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Symmetrization, green’s functions, harmonic measures and difference equations Pruss, Alexander Robert 1996

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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. Bef