN+1) - a^^N-i(03, • . .,ajv+i)> (9.6) for N > 1. This identity is central to our work. The proof of the identity is not very difficult. For, take one of the terms in ^ J V + I • • •, a;v+i). It will be either of the form (-l)na;iazi+iaJ 2a;2 +i .. . a;na2„+i with 1 < n < and 1 < ii < ix + 1< i 2 < i 2 + 1 < • • • < in < in + 1 < N + 1, or else it will be identically 1. If 0 1 occurs in this term then i\ — 1 so that a2 must also occur in it. It is easy to see by the definitions that it must then also be a term of —(11(12$!N-\{a3, • • o/v+i). On the other hand, if a\ fails to occur in the term, then this term must be a term of \IJyv(«2) • • • > &N+I)- Conversely, it is easy to verify that any term of the right hand side of (9.6) is also a term of the left hand side, and the proof of the claim is complete. As a corollary of (9.6), we can see that VN+1(0,a1,...,aN) = *jy(ai,.. .,ajv), (9.7) 313 Chapter IV. Radial rearrangement for N > 1. For TV = 0 this also holds trivially, and hence (9.7) is valid for all N > 0. Also, by (9.6) and (9.7) we obtain $N+i(ai,0,a 3 , . . .,ajv+i) = * J V (0, a3,..., aN+i) = *^_i(a 3,..., ajy+i), (9.8) for N > 1. Note that \P;v(ai,..., a;v) = \Pjv(aAA,..., a i ) , so that *jv(ai, • • •, ajv) = *jv_i(ai,.. - ,ajv-i) - aNaN-i^N-2{ai, • • -,aN-2), (9.9) whenever > 2, by (9.6). Now, define M P ) = < p, if n is even, 1 — p, if n is odd. Note that d>n+i(p) = <f>n(l —p) = l — <f>n(p) for every n and p. Because the expressions that will be involved would be unmanageable otherwise, it will be useful to have two more abbreviations. Let *^(ai,...,ajv) = VN+1(l,d>i(p)ai,.. .,d>N(p)aN) and ypN(ai,...,aN) = *Ar(<Mp)ai,...,<Mp)ajv)-At times the reader will be implicitly expected to be able to use the definitions to mentally rewrite the WN and ^ P N in terms of the *PJV The following result then gives a formula for the probability of traversal; a proof will be given in §9.3. Theorem 9.6. For p 6 (0,1] and s\,..., sjy € [0,1] we have Pf,{s1,...,sN) = =j!- -. (9.10) Moreover, the denominator is always strictly positive under the above conditions. 314 Chapter IV. Radial rearrangement Assuming Theorem 9.6, I claim that WN(l,a2,.,..,aN)=p¥N_1(a2,.:.,aN) (9.11) For, if p is fixed then both sides are linear in any one variable when the others are fixed, so that it is enough to verify (9.11) for a 2 , . . . ,a^ £ (Oil]- Moreover, both sides of (9.11) are continuous in p and hence it suffices to consider p £ (0,1]. But under such circumstances (9.11) follows from (9.4) and Theorem 9.6. Note that if p = \ then (9.11) takes the particularly simple form yN+i(l,±,x2,.. .,xN) = i$jv(l,x2, ...,xN). Lemma 9.1. Let N > 1 and fix a i , £ [0,1]. Suppose p £ (0,1]. Then *AT + 1 (x, 4>i(p)ai, • • •, <f>N{p)aN) is strictly positive for every a; £ [0,1]. Proof. Fix ai , . . . , a/v- Now, x H 4 *N+I(X, <f>i(p)ai,..., 4>N(p)aN) is a linear function and hence it suffices to verify its strict positivity for x £ {0,1}. If x = 1, then the strict positivity immediately follows from the "moreover" in Theorem 9.6. Now, for x = 0, by (9.8) we may write ^N+i(0,4>1(p)a1,...,4>N(p)aN) = *Ar+2(l,0,<^i(p)a1,.. .,(j>N(p)aN) = *TV+ 2(1,^I(1 -p) -0,<?!>2(1 - p)a1,...<f)N+1(l -p)aN). The strict positivity of this again immediately follows from the "moreover" of Theorem 9.6. • We also note that 0, h,..., bN) = * M ( « I , • • •, aM)^N(bu ...,bN). (9.12) 315 Chapter IV. Radial rearrangement The easiest way to prove this is to note that every term of the right hand side is a term of the left hand side and vice versa, much as in the proof of (9.6). F ina l ly , it is easy to use the fact that (f>n(p)(f>n+i(p) = p(l — p) = 4>n(l — p)4>n+\(1 — p) for every n together wi th the way that \ P M is defined to show that ®M(<f>i(p)ai,...,4>M{p)aM) = *M(<?!>I(1 - p)ax,... ,(f>M(l -p)aM). (9.13) We can wri te this concisely as = ^ ] ^ p - Now, recall ing that 1 — <j)n(p) is either p or 1 - p for any n, and apply ing (9.12), followed by (9.13) if necessary, we see that * M + J V + i ( a i ) • • •, a M , 0, h,..., bN) = W M ( a 1 } a M ) ^ N ( b i , • • •, 6JV) ( 9 J 4 ) = * M ( O I , • • . , C t M ) * ^ ( ^ l , • • -,bN), where r — 1 - 4>M+2 (P) • L e m m a 9 .2 . Let o i , . . . , am and b\,..., bn be in [0,1]. Let p £ (0,1) . Suppose that m i n ( a i , . . . , am) > m a x ( 6 i , . . . , bn). (9.15) Then (« i a m _ ! ) * P ( 6x , . . . , 6„) >Wm(a1,...,am)^pn_1(b2,...,bn). (9.16) Moreover if equality holds then at least one of the aj vanishes. Proof. We proceed by induct ion on max(m,n). If max (m, n) = 1 then (9.16) becomes 1 > f ? ( o i ) = 1 - <f>1{p)a1. This is clearly true, and str ict inequali ty holds unless a\ = 0. Now suppose that L e m m a 9.2 has been proved when max (m, n) = N — 1 and also assume that we have m a x ( m , n) = N > 1. B y (9.6) and (9.9), we see that (9.16) is equivalent to the 316 Chapter IV. Radial rearrangement inequality Wm_1(au...,am^1)^_1(b2,...,bn) -Wm_1(a1,...,am_1)cp1(p)<p2(p)blb2^pn_2(b3,...,bn) > ^li-i ( a i , • • • , flm-i)^_i (h,..., bn) - cpm-i (p)c()m(p)am^1am^Pn_2(a1,am_2)*£_i(&2, • • •, M -Note that we have implicitly used (9.13) after applying (9.6). Clearly our last inequality is equivalent to p(l - p) Wm_2 (oi,..., am_2)vp^_1(62,..., bn) > p(l - p)6i62*^_i(ai, • • •, am_i)^_2(63, • • •, &„)• But this is true by the induction hypothesis since max(ra — l,n — 1) = N — 1 and since a m _ia m > 6i62 because of (9.15). Moreover, if equality holds then either am-\am vanishes or, again by the induction hypothesis, one of ai , . . . ,a m _i vanishes. • The following lemma is an exact equivalent of Essen's [43, Lemma 1], and indeed our strategy for the proof of Theorem 9.1 is quite similar to that of Essen. Of course we use the convention that the infimum of an empty set is equal to +oo. Lemma 9.3. Fixp € (0,1). Suppose that a i , . . . , a^ v G [0,1] and assume that i G {1,..., N — 1} has the property that inf {ai,..., aj_i} > max(a;,..., a;v) (9-17) (this condition on i is trivially satisfied if i — 1). Finally suppose that a, < max(a,',..., a/y) (9.18) and that j G {i + 1,..., N} is such that aj = max(o„ . . . , a^)- Then WN(ai,'. ..,aN) > *^(ai,..., a,-_i, aj, ai,ai+i ..., aj^, aJ+1,..., aN). (9.19) For the rest of this section, in interpreting (9.19) and similar expressions we use the convention that a sequence of the form afc,...,an- is empty and omitted if n < k. We shall assume 317 Chapter IV. Radial rearrangement Lemma 9.3 for now and show how it implies Theorems 9.1 and 9.3. A more elementary method of proof of Theorem 9.3 was kindly communicated to the author by Mr. Ravi Vakil. His approach in effect reduces the question to consideration of the movement of the system between the points j — 1, j, i+1, N and oo, where oo indicates that the random walk has been terminated by having fallen into one of the dangers. This new system is sufficiently small that explicit computation can be used to prove the desired result (cf. the outline of proof of Theorem 9.5, above). However, since we have Lemma 9.3 available (and we will definitely need it for Theorem 9.1), we proceed as follows. Proof of Theorem 9.3. Assume that S \ , . . . , S j - i , S j + i , . . . , SJV £ (0,1]. (If one of these vanishes then the result is trivial.) The result is easy if p 6 {0,1} so assume 0 < p < 1. It is clear on probabilistic grounds that we may assume that SJ = 1 since changing Sj = 1 to Sj < 1 would strictly decrease the left side of (9.2) and leave the right side unchanged. By Theorem 9.6, we need only show that p~1WN(s1,...,sN) > * ^ _ 1 ( s 1 , . . . , S j _ i , S j + i , . . . , a ^ ) (9.20) and that equality holds if and only if si = • • • = Sj• = 1. We shall prove this by induction on N. If N = 1 then the result follows immediately from the definition of the y^v- Suppose that N > 1 and the desired result has been proved for N - 1. Assume first that S i = 1. If j — 1 then by (9.11) we do have equality in (9.20) as desired. If j > 1, on the other hand, then we may apply (9.11) to both sides of (9.20) and the desired result will then follow by the induction hypothesis. Hence we may assume that si < 1. Then, the hypotheses of Lemma 9.3 are satisfied with i = 1 and j as above so that * J V ( S I , • --,SN) > ^ J V v 5 . ? , S!,S2, • • S j + i , . -.,sN). Now since Sj = 1, an application (9.11) to the right hand side of the above inequality proves that strict inequality holds in (9.20) as desired. • Proof of Theorem 9.1. Again, we may assume that 0 < p < 1 and that the sk are all strictly 318 Chapter IV. Radial rearrangement positive. Then, assuming Lemma 9.3 and given S i , . . . , SJV £ (0,1], I claim that WN(s1,...,sN) > WN(s^,...,sN) with equality if and only if Si > s2 > • • • > sjy. For, if it is not true that S i > s2 > • • • > SJV, then we may let i be the maximum of the numbers i\ £ {1,..., N} which have the properties that s\,..., SJ-J-I are in decreasing order and that whenever 1 < k < i\ then sk > max^, . . . , SJV) (note that the conditions on i\ are automatically satisfied for i\ = 1). Because S\,...,spi are not all in decreasing order, it follows that i < TV and the maximality of i implies that Si < max(sj,..., SJV). We may then apply Lemma 9.3, and let (s[, . . . , Sjy) = ( S i , . . . , S 8 - _ l , Sj, S{, S i + i , . . . , Sj-i, Sj+i, . . . , SJV). Note that s[,..., s'N are a permutation of s\,..., SJV- Hence, if s[,.. .,s'N are in decreasing order then we are done. Otherwise, proceed just as before and define i' in terms of the s'k just as i was defined in terms of the s^ . Then it is easy to see that i' > i. We may iterate this procedure at most N — 1 times until we have sorted the sk into decreasing order, and so the claim is proved. Theorem 9.1 then follows from Theorem 9.6 and this claim. • We now prove Lemma 9.3 by exploiting the linearity properties of the \Pjv, using a reduction reminiscent of Hardy and Littlewood's [57] reduction of a certain rearrangement inequality to the case where all the variables were in {0,1}. Proof of Lemma 9.3. Throughout p £ (0,1) shall be fixed. Let j be as in the statement of the Lemma and set A = aj. Note that by (9.18) we have A > 0. Fix aj as well as a\,..., a , -_ i . What we must prove is that *jv(ai,..., ajv) - * j v ( a i , . . . , a i _ i , a j , ai,al+l,... ,aj_x,aJ+1,... ,aN) > 0 (9.21) whenever a;+i,..., a j - i , flj+i,..., ajv £ [0, A] and 0 < a 8 < A. We first consider the case when j = i + 1. In that case, the two TV-tuples serving as arguments to the W*N in (9.21) will only differ by an exchange of their ith and jth elements. Moreover, if all variables other than ai are 319 Chapter IV. Radial rearrangement fixed, then the left hand side of (9.21) will be a linear function of at. If we had a; = A then the left side of (9.21) would vanish since aj = A too. On the other hand if we had a; = 0 then this left hand side would become $^(ai,..., a;_!, 0, aj,a^j) - *^ + 1 (ai , . . . , at_i, aj, 0, a J + i , . . . , a )^-But applying (9.14) to both terms and then using Lemma 9.2, we see that this is strictly positive. Note that Lemma 9.2 is applicable since by choice of j and by (9.17), we have min(ai,..., a,-_i, aj) = A > A = max(aj,..., a;v), and moreover A > 0 so that strict inequality must hold. Hence, the left side of (9.21) is strictly positive if a; = 0, vanishes if a, = A and hence by linearity is strictly positive if a,- £ [0, A). This completes the proof if j = i + 1. Now suppose j > i + 1. By linearity considerations we need only verify (9.21) when . ffli-i-li • • •, fflj-i, aj+i,. • • •, aN € {0,A} and the conclusion for them lying in the full interval [0, A] will immediately follow. Of course we always have aj = A. Thus from now on we assume that a;+i,.. . ,a;v £ {0, A}. Now, take the least integer ji £ {i + 1,..., j} with the property that A = ajl = a^+i = • • • = aj. Then, the JV-tuple (ai,..., a,-_i, aj, a,-, a,-+ 1,..., a,_i, a J + i , . . . , ajv) (9.22) will not at all change if we replace j by ji throughout its definition, since when we are moving one of the A's from the string aj1,..., aj, then it clearly does not matter which one we move (see Figure 9.1). Thus, we may replace j by ji and by minimality of ji assume that either j = i + 1 or that aj_i ^ A (or both). We have already handled the case j = i+ 1. Hence, we have a,-_i ^ A and j > i + 1. Moreover aj_i £ {0, A} so that aj_i = 0. Now keep a\,..., a,-_i, a;+i,..., a/v fixed. We shall show that in our present case (9.21) holds whenever ai £ [0, A]. By linearity it suffices to consider a,- £ {0, A}. We first note that we can reduce that 320 Chapter IV. Radial rearrangement i u 31 3 I 4- I I a-i a2 a-3 A A A 0 A O A A O A A O A A A a i 9 a2o Figure 9.1: An example of the original A -^tuple ai,...,ffljv for N = 20, i = 4 and j — 18. The new ./V-tuple (9.22) will be formed from this AT-tuple by cutting out the jth element and pasting it to the left of the ith. Clearly the result of this operation will be the same whether it is the element in position j or the element in position j\ that we cut out. The result will also be the same whether it is to the left of position i or to the left of position i\ that we paste this element. case a, = A to the case ai = 0 as follows. Suppose ai — A. Then, let i\ be the greatest integer ix € {i,...,N} with the property that a; = a t + i = • • • = a t l = A. Since aj_i = 0, we have i\ < j — 1. Just as in our work with j\ w e c a n s e e that the A -^tuple (9.22) will not change if i is replaced by z'i + 1 (this is so because a,-,..., a2l is a string of A's and it does not matter on which side of this string we insert aj = A; see Figure 9.1). But the maximality of ii implies that a; 1 +i ^ A, hence a; 1 +i = 0. Hence, indeed, replacing i by i\ + 1 if necessary, we may assume that a,- = 0. We now thus need only consider the case where at- = aj_i = 0 and aj = A. The case j = i + 1 was already handled, so we may still assume that j > i + 1. Then, we may rewrite the left hand side of (9.21) as WN+1(ai,..., at_i, 0, ai+i,..., O j _ 2 , 0, aj,..., ajv) — W^ + 1 (ai, . . . , aj_i, aj, 0, a J + i , . . . , aj_2, 0, a J + i , . . . , a^ r). Applying (9.14) twice in each of the two terms, we see that this equals * i _ 1 ( a i , . . . , a j _ i ) ^ _ 2 _ i ( a i + i , . . . , a j _ 2 ) * ^ _ J - + 1 ( a i ) . . . , a j V ) - tf?(ai,..., o,-_i, a,-)* j?_2_j(a,-+i,..., a i_2)#^_ j(a j + 1,..., ajv). But the middle factor in both terms is the same, and by Lemma 9.1 it is strictly positive. Moreover, min(ai,..., a4_i, aj) = A > A = max(aj,..., a^ v) and A > 0 so that the left hand side of (9.23) is strictly positive by Lemma 9.2. • 321 Chapter IV. Radial rearrangement Fina l l y we give a proof of Theorem 9.2. Proof of Theorem 9.2. F i x p 6 (0,1) . To obtain a contradict ion, suppose that en I 0 as n —> oo, wi th en € (0,1] and -PfO r^ai 1) £»u 1) ^ -^(e^, £ n , 1,1). Then , by Theorem 9.6 P 4 4 > P 4 4 ¥ ^ ( e n , l , £ „ , l ) ^ ( e n , e n , l , l ) ' Hence, ^ ( e n , e „ , l , l ) > ^ ( e n , l , £ n , l ) . Tak ing the l imi t as n oo, we see that ^ ( 0 , 0 , 1 , 1 ) > ¥ ^ ( 0 , 1 , 0 , 1 ) . (9.24) B u t by (9.14) and the definit ions of the Wk and the we have ^ ( 0 , 0 , 1 , 1 ) = ¥ ^ ( 0 ) ^ ( 1 , 1 ) (9.25a) (9.25b) = # ( l , 0 ) # 2 ( l - p , p ) = l - ( l - ( l - p ) p ) = l - p ( l - p ) < l . O n the other hand, by the definit ion of and by two appl icat ions of (9.12), ^ ( 0 , 1 , 0 , 1 ) = * 5 ( 1 , 0 ,p ,0 ,p) = * 1 ( l ) * 1 ( p ) * 1 ( p ) = l . Clear ly (9.25a) and (9.25b) contradict (9.24), and so the proof is complete. • 9.3. Proof of the formula for the probability of safe traversal Instead of giv ing a probabi l ist ic proof, we give one coming f rom a solution of an associated system of simultaneous equations. 322 Chapter IV. Radial rearrangement Proof of Theorem 9.6. If p = 1 then = 1 for all N > 1 by a repeated application of (9.8), so that the content of the Theorem for p = 1 follows from (9.1). From now on we assume that p G (0,1). Let q = 1 - p. Consider a random walk with the same transition probabilities as {rp}, with the same boundary condition at 1, but not necessarily starting at the point 1. Let pn be the probability that when started at n, it arrives at iV without having fallen into any of the dangers along the route. Then, Pi = P N ( S ! , . . . , S N ) . The following equations are easy to verify: Pi = si(qpi + pp2) P2 = s2(qpi +PP3) P3 = s3(qp2 +PPA) PN-i = sN_i (qpN-2 + PPN) PN = sN(qpN_i +p). This is a tridiagonal system of TV equations in the TV unknowns pi,.. - ,PN- If P = Q = \ then all but the first and last equations can be rewritten as D2p3 - SjPj = 0 where 2 < j < N — 1, D2pj — |(pj-i + Pj+i) — Pj and Sj — sj1 — 1. This shows the similarity with the work of Essen [43] who considers a similar question but with different boundary conditions and with D2 replaced by A 2 , where A2pj = 2D2pj_\ so that A2pj — A(Apj) where APj =Pj-Pj-i-323 Chapter IV. Radial rearrangement In fact, for general p G (0,1), our system can be solved by a simple and standard elimination scheme. First we transform it into the upper triangular system of equations (Ai psx 0 0 A 2 ps2 0 0 0 \ 0 0 0 0 0 0 0 0 A;v-i psN-i 0 0 AN J W PI \ / o \ 0 0 \-psNJ P2 PN-1 \ P N J where the A; are inductively defined by Ai = qs! - 1 and by An+1 = -1 pqsnsn+1 A ' for n = 2,..., TV. It is easy to inductively verify that we will have An < - min(p, g) < 0 for n = 1,..., 7V so that everything is well defined. Then, a further reduction which transforms the system starting from the bottom up into a diagonal system easily shows that / ^jyQi)(ps2) • • • ( P S N ) P l = ( - 1 } AlA2...AN • Comparing this with (9.10), we see that we will be done as soon as we show that (-Ai)(-A 2 ) . . . (-A/y) = *JV+I(1, <f>i(p)si,. ..,4>N(P)SN)-(9.26) Since we have seen that An < 0 for n = 1,..., TV, the positivity of the denominator in (9.10) will also follow from (9.26). Let Bn — — AjV-n+l) 324 Chapter IV. Radial rearrangement for n = 1,. . . , N and set tn = VqSN-nSN-n+\, for 7i = 1,. . . , N — 1. Define i/v = <7Si- Then from the inductive definition of the AN we find that BN = 1 - £/v while B n = l - t n 5 - | 1 ) ( 9 . 2 7 ) for n = 1,. . . , N - 1. Let BN+I — 1-Then ( 9 . 2 7 ) also holds for n — N. We then have BNBn+i = Bn+i - tn, ( 9 . 2 8 ) for n = 1, . . . , N. Let TN = JB1B2 • • -BN. Then since BN+I — 1, and since \P/v_|_i(ai,..., ajv+i) = \&/v+i(a7V+i) • • • i we see that ( 9 . 2 6 ) is equivalent to the assertion that TJV+1 = i&N+1(a1,a2,...,aN+1), ( 9 . 2 9 ) where a n = <f>N-n+i(p)sN-n+i for n = 1,..., N and a/v+i = 1. Recall that (t>n(p)4>n+i(p) = pq for every n and that <?f>i(p) = q so that i n = anan+i for n = l , . . . , iV. We shall now work exclusively in terms of the an, tn, BN and F N . To compute T n , note that r i = 5i. Suppose that L n — oiNBN -\- f3n. 325 Chapter IV. Radial rearrangement Then — ®nBnBn+i + 8nBn+i = otn(Bn+i - tn) + 8nBn+i = ( « n + 8n)Bn+i - tnan, where we have used (9.28) to obtain the second-last equality. Thus, if we define an and 0n inductively by «i = 1 0i = 0 and an+1 = an + 0n (9.30a) Pn+l = -tn<Xn, (9.30b) for n — 1,..., N, then we will always have F n — CtnBn "T" 0n-Since BN+I = 1, it follows that TN+1 = (XN+1 + 0N+I- (9.31) I claim that a n = * n_ 1(ai,...,a n_i) (9.32a) 0n = * n ( « i , • • • , « « ) - * n_i(ai,. ..,an_i), (9.32b) for n = 1,..., JV + 1. If this were true then (9.29) would immediately follow from (9.31). We prove (9.32a) and (9.32b) by induction. For n = 1 they are true since $i and $o are both identically 1. Suppose that they hold for n. Then by applying (9.30a) to (9.32a) and (9.32b), we see that <*n+l — * n ( a i , • • • , An) 326 Chapter IV. Radial rearrangement as desired. Now applying (9.30b) to (9.32a) we find that Pn+l = —tn^n-1 ( ^ l i • • • , &n-l) • Thus, to obtain (9.32b) for n+lwe must show that * n + i (a i , . . . ,a n +i) - * n(ai,...,a n) = (a l 5 . . . , a„_i). (9.33) But i n = a n a n + 1 so that (9.33) follows from (9.9). • 9.4. The one-dimensional continuous case We now suppose that p=\. For a sequence pj, let D2pj — ^(pj-i +Pj+i) — Pj- Then, it is not 1 /2 difficult to verify (cf. the proof of Theorem 9.6, above) that to find P^ (si,.. . ,SJV) one needs to solve D2p3 - SjPj, for j G { — A'" + 1, . . . , AT}, subject to the conditions PN+l = P-N — 1 and Po =Pi, where 5j = <5i_j = sj1 — 1 if j G {1,..., N}. Then one will have Po = PpJ [SI,---,SN)-The symmetry of the problem easily shows that the solution will have the property that if j G { 1 , . . . , N} then pj = Pi-j, and this symmetry easily shows why this system is equivalent to the one exhibited at the beginning of the proof of Theorem 9.6. (Note that we are in effect now considering a random walk on { — + 1,..., A^} in place of our reflecting walk on {1,..., Af}.) The reason for writing the system as above is that it suggests as a continuous analogue the differential equation p"(x) - S(x)p(x) = 0 (9.34a) 327 Chapter IV. Radial rearrangement on [— L, L], where 5 is even, and where p is subject to the conditions that p(L) = p(-L) = l and p'(O) = 0. (9.34b) To solve this, by symmetry we need only solve (9.34a) on [0, L] subject to (9.34b) and to the condition that We now define the function 5X on [0, L] to be the equimeasurable increasing rearrangement3 of the restriction of 8 to [0,L]. (Note that we are rearranging in the opposite order from the way we rearranged the pj because S(x) corresponds to pj1 — 1.) The following result is then an exact continuous equivalent of the p = | case of Theorem 9.1. Theorem 9.7 (special case of Essen [42, Thm. 5.2]). Let 6 be a nonnegative lower semi-continuous piecewise constant function on [0,L], and let p be the solution of (9.34a), (9.34b) and (9.34c). Let P be the solution of (9.34a), (9.34b) and (9.34c) after replacing 8 with 8X. Then P(0) > p{0). It is not unlikely that the above theorem can be given some probabilistic interpretation in terms of Brownian motion, but such an interpretation is not as interesting as the probabilistic interpretation of our discrete results. 10. Horizontal convexity of extremals for some least harmonic majorant functionals Conjecture 1.2 can be rephrased as follows. 3For S C (0,L), put Sx = (L - \S\,L) and use (1.2.1) to define fx on (0,L). Extend fx to [0,L] by requiring continuity at the endpoints. p(L) = 1. (9.34c) 328 Chapter IV. Radial rearrangement Conjecture 1.2'. Let 3>(z) be a subharmonic function on C which depends only on |z|. Then given any domain U which contains the origin and has finite area, there exists a domain U with the following properties: (i) 0 G U (ii) AreaC/ < AreaC/ (iii) T*(U) > U{U) (iv) U is "radially convex", i.e., the intersection of U with any ray starting at the origin is a convex set. What is worth noting is the connection between (iv) and the assumption that 3>(z) depends only on \z\. This connection can be summarized in effect as saying that U has a convex intersection with all lines which orthogonally intersect the level sets of A natural analogue of Conjecture 1.2' is then to consider a subharmonic function $(z) which depends only on Rez and to look for a domain U satisfying (i)-(iii) and (iv'), where (iv') is the requirement that U be horizontally convex, i.e., that its intersection with any horizontal line be convex. Since the level sets of this $ are vertical lines, it is the horizontal lines that intersect the level sets of <h orthogonally. Under an auxiliary growth condition on this modified conjecture is actually true as the following theorem states, and by analogy this truth provides evidence for the likelihood of the truth of Conjecture 1.2'. Theorem 10.1. Let 3>(z) be a subharmonic function on C depending only on Rez. Assume that $(Rez) = o(e(Re2>2) as |Rez|-+oo. (10.1) Then for any domain U of finite area containing the origin, there exists a horizontally convex domain U also containing the origin and satisfying AreaC/ < AreaC/ and T$(C/) > T$(C/). Moreover, the domain U can be taken to be Steiner symmetric. 329 Chapter IV. Radial rearrangement There is really nothing new in the "Moreover"; it follows from Steiner analogues of known results of Baernstein [7]. Conjecture 10.1. Theorem 10.1 remains true even if the assumption (10.1) is dropped. The idea of the proof of Theorem 10.1 is roughly as follows: (I) Use either Steiner symmetrization or Baernstein's sub-Steiner rearrangement (see §11.6) to reduce the problem to considering Steiner symmetric domains which are necessarily simply connected. This will allow us to consider A$ functionals instead of T$ functionals. (II) Use an approximation argument to allows us to assume that A$ has an extremal over 23 and $ is in nice enough that we can analyze that extremal via the variational equation in §4.4. (Ill) Prove from the variational equation that this extremal in fact has a horizontally convex image. First we recall that cp(Rez) is subharmonic on C if and only if <f> is convex on E (Theorem 1.4.3). We shall actually go through the three steps of the proof in reverse order. 10.1. Step III of the proof of Theorem 10.1 Step III is encapsulated in the following Theorem which may of some independent interest. Theorem 10.2. Let $(z) = <f>(Rez) for a convex function <f> which is differentiate on E and whose derivative satisfies \<j>'(t) \ < CeCt2 for all t G E and some finite C. Assume also that <p is strictly convex at 0 and that A$ attains a maximum over 23 at f G 2$. Then: (i) f is absolutely continuous on D; in fact, f G A*(T) (ii) f is univalent and its image / [D] is Steiner symmetric about the real axis and horizontally convex. 330 Chapter IV. Radial rearrangement Moreover, there exists w G T such that if gr(z) — f(rwz) then: (iii) for every fixed r G (0,1), the function z ^ Regr(z) is symmetric decreasing on T (iv) for every fixed r G (0,1) the function z 1-4 Re(zg'r(z)) is symmetric decreasing on T The proof will require the following Lemma which will also be useful in Step II, below. Lemma 10 .1 . Let g be a holomorphic function on D satisfying (iii) and (iv) of Theorem 10.2, def where gr(z) = g(rz). Assume that g(0) = 0 but that g does not vanish identically. Then, g satisfies the same conditions that (ii) imposes on f. Open Problem 10 .1 . Must a holomorphic function g satisfying the conditions of the Lemma necessarily be convex? Proof of Lemma. Write gr{z) — d{rz)- Fix r G (0,1). It suffices to prove that gr satisfies (ii), since if gr is univalent for all r G (0,1) then likewise g is univalent, while the union of an increasing sequence of horizontally convex Steiner symmetric domains is evidently horizontally convex and Steiner symmetric. Put / = gr. The Steiner symmetry of the image of / and the univalence of / follow from Proposition III.5.1. Write u = Ref and v = Im / . Then for z = e,e we have ,/, *. ~ df(eie) dv(eie) Now, Re zf is symmetric decreasing on T. Hence, ^ is decreasing for 6 G [0, TT] and increasing for 6 G [n, 27r ] . In other words, v(el6) is a concave function of 6 for 6 G [0,7r] and is convex for 0 e [ 7 r , 2 7 r ] . Moreover, for all z G O we have v(z) = —v(z), v(l) = v(—1) = 0 and v(elS) is positive on [0,7r] and negative on [7r,27r] (see the proof of Proposition III.5.1). This, together with our concavity/convexity conditions on v(eie) implies that there is an cv G (0,7r) such that v(elS) is increasing for 6 G [0,cv] and decreasing for 6 G [ce,7r]. By the reflection symmetry condition v(z) = —v(z) we have v(el8) decreasing for 6 G [n, 2n — a] and increasing for 9 G [27r — a, 2n]. 331 Chapter IV. Radial rearrangement / ( - I ) Figure 10.1: The graph of f(eie). Together with the symmetric decreasing character, this lets us draw a rough sketch of the graph of f(et9) = u(el9) + iv(ete); see Figure 10.1. The graph makes it intuitively clear that the image of / must be horizontally convex, but we shall prove it rigorously. Fix s e K - l ) , t t ( l ) ] . Define g(x) = mf{v{eiB) : 6 £ [0, TT], u(eie) = x}. Note that u(eie) is continuous and decreasing for 6 £ [0, TT]. The fact that v(elS) is increasing for 6 £ [0, a) and decreasing for 6 £ [a, TT] then implies that g(x) is increasing for x £ [u(—1), u(eia)] and decreasing for x £ [u(eia), u(l)]. Define U = {x + iy : x £ (u(-l),u(l)),\y\ < g(x)}. The increasing character of g on [u(—1), u(a)] and the decreasing character of g on [u(a), u(l)] implies that U is horizontally convex. 332 Chapter IV. Radial rearrangement I claim that U = /[D]. To see this, because both U and /[D] are reflection symmetric, it suffices to verify that U+ and /[D]+ are equal, where for a set D C C we write D+ = {z G D : Im z > 0}. Fix z G t/+. We shall show that z G /[B>]. Write 2 = a; + iy with y > 0. Now, because u(—l) = /( — 1) < a; < /(l) = u(l) and we have Steiner symmetry, it follows that x G /[B1]- We may thus dismiss the case y = 0 and assume y > 0. Let yi = sup{y' : x + iy' G /[D]}. We have yi > 0 since x G /[B]. I claim that yi > y. If so, then we are done, since it follows immediately by the Steiner symmetry and openness of /[B] that y G /[B]. We now prove the claim. Hence, assume that yi < y. But x + iy\ G c*/[B] as is easy to see, and hence x + iy\ = f{e%e) for some # G [0, 27r] as can be seen by the methods that were so often used in the proof of Proposition III.5.1. Since yi > 0 it follows in fact that 6 G (0,7r). We have u(el&) = x. Thus, v(e%e) = yi > g(x) > \y\ and so the claim is proved. Hence, U+ C /[0]+. We now prove the opposite inclusion. Suppose that x + iy G /[ID1]"1" for y > 0. Then u( — 1) < x < w(l) as is easily seen, since w takes on its minimum and maximum at —1 and 1, respectively. By continuity of v, there exists 9 G [0,7r] such that u(etS) = x and v(e10) = y(a;). Now, f(el6) G 9/[D] by the univalence of / and the continuity of / on D. Thus, x + ig(x) ^ /[D]- By Steiner symmetry of /[D], since y(x) > 0 it follows that since x + iy G /[D], we must have y < g(x). But if y < y(x) then x + iy G C/+ as desired, and the desired inclusion is proved. Thus, U+ — /[1D)]+ which combined with previous remarks completes the proof. • Proof of Theorem 10.2. We have already proved (i) in Theorem III.4.4, since the strict convex-ity of 0 at 0 implies SSARIP of $ by Example III.4.4. Moreover, the univalence of / and Steiner symmetry of its image follows from Corollary III.5.2. That same Corollary implies that we may choose w so that Rey is symmetric decreasing on T if g(z) = f(zw). Without loss of generality we may assume that in fact w = 1, since otherwise we need only replace g by / . 333 Chapter TV. Radial rearrangement Wri te / = u + iv. Theorem III.4.3 then says that Qzf' = -P0[{VQ)(f)] o n T , for a real constant Q which is in fact str ict ly positive by S S A R I P . Now, V $ ( / ) = cp'(u) is real. Thus , VoV<&(/) = 4>'(u) + C + iw on T for some real funct ion w on T wi th mean zero and a constant C chosen so that fT(cp'(u) + C) = 0. Thus , QRezf = <j)'(u) + C o n T . Now u is symmetr ic decreasing on T and cp' is a monotone increasing funct ion as cp is convex. Hence, z 1-4 cp'(u(z)) + C is symmetr ic decreasing on T and hence so is Re zf. def B u t W(z)= Rezf'(z) is a harmonic funct ion on D. Hence if it is symmetr ic decreasing on T , l ikewise we must have Wr(z) = W{rz) symmetr ic decreasing on T for all r < 1 because of Coro l la ry 1.6.3. Bu t the symmetr ic decrease of wr is precisely equivalent to statement (iv). In the same way since u is symmetr ic decreasing, so is ur(z) = u(rz) and this proves statement (i i i). F ina l ly , statement (ii) follows f rom L e m m a 10.1. • 1 0 . 2 . S t e p I I o f t h e p r o o f o f T h e o r e m 1 0 . 1 Step II is encapsulated in the proof of the fol lowing Propos i t ion . Th is proof wi l l in an essential way make use of Theorem 10.2. P r o p o s i t i o n 1 0 . 1 . Let $ ( z ) = <p(Rez) for a convex function cp on R with cp(t) = o(e* 2) as \t\ —> 00. Then, there exists an extremal / € 23 for A $ on 23 such that f is univalent while its image / [D] is Steiner symmetric about the real axis and horizontally convex. The proof of this wi l l need the fol lowing elementary lemma whose proof we leave to the reader. L e m m a 1 0 . 2 . Let cp be a convex non-linear function on R . Then, there is a sequence of continuously differentiable convex functions such that: (a) <P\ < 4>2 < • • • on R 334 Chapter IV. Radial rearrangement (b) 4>n(x) - 4 4>{x) for every x € R as n —> oo (c) cp is differentiate on R and for every n there is a finite constant Cn such that \cp'n(x)\ < C „ ( l + \x\)c" for allx^R (d) cpn is strictly convex at 0. Proof of Proposition 10.1. If cp is l inear then we are done since then A * ( / ) = # 0 ) for every / and so we may set f(z) = z and we wi l l have the max imum achieved at / . Hence suppose that cp is non-l inear. Then , choose <i sequence of functions <f>n as in L e m m a 10.2. Let $n(z) = cpn(Rez). Ex t remals for A $ n over 23 exist (Theorem III.3.4). Let fn 6 23 be an extremal for A $ n . Moreover , $ n satisfies the hypotheses of Theorem 10.2. Replacing fn(z) by fn(zw) if necessary (for some w € T ) we may assume that w can be taken to be 1 in the conclusions of that Theorem. Now, fn £ 23 and 23 is weakly compact . Passing to a subsequence if necessary, we may assume that / „ converges weakly to some / € 23. Then , / „ converges to / uniformly on compact subsets of D as is well known (see L e m m a 10.3, below). It follows that because / „ satisfied conclusions (iii) and (iv) of Theorem 10.2, likewise / must satisfy them, since these conclusions are clearly preserved under uni form convergence on compacta. Hence, by L e m m a 10.1 it follows that / is univalent and has a Steiner symmetr ic horizontal ly convex image providing / does not vanish identical ly. We shal l take care of this last proviso later. We now show that A $ actual ly attains a max imum over 23 at / . To see this, suppose that on the contrary there is an h 6 23 wi th A$( / i ) > A $ ( / ) . Now, by monotone convergence we have A $ n ( / i ) —>• A$ ( / i ) . Hence there exists e > 0 such that for sufficiently large n we have A * „ ( / i ) > e + A * ( / ) . Bu t , A<s(/) > l im sup A $ ( / n ) n 335 Chapter IV. Radial rearrangement by Theorem III.3.2 applied with *(t) = e<Ret)2 (of course A$ is bounded on 23 by the Chang-Marshall inequality). Hence, for sufficiently large n we have A$(/n) < A$(/) +e/2. Thus, for sufficiently large n we have A<sn(/i) > e/2 + A$(/„). But A$(/n) > A < 5 n ( / „ ) . Hence, A$ n ( / i ) > A$n(/n) for all large n, which contradicts the fact that /„ maximizes A$„. Thus, indeed A$ achieves a maximum over 23 at / . If / ^ 0 then we are done by previous remarks. Suppose now that / = 0. Then, A$(/) = .$(0) . Set F(z) = z. Then, A$(F) > $(0) by the subharmonicity of <&. Hence, we are also done, since in that case the maximum of A$ must be achieved at F and F is univalent and has a Steiner symmetric and horizontally convex image. • The following lemma is well-known and is a consequence of a more general proposition of Cima and Matheson [36]. We shall give a very simple proof of the lemma for the reader's convenience. L e m m a 1 0 . 3 . Let fn be a sequence of functions in "D converging weakly to a function / 6 I) . Then fn—tf uniformly on compact subsets of D. Proof. We have fn -4 / in I,1(T) by Andreev and Matheson's [5] result on the weak continuity of LP(T) norms (finite p) on X>. By Theorem 1.3.3, since C H1, we have fn(z) = (P*fn)(z), and /(z) = (P */)(*), for all z € D. It is clear from the form of the Poisson kernel that i 1(T) convergence implies convergence on compacta within D. • 336 Chapter IV. Radial rearrangement 10.3. Step I and the rest of the proof of Theorem 10.1 Proof of Theorem 10.1. Without loss of generality, by scaling assume that AreaC/ = TT. Let V = C/B, as in §111.6. (One could also use Steiner symmetrization here and Theorem 1.6.6.) Then, UB has area not exceeding TT by Corollary III.6.1. Moreover, C/B is Steiner symmetric and hence simply connected. Let fi be the Riemann map sending D onto C/B and satisfying /i(0) = 0. Then / i e 23 and A$(/x) = T$(C/B) = T9(U) by Theorems III.1.2 and III.6.1. Apply Proposition 10.1 to obtain a univalent function / € 23 with Steiner symmetric and horizontally convex image such that A$ attains its maximum over 23 at / . In particular A$(/) > A$(/i) = r$(C/). Let U = /[D]. Then, by Theorem III. 1.2 we have r * ( l 7 ) = A9(U) > T*(U) as desired. • 337 List of notations and symbols 1. Rearrangement-type operators Notation First use Description # P- 5 a generic rearrangement ® P- 9 Schwarz symmetrization B P- 9 Steiner symmetrization about the real axis ® P- 11 circular symmetrization 0 P- 44 the circular *-function; = J(f®) F® P- 244 a symmetrized version of a holomorphic F * P- 263 radial rearrangement < P- 275 a cylindrical version of radial rearrangement P- 282 a lengthwise Steiner-type *-function ^ P- 276 two-sided lengthwise Steiner rearrangement < ^ P- 295 I*~ is a single interval with same logarithmic length as I Y P- 307 decreasing rearrangement X P- 328 increasing rearrangement TJ(M) P- 284 one-sided Steiner rearrangement of U at abscissa M P- 279 a certain cutting operation applied to U UB P- 251 Baernstein's sub-Steiner rearrangement of U jy P- 61 a certain set containing D P- 75 partial rearrangement of / with respect to involution p 9 P- 131 a "de-rearrangement" of g 2. Some other operators and relations for sets and functions Notation First use Description Ac P- 3 complement of A \A\ P- 3 measure or cardinality of A 2A P- 3 power set of A I M P- 3 union of all elements of A A\B P- 3 set of elements of A not lying in B 1A P- 3 indicator function of a set A fx P- 5 level set of function / at height A P- 144 a domination between functions (only in §11.9) < P- 165 another domination between functions L* P- 138 L*(x,y) = L(y,x) f*9 P- 48 convolution of two functions / and g on T K *g P- 74 the function x i-» ^ 2y K(d(x, y))g(y) D P- 34 the universal covering surface of a domain D 338 List of notations and symbols 3. Numerical operators N o t a t i o n F i r s t use D e s c r i p t i o n t+ p. 3 positive part of a number or function t t~ p. 3 positive part of a number or function t \t\ p. 3 absolute value of a number or function t $j p. 4 partial derivative of $ with respect to Xj 4. Miscellaneous N o t a t i o n F i r s t use D e s c r i p t i o n [v, w] p. 88 geodesic joining v with w 1{P} p. 3 indicator function of a proposition P Kz ® Kx p. 141 product kernel on Z X X 5. Greek alphabetical index N o t a t i o n F i r s t use D e s c r i p t i o n r$ P- 192 a certain functional on domains A P- 39 continuous Laplacian operator T^J"=i ~§^? o n A , A G P- 94 discrete Laplacian operator on a graph G P- 3 Kronecker delta function 8(v), 6G{v) P- 82 degree of vertex v in graph G 9(r;U) P- 11 angular measure of U D T(r) A* P- 231 Zygmund class A$ P- 188 a functional on measurable functions M$ P- 39 Riesz measure of $ »i(G) P- 94 first nonzero Dirichlet eigenvalue of — A on G C Tp TTl P- 151 projection of V = Z x X onto Z 7T2 P- 151 projection of V = Z x X onto X P(I) P- 295 logarithmic length of I ^ ( / ) P- 166 Y.Y9-I TD P- 32 first exit time of Brownian motion from domain D Ts P- 143 first time that the random walk does not survive a time step <t>n{p) P- 314 p for even n and 1 — p for odd n P- 313 a useful auxiliary function P- 314 another useful abbreviation P- 314 yet another useful abbreviation P- 312 an auxiliary function P- 46 Haar measure on T co(z,A;D) P- 28 harmonic measure of A at z in D 339 List of notations and symbols u(z,A;s) p. 144 generalized harmonic measure w? p. 28 u(z,-;D) 6. Latin alphabetical index N o t a t i o n F i r s t use D e s c r i p t i o n arg P- 3 argument of a complex number A P- 126 fibres of our Steiner type rearrangement # B P- 206 unit ball of Hilbert space (in §111.3.2) B(r) P- 206 ball of radius r in Hilbert space (in §111.3.2) Ba P- 196 Beurling function Bt P- 32 Brownian motion process BM P- 197 cut-off Beurling function BMO P- 21 the space BMO on T BMOA P- 21 the analytic functions from BMO B P- 192 set of all domains of area < TT containing 0 23 P- 190 unit ball of 1) 23a P- 190 unit ball of Va b P- 190 unit ball of d ba P- 190 unit ball of oa Children v P- 90 the set of children of a vertex v of Tp Cnf P- 189 the nth Fourier cosine coefficient of / D P- 165 the difference operator K — 1 (only in §11.11) DM P- 197 image of cut-off Beurling function Desc v P- 90 the set of descendants of a vertex v of Tp D(G) P- 94 set of all functions vanishing outside G but not identically zero P- 190 holomorphic Dirichlet space IDi P- 189 a-weighted holomorphic Dirichlet space 0 P- 190 real harmonic Dirichlet space Oi oa P- 189 a-weighted real harmonic Dirichlet space D P- 3 open unit disc D(r) P- 3 open disc of radius r about the origin B(z; r) P- 3 open disc of radius r about z P- 165 the difference operator L — 1 (only in §11.11) EM(f) P- 200 the measure of the level set fw E-°°[-] P- 285 a limiting case of Ez[-] E*[.) P- 32 expectation conditioned on the process starting at z EdgeG P- 81 set of edges of a graph G Fix p P- 72 fixed point set of involution p T P- 211 a set of measurable functions on I (in §111.3.3) T P- 260 set of $ with $ ( |z | ) subharmonic (in Chapter IV) GD P- 60 Green's function of D with pole at 0 G P- 217 a subset of ^[0, oo) (in §111.3.3) <SP P- 119 collection of graphs on which there is an ordering for which P holds 340 List of notations and symbols g(z,w;D) P- 38 Green's function of D evaluated at z and with pole at w g(z,w;D) P- 53 circular rearrangement of g(z, w; D) with respect to z g(z,w) P- 56 g(z,w;J}) H P- 205 Hilbert space (in §111.3.2) H8 P- 84 edge graph of the octahedron n o P- 226 {/ e 772(D) : /(0) = 0} H P P- 18 holomorphic Hardy space h? P- 18 harmonic Hardy space h{v) P- 90 height of a vertex v in Tp nD P- 28 cr-algebra of sets for which harmonic measure exists * P P- 72 the set of points mapped by p to something strictly -^larger P- 72 a fixed set of involutions Jg(reie) P- 44 jLe)e]g(re^)d<p K P- 138 kernel function (in §11.9) Kx P- 141 simple random walk kernel on the graph X Kz P- 141 any kernel on Z Kf P- 141 trivial kernel on Z Ki r P- 141 simple random walk kernel on Z K1 eg) Kx P- 141 product kernel o n Z x I L{t-W) Ll{uD) P- 276 a measure of W along a line P- 28 functions integrable with respect to harmonic measure Ls P- 309 first time random walk fails to survive a step L H M ( 2 , $ ; D ) P- 31 value at z of the least harmonic majorant of $ on TJ M P- 166 a certain collection of positive symmetric functions N(v) P- 94 set of vertices adjacent to v NF P- 254 Nevanlinna counting function of F n. t. lim P- 19 nontangential limit operator 0 P- 90 root of Tp pp P- 307 a certain survival probability Pr P- 19 Poisson kernel p - ° ° ( - ) P- 285 a limiting case of Pz(-) Pz(-) P- 32 probability conditioned on the process starting at z p P- 88 degree of our tree Tp Po P- 226 projection operator from L2(T) onto T T ^ T ) Q(f,g;*,K) P- 71 Z*,v*(\n*)-f{y)\)K(d(*,y)) Re(0;U) P- 262 logarithmic measure of a ray intersected with U Rn P- 142 the random walk on V P- 306 a reflecting random walk on Z+ P- 94 Rayleigh quotient for / S(r) P- 206 S(r;l) (in §111.3.2) S(r;R) P- 206 {/ : r < ll/H < R} (in §111.3.2) s P- 143 survival probabilities on V (in §11.9) sgn z P- 3 z N supp/ P- 3 support of a function / 3nf P- 189 the nth Fourier sine coefficient of / 341 List of notations and symbols p- 3 0 7 first time random walk reaches N + 1' p- 8 8 p-regular tree p- 9 0 the height k subtree of Tp T . p- 3 unit circle T ( r ) p- 3 circle of radius r about the origin T(z;r) p- 3 circle of radius r about z ua p- 5 4 D ( a ; 1 ) uab p- 5 5 disc with a slit and with a piece sliced off Uabcd p- 5 5 B\([-d,-c]U[a,b]) V p- 1 3 8 Z x X (in § 1 1 . 9 ) VertG p- 8 1 set of vertices of a graph G v(r;V) p- 2 6 6 a certain harmonic measure functional v- p- 2 7 5 the semi-infinite cylinder (—oo, 0 ) x T W(I;D) p- 2 9 4 u(0,dD;D\I) WM p- 2 8 4 a certain harmonic measure functional w(X) p- 1 6 6 set of functions on X satisfying a certain size property wr p- 2 6 7 a certain harmonic measure functional X p- 1 3 8 a discrete set on which a Steiner type symmetrization is given (in § 1 1 . 9 ) X{u;W) p- 2 7 5 a measure of W along a line Xn p- 1 4 3 sequence of i.i.d. random variable uniformly distributed on ( 0 , 1 ] Y(x;U) p- 9 linear measure of {z : Re z = a;} fl U Z p- 2 the set of integers z- p- 2 { - 1 , - 2 , . . . } ZQ p- 2 { 0 , - 1 , - 2 , . . . } Z + p- 2 { 1 , 2 , . . . } Z+ p- 2 { 0 , 1 , 2 , . . . } zf p- 1 2 0 edge graph of cube p- 1 2 0 ternary plane graph p- 8 6 the circular graph on n vertices 3 4 2 Bibliography [1] Adimurthy, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-laplacian, Ann. 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Zygmund, Trigonometric series, second ed., Cambridge Univ. Press, London and New York, 1968. 348 Appendix A Source code for cubetern.c /* cubetern.c by Alexander R. Pruss <pruss@math.ubc.ca>, 1995. */ /* Turbo C++ (C mode) code */ /* Check whether there exists a measure preserving rearrangement # under which we have sum_{v\in G} f(v) Nf(v) <= sum_-[v \ i n G} f~#(v) Nf#(v), where Nf=sum_{w \ i n N(v)} f(w). We do th i s f o r the cube graph G=Z_2~3 and for the ternary plane graph G=Z_3~2. */ #define CUBE_GRAPH /* comment t h i s out to do the ternary plane instead */ #include <stdio.h> #include <stdlib.h> #include <time.h> #ifdef CUBE_GRAPH /* case of the cube graph */ # define NAME "cube" # define ALL_VERTICES_EQUIVALENT # define DEGREE 3 /* graph has degree 3 */ # define N_VERT 8 /* graph has 8 vertices */ enum vertices { v000=0, v001=l, v010=2, v011=3, vl00=4, vl01=5, vll0=6, vlll=7 /* name the vertices by t h e i r coordinates */ int N[N_VERT][DEGREE] /* neighbour l i s t */ = { { vOOl, vOlO, vlOl >, /* neighbours of vOOO */ { vOOO, v O l l , vlOl }, /* neighbours of vOOl */ { v O l l , vOOO, vllO }, /* neighbours of vOlO */ { vOlO, vOOl, vlOO }, /* neighbours of vOll */ { v l O l , v l l O , vOOO }, /* neighbours of vlOO */ { vlOO, v l l l , vOOl }, /* neighbours of vlOl */ { v l l l , vlOO, vOlO }, /* neighbours of vllO */ 349 Appendix A. Source code for cubetern. c { v l l O , v l O l , vOll } /* neighbours of v l l l */ >; #else /* case of the ternary plane graph */ # define NAME "ternary plane" # define ALL_VERTICES_EQUIVALENT # define DEGREE 4 /* graph has degree 4 */ # define N_VERT 9 /* graph has 9 vertices */ enum vertices { vOOO, v01=l, v02=2, vl0=3, vll=4, vl2=5, v20=6, v21=7, v22=8 }; /* name the vertices by t h e i r coordinates */ int N[N_VERT][DEGREE] /* neighbour l i s t */ = { •c vOl, v02, vlO, v20 >, /* neighbours of vOO */ { v02, vOO, v l l , v21 }, /* neighbours of vOl */ { vOO, vOl, vl2, v22 }, /* neighbours of v02 */ •c v l l , vl2, vOO, v20 }, /* neighbours of vlO */ •c vl2, vlO, v21, vOl }, /* neighbours of v l l */ •c vlO, v l l , v22, v02 }, /* neighbours of vl2 */ •c v21, v22, vOO, vlO }, /* neighbours of v20 */ •c v22, v20, vOl, v l l }, /* neighbours of v21 */ •c v20, v21, v02, vl2 } /* neighbours of v22 */ >; #endif /* the rest of the code i s the same for both graphs */ #define LARGEST.VAL 20 /* t r y values of functions from 0 to 20-1 */ ttdefine NUM.TRIES 20000 /* don't bother trying more than 20000 functions per order */ unsigned f [N_VERT]; /* the input function f */ int order[N_VERT]; /* the order we are sorting i n : t h i s defines our measure preserving rearrangement # */ unsigned sorted_f[N_VERT]; /* the function f sorted according to order [] */ unsigned sum_fNf(unsigned *F) /* compute sum_{v\in G} F(v)NF~#(v) */ { /* With LARGEST_VAL=20, the biggest sum_fNf can be i s 19*19*DEGREE*N_VERT, which, i n the case of the ternary plane i s 12996, and i n the case of the cube i s 8664. Both values w i l l n i c e l y f i t i n an unsigned variable. */ unsigned sigma=0; int v,w; for(v=0; v<N_VERT; v++) { int NF=0; for(w=0; w<DEGREE; w++) NF+=F[N[v] [w] ] ; 350 Appendix A. Source code for cubetern. c s igma+=F[v ] *NF; } r e t u r n s i g m a ; } v o i d d o _ s o r t ( u n s i g n e d * F ) / * S o r t t h e i n p u t i n d e s c e n d i n g o r d e r a c c o r d i n g t o o r d e r [ ] . Use t h e b u b b l e s o r t . * / { i n t i , j ; f o r ( i = 0 ; i<N_VERT; i++) / * a t t h e b e g i n n i n g o f t h e i - t h l o o p , t h e f i r s t i e lements a r e s o r t e d c o r r e c t l y * / f o r ( j = i + l ; j<N_VERT; j++) / * pu t t h e ( i + l ) s t l a r g e s t e lement i n t h e r i g h t p l a c e * / i f ( F [ o r d e r [ j ] ] > F [ o r d e r [ i ] ] ) { u n s i g n e d swap; s w a p = F [ o r d e r [ j ] ] ; F [ o r d e r [ j ] ] = F [ o r d e r [ i ] ] ; F [ o r d e r [ i ] ] = s w a p ; } } u n s i g n e d max_num_tr ies=0; i n t p r o v e d = l ; v o i d t e s t _ o r d e r ( v o i d ) / * check i f t h e c u r r e n t o r d e r [ ] i n d u c e s t h e r i g h t i n e q u a l i t y * / / * I f t h i s f u n c t i o n r e t u r n s w i t h o u t p r i n t i n g any messages , t h e n i t i s g u a r a n t e e d t h a t o r d e r [] does no t i nduce t h e r i g h t rea r rangemen t i n e q u a l i t y . I f t h i s g u a r a n t e e cannot be g i v e n , t h e n p r o v e d i s s e t t o 0 . * / { u n s i g n e d t r i e s ; i n t i ; f o r ( t r i e s = 0 ; t r i es<NUM_TRIES; t r i e s + + ) { f o r ( i = 0 ; i<N_VERT; i++) s o r t e d _ f [ i ] =f [ i ] =rand() 7,LARGEST_VAL; d o _ s o r t ( s o r t e d _ f ) ; i f ( s u m . f N f ( s o r t e d . f ) < s u m _ f M f ( f ) ) { / * t h i s o r d e r d o e s n ' t g i v e t h e d e s i r e d i n e q u a l i t y * / i f ( t r i e s + l > m a x _ n u m _ t r i e s ) m a x _ n u m _ t r i e s = t r i e s + l ; r e t u r n ; 351 Appendix A. Source code for c u b e t e r n . c > } /* we haven't found a counterexample! */ printf("We haven't found a counterexample for order:\n"); for(i=0; i<N_VERT; i++) p r i n t f ('7,d ", order [i] ); putchar('\n'); proved=0; > void loop_orders(int depth) /* recursively generate permutations on ( 0,...,N_VERT-1 ) and run test_order on them */ •c int i , j ; if(depth==N_VERT) { test_order(); return; } for(i=0; i<N_VERT; i++) •c for(j=0; j<depth; j++) if(order[j]==i) goto TRY_NEXT_i; /* t h i s value already chosen e a r l i e r i n order[] */ order[depth]=i; loop_orders(depth+1); TRY_NEXT_i: ; # ifdef ALL_VERTICES_EQUIVALENT if(depth==0) break; /* a l l vertices are equivalent, so only need to t r y one value for order[0] */ # endif } if(depth==0) fprintf(stderr,"[done!]\n"); > int mainQ { time_t t _ s t a r t ; srand(317); p r i n t f ("Computing for the '/.s . . . \n" .NAME) ; t_start=time(NULL); loop_orders(0); p r i n t f ("Run time: '/Id seconds . \n" , time (NULL)-t_start); 352 Appendix A. Source code for c u b e t e r n . c i f ( p r o v e d ) { p r i n t f ( " W e have p r o v e d t h a t t h e r e i s no o r d e r g i v i n g t h e " " i n e q u a l i t y . \ n " ) ; p r i n t f ( " T h e maximum number o f random f u n c t i o n s p e r o r d e r " "was ' / , u . \ n " , m a x _ n u m _ t r i e s ) ; > r e t u r n 0 ; 353 Appendix B Source code for cm.f c C FORTRAN 77 source code "cm.f" C C Check whether the Chang-Marshall functional i s r e a l l y less C than e as i t i s conjectured to be. C Copyright (c) 1994-1995 Alexander Pruss. C program chmrsh rea l p(l:100) re a l L integer iy.niter,nterms open(unit=80,file='cmdcfg') read(80,*) i y read(80,*) n i t e r read(80,*) nterms close(unit=80) p r i n t * , 'iy=',iy p r i n t * , 'niter=',niter p r i n t * , 'nterms=',nterms istop=l nstops=0 do 1 i = l , n i t e r do 2 j=l,nterms 2 p(j)=urand(iy) c a l l cmfune(nterms,p,L) i f ( l o g ( L ) .ge. 1) p r i n t * , L,(p(k),k=l,nterms) i f ( i . e q . i s t o p ) then nstops=nstops+l istop=nstops/20.*niter p r i n t * , ' [',real(i)/niter*100. , " / . f i n i s h e d ] ' endif 354 Appendix B. Source code for cm.f 1 c o n t i n u e p r i n t * , ' F i n i s h e d ' s t o p end complex f u n c t i o n c r p o l ( n , p , z ) C e v a l u a t e t h e r e a l - c o e f f i c i e n t p o l y n o m i a l C p ( 0 ) + p ( l ) z + . . . + p ( n ) z ~ ( n ) a t complex z i n t e g e r n complex z r e a l p ( 0 : * ) i n t e g e r j c r p o l = p ( n ) do 1 j = n - l , 0 , - l c r p o l = p ( j ) + c r p o l * z 1 c o n t i n u e r e t u r n end s u b r o u t i n e d i r n o r ( n , p ) C n o r m a l i z e (wr t t h e D i r i c h l e t norm) t h e r e a l - c o e f f i c i e n t p o l y n o m i a l p C (p(0)=0) i n t e g e r n r e a l p ( l : * ) r e a l dnor i n t e g e r j dnor=0 do 1 j = l , n dnor=dnor + j * p ( j ) * * 2 1 c o n t i n u e i f ( d n o r . e q . O ) r e t u r n d n o r = s q r t ( d n o r ) do 2 j = l , n p ( j ) = p ( j ) / d n o r 3 5 5 Appendix B. Source code for cm.f 2 c o n t i n u e r e t u r n end s u b r o u t i n e c m f u n c ( n , p , L ) i n t e g e r j i n t e g e r n r e a l p ( l : * ) r e a l L r e a l s impsn ,Lambda e x t e r n a l Lambda r e a l d O . d l , d 2 , d 3 , d 4 , m 4 r e a l e p s i l o n i n t e g e r n i t e r r e a l p i , t w o p i , e pa rame te r (two p i=2*3 .141592653589793) pa rame te r (p i=3 .14159 26535 89793) pa rame te r (e=2.71828 18284 59045 23536) C n o r m a l i z e ou r p o l y n o m i a l c a l l d i r n o r ( n , p ) C l o a d a l l p o l y n o m i a l s c a l l l o a d p ( n . p ) C compute d e r i v a t i v e s o f p ( z ) / z a t z = l o f o r d e r s 0 , 1 , 2 , 3 and 4 dO = 0 d l = 0 d2 = 0 d3 = 0 d4 = 0 do 10 j = l , n dO = d0+p( j ) i f ( j . g e . l ) d l = d l + p ( j ) * ( j - l ) i f ( j . g e . 2 ) d2 = d 2 + p ( j ) * ( j - l ) * ( j - 2 ) i f ( j . g e . 3 ) d3 = d 3 + p ( j ) * ( j - l ) * ( j - 2 ) * ( j - 3 ) i f ( j . g e . 4 ) d4 = d 4 + p ( j ) * ( j - l ) * ( j - 2 ) * ( j - 3 ) * ( j - 4 ) 10 c o n t i n u e C Maple g e n e r a t e d f o r m u l a m4 = 2 * e x p ( d 0 * * 2 ) # * ( 3 * d 2 * * 2 + 3 6 * d l * * 2 * d 0 * d 2 + 4 * d l * d 3 + 6 * d l * * 4 + 2 4 * d l * * 4 # *d0* *2+d0*d4+8*d0* *2*d3*d l+6*d0* *2*d2* *2+24*d0* *3 356 Appendix B. Source code for cm. f # * d 2 * d l * * 2 + 8 * d 0 * * 4 * d l * * 4 ) / 1 8 0 . C End Maple g e n e r a t e d f o r m u l a e p s i l o n = . 0 1 do 1 j = l , 1 6 i f ( j . n e . 1 ) t h e n i f ( a b s ( L - e ) . g e . l e - 5 ) t h e n e p s i l o n = m i n ( e p s i l o n , a b s ( L - e ) ) / 2 e l s e e p s i l o n = e p s i l o n / 8 . e n d i f e n d i f C S i m p s o n ' s r u l e g u a r a n t e e s t h a t n i t e r w i l l be a s u f f i c i e n t C number o f t i m e s t e p s t o o b t a i n p r e c i s i o n e p s i l o n . n i t e r = 2 * i n t ( l . + p i * ( m 4 / e p s i l o n ) * * . 2 5 ) i f ( n i t e r . g t . 130) n i t e r = 1 3 2 L=s impsn (Lambda ,0 . , two p i , n i t e r ) / t w o p i C C The 1.1 i n t h e n e x t l i n e i s a s a f e t y f a c t o r t o t a k e c a r e o f C p o s s i b l e r o u n d o f f e r r o r . C i f ( a b s ( L - e ) . g e . l . l * e p s i l o n . o r . m4 . e q . 0 . o r . n i t e r + . e q . 132) go to 2 1 c o n t i n u e C R e t u r n an upper bound f o r L C The 1.1 i n t h e n e x t l i n e i s a s a f e t y f a c t o r t o t a k e c a r e of C p o s s i b l e r o u n d o f f e r r o r . 2 L = L + 1 . l * e p s i l o n r e t u r n end s u b r o u t i n e l o a d p ( n . p ) r e a l p ( l : 1 0 0 ) i n t e g e r n ,nn r e a l p p ( l : 1 0 0 ) common/mainp/pp,nn i n t e g e r j nn=n Dw=0 do 1 j = l , n P p C j ) = p ( j ) 1 c o n t i n u e 357 Appendix B. Source code for cm. f return end r e a l function Lambda(t) rea l t,abs2 complex z,crpol r e a l p(l:100) integer n common/mainp/p,n abs2(z)=real(z)**2+imag(z)**2 C our integrand Lambda=exp(abs2( crpol( n-l,p,exp(cmplx(0.,t)) ) )) return end r e a l function simpsn(f,a,b,niter) C C Find $\int_a~b f(x) dx$ v i a Simpson's rule with n i t e r intervals C rea l f,a,b integer n i t e r real h integer j C number of intervals must be even if(mod(niter,2).eq.1) niter=niter+l h=(b-a) / n i t e r C here we have f(x_0) + 4f(x_l) + f(x_n) simpsn=f(a)+4*f(a+h)+f(b) do 1 j=2,niter-2,2 simpsn=simpsn+2*f(j *h)+4*f((j + 1) *h) 1 continue simpsn=simpsn*(b-a)/(3*niter) return end C C The following function was obtained v i a the Internet C from NETLIB 358 Appendix B. Source code for cm.f C REAL FUNCTION URAND(IY) INTEGER IY C C URAND IS A UNIFORM RANDOM NUMBER GENERATOR BASED C ON THEORY AND SUGGESTIONS GIVEN IN D.E. KNUTH C (1969), VOL 2. THE INTEGER IY SHOULD BE C INITIALIZED TO AN ARBITRARY INTEGER PRIOR TO THE C FIRST CALL TO URAND. THE CALLING PROGRAM SHOULD C NOT ALTER THE VALUE OF IY BETWEEN SUBSEQUENT C CALLS TO URAND. VALUES OF URAND WILL BE RETURNED C IN THE INTERVAL (0,1) . C [ the source code f or t h i s standard NETLIB function i s omitted ] 359 Index adjacent vertices, see graphs, vertices of, adjacent analytic functions, see functions, holomor-phic ancestors, see trees, regular, ancestors in automorphisms of graphs, see graphs au-tomorphisms of backtracking, see graphs, paths in, back-tracking of Baernstein *-functions circular, 44 -46 lengthwise, 282-283 Beurling shove theorem, see Theorem, Beurl-ing shove B M O , see spaces, B M O B M O A , see spaces, B M O A bounded mean oscillation, see spaces, B M O Brownian motion, xi i , 32-34 and r $ functionals, 285-291 and harmonic measure, 32 -33 , 285-291 and PWB solutions, 32 -33 conformal invariance of, 33 exit times of, 32 and radial rearrangement, 291-294 C 1 domains, see domains, C 1 Chang-Marshall inequality, see inequal-ity, Chang-Marshall children, see trees, regular, children in circle graphs, see graph, circle circular symmetrization, see symmetriza-tion, circular circular symmetry, see symmetry, circu-lar conjugate functions, see functions, conju-gate convex functions, see functions, convex convex sets, see sets, convex convolution-rearrangement inequality, see inequality, convolution-rearrangement critically sharp inequalities, 210-225 cube, see graph, cube cylinder continuous, xiv, 275-280, 282-283, 285-291 Brownian motion on, xvii, 285-286 drifting Brownian motion on, 287-291 radial rearrangement lifted to, 275-276 discrete random walk on, 143, 158-164 shove theorem on, 158-164 symmetrization on, xiv, 139 decreasing rearrangement, see rearrange-ment, decreasing degrees of vertices, 82 descendants, see trees, regular, descen-dants in difference inequalities and discrete rearrangement, 168-185 Dirichlet integrals and radial rearrangement, 271-272 and Steiner symmetrization, 270-271 Dirichlet principle, 2 7 4 Dirichlet problem, 27 PWB solution of, 27-29 and Brownian motion, 32-33 and uniformizer, 35-37 Dirichlet spaces, see spaces, Dirichlet disc algebra, 21 domains, 2 C 1 , 30-31 circularly symmetric, 41-42 convex, 2 Greenian, 27 -30 , 33-35 HP, 194 HP, 18 horizontally convex, 2, 329-337 Nevanlinna, 22 regular, 29-31 simply connected, 52 -53 Green's functions for, 39 star-shaped, 2 Steiner symmetric, 51-52 edges, see graphs, edges of equimeasurable functions, see functions, equimeasurable Essen inequality, see inequality, Essen exit times of random walk, see random walk, exit times of extremals existence of, xvi, 199-205, 210-225 360 Index existence of (for the 1$), 252-254, 261 existence of ones with special properties, 328-337 image of, xvi i , 246-248 nonexistence of, xvi , 210-225 properties of, 225-237, 240-242, 246-248 regularity of, xvi i , 231-237, 241-242 univalence of, xvi i , 246-248 variational equation for, xvi , 225-231, 240-242 fibres of Steiner type rearrangement, 126 uniqueness of, 128 Fourier coefficients, 19 f u l l subgraphs, see subgraphs, full func t iona l s extremals of, see extremals T$, x i i -xv i i i , 192-194 and Brownian motion, 285-291 A * , xv -xv i i , 188-189, 192-194, 198-242 finiteness of, 206-207, 210 upper semicontinuity of, 199-205, 207-209 functions *, see Baernstein ^-functions analytic, see functions, holomorphic conjugate, 20-21 convex, 4 equimeasurable, 47-48 Green's, see Green's functions harmonic, 17 holomorphic, 3-4 kernel, 138-139 lower measurable, 6 Nevanlinna counting, see Nevanlinna counting function resolutive, 27 similarly ordered, 70-71 strictly convex, 4 approximation, 334-335 subharmonic, 22-26 maximum principle for, 22-23 superharmonic, 22 symmetric decreasing, 47, 49-50 and Poisson extension, 50 discrete, 70, 74-75 univalent, 4 geodesic balls in trees, see trees, regular, geodesic balls in geodesic distance on graph, 82 geodesies on graphs, 82 on trees, 88-90 geodesies in trees and spiral-like ordering, 91-92 graph circle, xv, 86-88 master inequality on, 86-88 cube, xv,120-123 octahedron, xv, 84-86 ternary plane, xv, 120-123 tree, see trees graphs automorphisms of, 82-83 connected, 82 constant degree, 82 definitions for, 81-83 discrete rearrangements on, 81-123 edges of, 81 geodesic distance on, 82 geodesies on, 82 isomorphisms of, 82-83 paths in, 81-82 backtracking of, 88 vertices adjacent, 82 degrees of, 82 vertices of, 81 graphs, tree, see trees Green's functions, 37-41 and circular symmetrization, x i i - x i i i , 44-46, 53-60 and discrete rearrangement, 148-151, 153 and harmonic majorants, 281 and harmonic measures, 281 and least harmonic majorants, 45-46 and Steiner symmetrization, 52 discrete generalized and rearrangement, 184 for simply connected domains, 39 generalized discrete, 148-151, 153 radial monotonicity of, 60-65 Greenian domains, see domains, Greenian Hp domains, see domains, Hp Hardy spaces, see spaces, Hardy Hardy-Lit t lewood inequality, see inequal-ity, Hardy-Lit t lewood harmonic functions, see functions, har-monic 361 Index harmonic majorants and Green's functions, 2 8 1 harmonic measure, xi i i , xvii , 28-29 and Brownian motion, 3 2 - 3 3 , 2 8 5 - 2 9 1 and circular symmetrization, xiii, 4 6 and discrete rearrangement, 1 4 5 - 1 4 6 , 1 5 2 - 1 5 3 and Green's functions, 2 8 1 and least harmonic majorants, 3 1 - 3 2 and lengthwise Steiner symmetrization, 2 7 6 -2 7 9 and Steiner symmetrization, 5 2 and uniformizer, 3 5 - 3 7 conformal invariance of, 3 3 generalized discrete, 1 4 3 - 1 4 6 , 1 4 9 - 1 5 3 and rearrangement, 1 8 0 - 1 8 3 on disc, 2 9 heights in trees, see trees, regular, heights in Hilbert spaces, see spaces, Hilbert holomorphic functions, see functions, holo-morphic horizontally convex sets, see sets, hori-zontally convex inequality Alexander-Taylor-Ullman, 2 6 1 - 2 6 2 Chang-Marshall, xvi, 1 9 4 - 1 9 8 , 2 0 9 - 2 1 0 , 2 6 2 convolution-rearrangement Baernstein, 4 8 - 5 0 Beckner, 4 8 - 4 9 discrete, 9 5 , 1 1 8 - 1 2 3 for some graphs, xiv-xv iterated, 1 5 3 - 1 5 4 , 1 5 6 - 1 5 8 none on %\ and 1 2 0 - 1 2 3 Riesz-Sobolev, 4 8 via discrete master inequality, 7 3 - 7 5 critically sharp, 2 1 0 - 2 2 5 Essen, xvi, 1 9 4 - 1 9 8 , 2 6 2 , 2 6 8 Faber-Krahn classical, xv, 9 3 on regular trees, xv, 9 3 - 1 1 8 Hardy-Little wood, 1 2 - 1 7 , 7 1 master, 7 1 - 8 1 discrete, 1 1 8 - 1 2 3 none on Z\ and Z § , 1 2 0 - 1 2 3 on circle graphs, 8 6 - 8 8 on octahedron graph, 8 4 - 8 6 on regular trees, 9 1 - 9 3 Moser-Trudinger, 1 9 4 - 1 9 8 inner radius, 281-282 and radial rearrangement, 2 6 3 , 2 8 1 - 2 8 2 interpolation of function spaces, 201-202 involution, 72 isometric involution, see involution isomorphisms of graphs, see graphs, iso-morphisms of kernel Poisson, see Poisson kernel kernel function, see functions, kernel A$-functionals, see functionals, A<j> Laplacian continuous on M " , 9 3 discrete, 9 3 - 9 4 , 1 6 5 first non-zero eigenvalue of, 9 4 - 1 1 8 in M 2 , 1 7 least harmonic majorants, 31-32 and circular symmetrization, 4 5 - 4 6 and Green's functions, 4 5 - 4 6 and harmonic measure, 3 1 - 3 2 and Steiner symmetrization, 5 2 Riesz decomposition of, 4 1 length logarithmic, see logarithmic length level set, 5 limits nontangential, 1 8 - 1 9 existence of, 1 9 , 2 2 logarithmic length, 295 lower measurable function, see functions, lower measurable majorants harmonic, see least harmonic majorants map Riemann, 2 8 universal covering, 3 4 - 3 7 master inequality, see inequality, master maximum principle, 22-23 measure harmonic, see harmonic measure Riesz, see Riesz measure metric space discrete, 7 1 - 8 1 two point, 7 5 Nevanlinna class, 22 362 Index Nevanlinna counting function, 254-255 and Steiner symmetrization, 254 Nevanlinna domains, see domains, Nevan-l inna nontangential l imits , see l imits , nontan-gential existence of, 19 octahedron, see graph, octahedron ordering Schwarz, 124-126 spiral-like, 90-91 and geodesies, 91-92 Steiner, 129 parents, see trees, regular, parents in paths, see graphs, paths in points regular, 29-31 Poisson extension, 19-20, 243 and symmetric decreasing functions, 50 integral, see Poisson extension kernel, 19 projection projection Szego, see Szego projection pseudotopology, a-, 4-5 P W B solution, see Dirichlet problem, P W B solution of radial rearrangement, see rearrangement, radial random w a l k exit times of and discrete rearrangement, 151-152, 184-185 in dangerous blind alley, 306-328 on discrete cylinder, 143, 158-164 with dangers, xvi i i , 143-156, 306-328 with geometric waiting times, 149 with respect to kernel function, 142-156 Rayleigh quotient, 94 rearrangement, 4-17 Baernstein's sub-Steiner, xvi i , 251-256 and F$ functionals, 251-252 and Steiner symmetrization, 252-256 decreasing, 8-9 and random walk in dangerous blind alley, 307-310, 317-322 discrete, 66-185 and difference inequalities, 168-185 and exit times, 151-152, 184-185 and Green's functions, 148-151, 153, 184 and harmonic measure, 145-146, 152-153, 180-183 on graphs, 81-123 discrete Schwarz type, 124 ordering with respect to, 124-126 discrete Steiner type, 126 decomposition of, 126 fibres of, 126 ordering with respect to, 129 reversing of, 131-134 uniqueness of fibres of, 128 one-sided lengthwise Steiner, 284-285 and harmonic measures, 284-285 radial, xvi i , 262-306 and Dirichlet integrals, 271-272 and exit times of Brownian motion, 291-294 and harmonic measure, 266-274 and inner radii, 263, 281-282 and Steiner symmetrization, 271 lifted to cylinder, 275-276 Schwarz type, see rearrangement, discrete Schwarz type Steiner type, see rearrangement, discrete Steiner type symmetric decreasing, 46-50 and Dirichlet norms, xvi i , 243-250 discrete, 70-71 for holomorphic functions, 244-248 various types of, see also symmetrization rearrangement, product, 134 regular domains, see domains, regular points, see points, regular regular trees, see trees, regular resolutive functions, see functions, reso-lutive Riemann map, 28 Riesz measure, 39-41 for C 2-functions, 39 for rotation invariant functions, 40 roots of trees, see trees, regular, roots of cr-pseudotopology, 4-5 S A R I P , 237-240 363 Index Schwarz symmetrization, see symmetriza-tion, Schwarz Schwarz type rearrangement, see rearrange-ment, discrete Schwarz type sets convex, 2 horizontally convex, 2 star-shaped, 2 similarly ordered functions, see functions, similarly ordered smooth strict analytic radial increase prop-erty, see SSARIP spaces B M O , 21, 201-202 B M O A , 21 Dirichlet, xv -xv i i , 189-192, 201-202 disc algebra, 21 Hp, see spaces, Hardy hp, see spaces, Hardy Hardy, 18, 194 Hilbert of measurable functions, 205-210 Nevanlinna class, 22 spiral-like well-ordering, see ordering, spiral-like SSARIP, 238-240 star-shaped sets, see sets, star-shaped Steiner symmetrization, see symmetriza-tion, Steiner Steiner symmetry, see symmetry, Steiner Steiner type rearrangement, see rearrange-ment, discrete Steiner type strict analytic radial increase property, see SARIP strictly convex functions, see functions, strictly convex subgraphs full, 82 subharmonic functions, see functions, sub-harmonic superharmonic functions, see functions, superharmonic symmetric decreasing functions, see func-tions, symmetric decreasing symmetric decreasing rearrangement, see rearrangement, symmetric decreas-ing symmetrization circular, x i i - x i i i , 11, 17, 41-46 and Green's functions, x i i - x i i i , 44-46, 53-60 and harmonic measure, x i i i , 46 and least harmonic majorants, 45-46 discrete Steiner type, 135 formal definition of, 17, 135 lengthwise Steiner, 276-279 and harmonic measure, 276-279 on discrete cylinder, xiv Schwarz, 9, 17 Steiner about real axis, 9-11, 17, 51-53 and Baernstein's sub-Steiner rearrangement, 252-256 and Dirichlet integrals, 270-271 and Green's functions, 52 and harmonic measure, 52 and harmonic measures, 52 and least harmonic majorants, 52 and Nevanlinna counting function, 254 and radial rearrangement, 271 lengthwise, see symmetrization, lengthwise Steiner various types of, see also rearrangement symmetrization theory, xii-xv symmetry circular, x i i , 41-42 and simple connectivity, 52-53 discrete Steiner type, 127-128, 130 Steiner about the real axis, 51-52 and simple connectivity, 52-53 Szego projection, 20 ternary plane, see graph, ternary plane Theorem Alvino, Lions and Trombetti, 282-283 Beurling shove, 158-164, 268-269, 294-306 Fatou, 19-20 Levy, 33 Riesz decomposition, 39-41 Riesz, F . and M . , 19-20 Riesz, M . , 20 Stein-Weiss, 201-202, 248 trees, 88 definitions for, 88 geodesies on, 88-90 regular, xv, 88-118 364 Index ancestors in, 90 children in, 90 definitions for, 88-91 descendants in, 90 Faber-Krahn inequality on, see inequality, Faber-Krahn, on regular trees geodesic balls in, 90-91 heights in, 90 master inequality on, 91-93 orderings on, see ordering, spiral-like parents in, 90 roots of, 90 uniform motion to the right, 287-291 uniformizer, 34-37, 194 and harmonic measure, 35-37 and PWB solutions, 35-37 univalent functions, see functions, univa-lent universal covering map, see map, univer-sal covering vertices, see graphs, vertices of 365
Thesis/Dissertation
1996-05
10.14288/1.0079810
eng
Mathematics
Vancouver : University of British Columbia Library
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
Graduate
Symmetrization, green’s functions, harmonic measures and difference equations
Text
http://hdl.handle.net/2429/4767