MAJORANT PROBLEMS IN HARMONIC ANALYSIS by MICHAEL ANTHONY RAINS B.Sc, University of Auckland, 1970 M.Sc.(Hons), University of Auckland, 1971 A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILMENT OF FOR THE DEGREE OF PHILOSOPHY in the department of MATHEMATICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1976 (a) Michael Anthony Rains < In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of Br i t ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1WS i i Thesis Supervisor: Professor J. Fournier ABSTRACT In various questions of Harmonic analysis we encounter the problem of deriving a norm inequality between a pair of functions when we know a (point wise) inequality between the transforms of these functions. Such problems are known as majorant problems. In t h i s thesis we consider two related problems. F i r s t , i n Chapter two, we extend the known results on the upper majorant property on compact abelian groups to noncompact l o c a l l y compact abelian groups. We show, using various test spaces and two notions of majorant, that a Lebesgue space has the upper majorant property exactly when i t s index i s an even integer or i n f i n i t y . Further-more, i f a Lebesgue space has the lower majorant property, then the Lebesgue space with conjugate index has the upper majorant property. In the f i n a l chapter we consider the second problem. Here-, we are concerned with deriving global i n t e g r a b i l i t y conditions from l o c a l i n t e g r a b i l i t y conditions for functions which have nonnegative transforms. Such a property holds only i n Lebesgue spaces whose index i s an even integer or i n f i n i t y . For Lebesgue spaces whose index i s not an even integer or i n f i n i t y the proof of the f a i l u r e of th i s property i s based on the f a i l u r e of the majorant property i n these spaces. i i i ACKNOWLEDGEMENTS I t i s a pleasure to thank Professor John Fournier for his encouragement and patience during the preparation of th i s work. I would also l i k e to thank Professors Fournier and J. Coury for their comments on the i n i t i a l d rafts. F i n a l l y , I thank the University of B r i t i s h Columbia for their f i n a n c i a l assistance. XV TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION 1 (1.1) Background 1 (1.2) Notations 5 (1.3) Groups: functions and measures thereon 6 (1.4) Preliminaries r e l a t i n g to subgroups and quotient groups 8 (1.5) Some properties of S(G) 10 CHAPTER 2 THE MAJORANT PROBLEM 13 (2.1) Definitions and examples 13 (2.2) Some reductions 18 (2.3) Examples derived from the compact case 22 (2.4) The case of the integers 26 (2.5) Examples derived from the integers 36 (2.6) Miscellany 40 (2.7) The lower majorant property 50 CHAPTER 3 FUNCTIONS WITH NONNEGATIVE TRANSFORMS 57 (3.1) Even integers again 57 (3.2) Failure of "good" behaviour when p i s not an even integer or 0 0 . 59 BIBLIOGRAPHY 67 CHAPTER 1 INTRODUCTION The f i r s t section of t h i s chapter i s devoted to providing background for the problems we treat i n t h i s thesis. The remaining sections are devoted to notations and various preliminaries. (1.1) Background Throughout t h i s section G i s an i n f i n i t e compact abelian group whose Haar measure has t o t a l mass one. The spaces ^p(^) a r e formed with respect to t h i s measure. (1.1.1) D e f i n i t i o n : If f.,g e ^ (Q) and |f | < g we say that g majorizes f (or g i s a majorant of f ) . Let 1 < p < 0 0 . We say Lp(G) has the upper majorant property i f there i s a positive constant D such that i f f,g e !• (G) and g majorizes f then | | f | | p < D | | g | | . We say L p(G) has the lower majorant property i f there i s a positive constant C such that every f e L p(G) has a majorant g e L (G) for which | | g|| < C||f | | . These properties w i l l be abbreviated to UMP and LMP respectively. (1.1.2) The majorant problem i s to determine for which p the space Lp(G) has the UMP or the LMP. This problem was i n i t i a t e d by Hardy and Littlewood [11] and from there evolved through C i v i n [5], Boas [3], and recently Bachelis [1] and Fournier [7]. The l a t t e r two authors extend the results of Hardy and Littlewood and of Boas for the c i r c l e group to a l l i n f i n i t e compact abelian groups. Bachelis completed the 2. the results for the c i r c l e . We now summarize thei r results as follows: (a) I f p i s an even integer or 0 0 , then L^(G) has the UMP with constant 1 . (b) I f p i s not an even integer or 0 0 t then ^(G) does not have the UMP. i (c) For 1 < p < co and — 4 — = 1 , the space L (G) has ' P I P the UMP i f and only i f L q(G) has the LMP, with the same constant. (d) I^CG) has the LMP with constant 1. (e) Consequently, ^(G) has the LMP i f and only i f P = 1>2, 2" »•••» 2k—l" '****' ^ ^ ^ " We can add a further statement i f we note that a continuous function on G which does not belong to A(G) (see (1.3) of t h i s thesis) has no majorant i n L^CG) • To prove t h i s use [13, (37.4) and (31.42)]. Hence we obtain (f) Lro(G) does not have the LMP. These s i x statements can be summarized i n the following way. If G i s an i n f i n i t e , compact abelian group, then: (A) Lp(G) has the UMP i f and only i f p i s an even integer or 0 0 ; and when L (G) has the UMP the constant i s 1 ; P (B) If 1 < p < °° and - + - = 1 , then L (G) has the - - P q P UMP i f and only i f L q(G) has the LMP with the same constant. The statements of (B) could be called "duality theorems". That L p(G) has the UMP implies L q ( G ) has the LMP was proved, for the c i r c l e group, by Hardy and Littlewood and t i d i e d up byBachelis [1]. This i s the more d i f f i c u l t of the two assertions. The other statement 3. i n (B) i s due to Boas [3]. Possible generalizations to noncompact l o c a l l y compact abelian groups are given by Ci v i n [5] and Boas [3]. Civ i n proved dual versions of the results of Hardy and Littlewood, while Boas extended Civin's work and considered various interpretations of the expression < g" i - n the d e f i n i t i o n of majorant. Recently Fournier [8, proof of theorem 3, p. 272] used the fact that L 2 k ( G ) , k e N and G noncompact, has the UMP (see chapter 2). In t h i s thesis we w i l l be concerned with analogues of these results for noncompact l o c a l l y compact abelian groups. We obtain analogues of (A) and half of (B). Even for the integer group and the r e a l l i n e , these results are new. (1.1.3) In t h i s section we are concerned only with the c i r c l e group T . A problem related to that of (1.1.2) was i n i t i a t e d by N. Wiener and taken up recently by S. Wainger [20] and H. Shapiro [18]. These results concern the derivation of global "good" behaviour from l o c a l "good" behaviour for functions belonging to L^(T) which have non-negative Fourier c o e f f i c i e n t s . By "good" behaviour, we mean that the function belongs to L ^ ( - 6 , 6 ) for some p , 1 < p < °° , and for some 6 > 0 . A f i r s t step i n th i s d i r e c t i o n i s the well known result that a positive d e f i n i t e function which i s continuous i n a neighbourhood of the identity i s continuous everywhere (see [13, (32.1) and (32.4)]). Furthermore, i f f i s continuous then f i s positive d e f i n i t e i f and only i f f > 0 (see [13, (34.12)]). Now consider general f belonging to L n(T) with f > 0 . A c l a s s i c a l r e s u l t , which may be found i n 4. [6, p. 144], states that If f i s also essen t i a l l y bounded i n a neigh-bourhood of the i d e n t i t y , then f belongs to ^ ( T ) with no increase i n norm. Boas [4, p. 242, Theorem 12.6.12] credits Wiener with an analogue of t h i s result which uses L„ instead of L . From Wiener's 2 «> result i t i s not d i f f i c u l t to derive the obvious analogue for L when P p i s an even integer. Indeed, suppose that p i s an even integer or 0 0 and that 6 > 0 . Then there i s a positive constant C , dependent on at most 6 and p , such that i f f e L 1(T) , f > 0 and f e L (-6,6) 1 p then f e L (T) and | | f | | < C | | f | | L (_ } . p P I t i s natural to ask what happens when p i s f i n i t e and not an even integer. I t has been shown by Wainger [20] and Shapiro [18] that the analogous statements f a i l i n these cases. The f i r s t examples were provided by Wainger and were phrased i n terms of theory. Thus, i f 0 < p < 2 and i f 0 < 6 < TT , then there i s a power series 00 F(z) = £ a^ z 1 1 , analytic i n the open unit disc with a_ > 0 (n=l,2,...) n=l such that sup ^ -6 -••fl P |F(re 1 0) | d0 < oo , while sup r<l |F(re 1 6)| Pd0 - T T I t was remarked by Shapiro [18, p. 10J that Wainger's results for 1 < p < 2. can be recast i n terms of two-sided trigonometric series, 5. as follows: there i s a trigonometric series with nonnegative coef f i c i e n t s which i s not the Fourier series of any function of L (T) , yet (in the sense of dis t r i b u t i o n s ) coincides with an L^-function i n the neighbourhood [ —T7 + 6 , TV — S ] of the id e n t i t y . For f i n i t e p > 2 which are not even integers Shapiro proved the following (see [18, p. 16]): i f 6 > 0 , then there exists g belonging to L^(T) which has nonnegative c o e f f i c i e n t s and which s a t i s f i e s ( i ) g e L (-6,6> ; and ( i i ) g \ L p(T) . In Chapter 3, we s h a l l obtain generalizations of these results on any i n f i n i t e compact abelian group. (1.2) Notations We denote the group of r e a l numbers by R , the c i r c l e group by T , and the integer group by Z . The pos i t i v e integers w i l l be denoted by N . The c y c l i c group of order r , r > 1 , i s denoted by Z(r) and w i l l l a t e r be realized i n two possible forms: either as the rt h roots of unity, a subgroup of T , or as {0,1,..., r-1} with CO* addition mod r . Z(r) i s the direct sum of countably many copies of ZCr). . If X i s any set, V a subset of X , and f any function on X , then f|^ denotes the r e s t r i c t i o n of f to V and 1^ denotes the c h a r a c t e r i s t i c function of V . Denote by sgn the signum function given by ' z 1 n sgnCz) = . 0 z. = 0 6. If 1 < p < °° , then q denotes the conjugate index given by (1.3) Groups: functions and measures thereon General references for the following preliminaries are [12], [13], and {15]. Throughout t h i s thesis, G w i l l denote a l o c a l l y compact abelian group (abbreviated LCAG) whose dual group we denote by G , the group of a l l continuous homomorphisms of G into T . If H i s a sub-group of G then the annihilator of H , denoted H"1, , i s {YEG|Y(X) = 1 , for a l l x e H} . When H i s a closed subgroup of G , we have (G/H) = H and H = G/H"1 . See [15, p. 96]. G carries a special measure, known as Haar measure, which i s translation invariant ^ l s o inversion invariant when G i s abelian) and i s uniqueue up to a positive multiple. A l l integrals are taken with respect to Haar measure. For G compact, the Haar measure i s usually assumed to have t o t a l mass 1; for G discrete, Haar measure i s generally counting measure. One exception, however, i s the case where H i s an open sub-group of a compact group G . In th i s case, the t o t a l mass of H i s IG:HJ - 1 , where [G:HJ i s the index of H i n G . We use Haar measure to construct ^ CG) (l<p<°°) , the Banach space of functions whose absolute value i s pth power integrable, and L^CG) , the space of ess e n t i a l l y bounded measurable functions. For f e L CG) (1<P<°°) , we denote i t s norm by | |f | | = {/ |f(x)| Pdx} ^ , P p G " where dx denotes Haar measure on G . For L CG) (l<p<<=°) , functions are i d e n t i f i e d i f they are equal a.e. with respect to Haar measure; 7. functions i n ^(G) are i d e n t i f i e d when they are equal l o c a l l y a.e. with respect to Haar measure. For f e L^(G) we define Fourier and inverse Fourier transforms of f by the formulae f(Y) = f V ( Y ) = f(x) y(x) dx G f(x) y(x) dx G r e s p e c t i v e l y , where y e G . For WC L ^ G ) , l e t w" = {f l|* feW} . We s h a l l use several other spaces of functions (and measures). CQ(G) denotes the Banach space of continuous functions vanishing at 0 0 . The dual space of CQ(G) i s denoted by M(G) and i s to be r e a l i z e d as a space of complex regular measures on G (see [12, section 14]). The space of continuous, p o s i t i v e d e f i n i t e functions on G i s denoted by P(G) (see [13, (132.1)]) and A(G) denotes the range of the Fourier transform; thus A(G) = J L ^ G ) ] = {f | fel^CG)} It i s well known that A ( G ) C T C Q ( G ) • The compactly supported members of A(G) form a space denoted A (G) . A clas s of functions we s h a l l use often c i s L ( G ) O A ( G ) , which for b r e v i t y we denote by S(G) . I f , for i = 1,2 , G^ i s an LCAG, then so i s G^ x G^ and i t s dual group i s G^ x G^ . The product of Haar measures on G^ and G^ 8 . gives a Haar measure on G i x G 2 • I f » f o r 1 = x > 2 » ^ i s a function on G; , we define a function f- ® f„ on G x G_ by f x ® f 2 (x,y) = f ^ x ) f 2 ( y ) for each (x,y) e G^ x G^ It i s easy to see that i f 1 < p < °° and f ^ e L p ( G i ) (1=1>2) , then f 1 ® f 2 e x and f n ® f || = | | f . | | | If 1 2 p 1 p 1'^2' 'p If f± e \(.G ) (1=1,2) , then f l 9 F 2 ( Y 1 ' V - f l ( V F 2 ^ 2 > * f l 0 F 2 ( V V for each C Y 1 » Y 2 ) e G I X G 2 ^ S E E ^ 1 3 ' ( 3 1 . 7 ) ( b ) ] ) . Consequently i f f ± e S(G i) , i = 1,2 , then f± ® f 2 e S(G ; L x G 2) . (1.4) Preliminaries r e l a t i n g to subgroups and quotient groups In t h i s section we l e t H be a closed subgroup of G , although i n most cases we s h a l l only be concerned with a compact and open subgroup. We are concerned with two s i t u a t i o n s : f i r s t l y , functions which are supported by H , and secondly functions constant on the cosets of H . These two notions are dual i n the sense of the following r e s u l t . (1.4.1) Lemma. Let u e M(G) . Then u i s supported by H if- and only i f u i s constant on the cosets of H . 9. For a proof, see [15, p. 118]. (1.4.2) I f TT i s the canonical projection of G onto G/H , then the map <f> |—> <j> o TT i s a p o s i t i v i t y preserving one - one correspondence between functions on G/H and functions on G which are constant on the cosets of H . (1.4.3) Suppose that H i s an open subgroup of G and l e t f e L..(H) rl X be continuous. If we extend f to a function f on G by l e t t i n g n f(x) = 0 i f x £ H , then f e L^(.G) i s continuous and f i s supported by H . Thus f i s constant on the cosets of H and so we can i d e n t i f y f with a function on G7H X = H . As i n [15, p. 122] we i d e n t i f y f with, f H (1.4.4) At the root of (1.4.5) and a l a t e r calculation i s a specialized version of Weil's formula (see [15, p. 70]) which says that i f H i s a closed subgroup of G , then I f f(acy)dy V d i >Gfal JH J f(x)dx G for £ E L^(G). ; here dy , dx , dx: denote Haar measure on H , G/H , and G respectively. We want In par t i c u l a r the special case where G i s compact and H. i s open, rf f i s constant on the cosets of H , l e t F be the function on Gyl'H for which F o TT = f . Then Weil's formula unveiled i s F(x)dx = [G:HJ G/H f (x)dx 10. (1.4.5) By combining (1.4.4) with 115, p. 118], we obtain, for compact open H , A(H) = ACG) | R = {£ | R | feA(G)} . (1.5) Some properties of S(G) In t h i s section we l i s t the properties of S(G) = L^(G)O A(G) which we w i l l use l a t e r . Unless otherwise stated, a l l proofs of these statements can be found i n [10, chapter 3]. (1;". 5.1) S(G) can be made into a Banach space v i a the norm l l f l l s - l l f l l x * l l f l l i • Then the Fourier transform maps S(G) isometrically onto S(G) . S(G) i s an ideal i n M(G) and thus also i s an ideal i n L^(G) S(G) i s dense i n L p(G) , 1 < p < 0 0 . (1.5.2) The Banach space dual of S(G) , denoted by S (G) , may be regarded as a space of di s t r i b t u i o n s on G. If 1 <' p < » , L (G) - P * can Be embedded i n S (G) , since i f f e L (G) the formula <f,u> = f(x) u(x)dx , u e S(G) , defines a continuous linear functional, denoted by L f , on S(G) Note that for 1 < r < » we have ||u|| < ||u|L when u e S(G) Similarly one embeds M(G) i n S*(G) by setting, for u e M(G) , <y,u> = u(x)dy(x) , u e S(G) G 11. (1.5.3) The Fourier transform for members of S (G) i s defined i n the usual dual manner. If L e S (G) we define L e S (G) by <L,u> = <L,u> , u E S(G) . A similar formula defines inverse transforms. We now have two ways of defining f when f e ^ p^) a n <* 1 < p < 2 ; one v i a the Hausdorff-Young theorem (see [13, (31.21)J) and the second v i a d i s t r i b u t i o n s . For f e ^ CG) a n a 1 1 P f 2 , these two d e f i n i t i o n s agree i n the sense that *L"^ * = L~ . For p > 2 we have, i n the noncompact case, only the d e f i n i t i o n as a d i s t r i b u t i o n . In t h i s case, by combining [10, 3.10] and [9, Chapter 6, Theorem 6.6], we see that there i s an f e L (G) for which f (that i s 1^) i s not defined P f by a measure. Xl.5.4) We need to know that S(G) contains functions with nonnegative transforms. This fact i s contained i n the following result (for a proof, see 113, C33.12)]). Lemma. L^(G) contains an approximate id e n t i t y {u^|iel} , each member of which belongs to S(G) , and s a t i s f i e s the following: (a) each u^ belongs to S(G)0 P(G) and i s nonnegative; (b) for each i e I , u. (x)dx = 1 ; G 1 (c) each u^ i s nonnegative and belongs to k^(G)(l'P(G) ; Cd) lim u^ = 1 pointwise and uniformly on compact sets. I t follows from [13, (32.33) Cb) and (32.4.8) (a)] that 12. Ce) for 1 < p < o> We have lim I|u.*f - f i l = 0 for i 1 1 i 1 'P f e LpCG) , and i f f e CQ(G) we have lim u.*f - f =0 l Cl.5.5) If f e SCG) , we have f e P(G) i f and only i f f > 0 This i s simply [13, (33.3) and (33.10)]. 13 CHAPTER 2 THE MAJORANT PROBLEM In t h i s chapter we present the main result of the thesis, along with some related topics. The main theorem i s stated i n section 1 and proved ,in the succeeding sections. Throughout t h i s chapter, unless otherwise stated, G w i l l be a noncompact LCAG. We are grateful to J. Fournier and M. Cowling for conversations regarding S(G). (2.1) Definitions and examples. Since we are dealing with noncompact groups we have problems regarding the Fourier transform on ^(G) when p > 2 7 S p e c i f i c a l l y , we want to interpret the inequality | f | < a when f , g e Lp(G) , p > 2 . As noted i n (1.5.3), f need not be a measure and so not a function. The d e f i n i t i o n s (1.1.1) were used by Boas and Bachelis, and i n [7], Fournier uses a d e f i n i t i o n involving only trigonometric polynomials. We state t h i s now. (2.1.1) D e f i n i t i o n . Suppose that G i s compact. We say ^(G) n a s the UMPT i f there i s a positive constant D such that whenever f , g are trigonometric polynomials ( f i n i t e l i n e a r combinations of characters) and g majorizes f , we have ||f||p<o||g||^ . We say ^ (G) has the LMPT i f there i s a positive constant C such that every trigonometric polynomial f has a trigonometric polynomial 14. majorant g for which ||g|| p < C||f||^ . We show that t h i s d e f i n i t i o n i s equivalent to that given i n (1.1.1) . (2.2.2) Lemma. Let G be compact. a) I f 1 < p < 0 0 , the UMPT as defined i n (2.1.1) i s equivalent to the UMP as defined i n (1.1.1). The constants are the same. b) I f 1 < p < oo , the LMPT as defined i n (2.1.1) i s equivalent to the LMP as defined i n (1.1.1). Proof a) I t i s obvious that the d e f i n i t i o n of (1.1.1) implies that of (2.1.1). Conversely, i f 1 < p < » a n d f , g e L p(G) are such that |f| < g , l e t (u^) be an approximate i d e n t i t y as i n (1.5.4). Then for every a , we have u * f < u *g ;• and thus | |u * f | | < D| | U *g| | , I I a i i p _ II a . O l |p where D i s the constant of the d e f i n i t i o n . Taking l i m i t s we obtain l l * l l p < D | . | 8 | | p • b) Let Cu ) be as i n (1.5.4) and l e t f e L (G) . Then a P u^ * f i s a trigonometric polynomial. Let g^ be a trigonometric 15. polynomial majorizing * f and s a t i s f y i n g i < C u•*f < C f =a 1 1 p = 1 1 a M p = 1 1 M p Here we use the fact that u L = 1 . 1 1 a 111 Then the net ^S a^ 1 S norm-boundediinttherBahach space L (G) and so has a weak * convergent subnet, {g } say, with weak * P p l i m i t g i n L^(G) . Then < lim g J < C f , lp = _ I l&gl i p = P and by weak * convergence we have g(y) = lim i (Y) > lim |u ( y ) f C Y ) I = | f ( Y > | 3 P B for every y e G . Hence L^(G) has the LMP i n the sense of (1.1.1). Conversely, i f f i s a trigonometric polynomial, then there i s g belonging to kp(CI) which majorizes f and s a t i s f i e s ||g||p = G l | f | l p • ^ e s n o w that g can be replaced by a trigonometric polynomial which changes only s l i g h t l y the constant for the LMP. For any e > 0 we can (according to [13, (31.37)] find an h e L^CG) with the properties ( i ) 0 < h < 1 ; ( i i ) h Is compactly supported and 1I(Y) = 1 for a l l Y such that I(Y) 4 0 '> ( i i i ) ||h|\± < 1 + e . 16. Then h * g i s a trigonometric polynomial majorizing f . Moreover, we have I |h*g| l p < I |h|\1 | |g| | p < U+O I |g| | p < d+e)C | |f | | p . This proves the lemma. It i s clear that the LMP implies the LMPT for L^G) as w e l l . For the moment, we can not give a direct proof, as i n (2.1.2)(a), for the converse of t h i s statement. However, that the converse i s true follows from the fact that the LMP and the LMPT hold for L^G) . The LMP i s easy to prove d i r e c t l y , and the LMPT for L^(G) may be derived from the LMP for L i(G) . Lemma (2.1.2) suggests that, when G i s not compact, we could use a suitable test space rather than a l l of L p ( G ) i - n o u r d e f i n i t i o n s of majorant and majorant properties. When G i s compact, the space of trigonometric polynomials coincides with [A c(G)] . We could use t h i s test space i n our d e f i n i t i o n s for noncompact G , but we prefer to use S(G) and return to the consideration of other spaces i n Section 6. Roughly speaking, any test space contained i n L^(G) and dense i n Lp(G) (p f i n i t e ) w i l l do. (2.1.3) D e f i n i t i o n . I f f , g e S(G) and | f | < g , we s h a l l say that g majorizes f or that g i s a majorant of f . We say L^(G) has the upper majorant property i f there i s a positive constant D such that whenever f , g e S(G) and g majorizes 17. f , we have | | f | | p < D||g||p . Due to the frequent appearence of t h i s phrase, upper majorant property w i l l be abbreviated to UMP from now on. (2.1.4) Proposition. I f p i s an even integer or 0 3 then L p(G) has the UMP with constant 1. Proof: F i r s t suppose p = 2 ; then the res u l t i s obvious by the Parseval formula, since i f f , g E S(G) and |f| < g w e h a v e i i f i i 2 - i i f i i 2 < i i i n 2 = i i g i i 2 . If p = 2k (keN) then h k e L £(G) for every h e L 2 k(G) If f , g e SCG) and |f| < g , then f k , g k e SCG) and |f I = |f* * f | < g* *g = g K There are k f's i n the f i r s t product and k g's i n the second. From the case we have 2k , 2 . 2 2k 11*11. " M f k H < Mg kM = ||g|| . 2k 2 2 2k that i s 2k 2k 18. If p = °° , we have S(G) C L (G) l<p<oo P and the r e l a t i o n h = lim h I I I 00 I I I I k-x» 2k holds for at least every h E ^ . The result follows from the L_, results, 2k Our main result i s , e s s e n t i a l l y , that the converse of (2.1.4) i s also true. We now state t h i s theorem. (2.1.5) Main Theorem. Suppose that G i s a noncompact LCAG and 1 < p < » . I f p i s not an even integer, then L (G) does not have the UMP. To prove t h i s theorem we must show that for every p o s i t i v e constant D , there exist f , g e S(G) such that a) | f | < g and b) | | f | | p > D||g||p . This w i l l be accomplished i n the next four sections. (2.2) Some Reductions. Our proof of the main theorem i s based on the structure theorem for LCAG's, and i n t h i s section we show that i t suffices to prove the theorem for certain classes of groups. '(2.2.1), Theorem. Suppose that G^ and G^ are LCAG's, and suppose that L (G,) or L (G 0) does not have the UMP. Then L (Gn x G„) does p 1 P 2 • p 1 2 not have the UMP. 19. Proof. For definiteness we assume that L (G.,) does not have the UMP. P 1 Hence, for any positive constant D there exist , g 1 e S(G^) which s a t i s f y a) f1 < g 1 on G1 and b) \\f1\\v > D||g1||p . Suppose that h e S(G 2) i s any n o n t r i v i a l function with h > 0 . Set F = f, ® h , G = g. ® h ; then F., G e S(G1 x G„) and 1 1 A 1 Z we have |F| = l ^ s h l = \f±\ ® h < g 1 ® h = G . Furthermore, H ' l l p - I M , I N I , * " ! ! * ! ! ! , I N I , - » l l 6 | | , • Thus L (G.. x G.) does not have the UMP. p i L We now r e c a l l the structure theorem for LCAG's (see[12, (.24.30)]). This theorem states that any LCAG i s of the form R n x GQ , where n i s a nonnegative integer and G^ i s an LCAG containing a compact open subgroup. We apply Theorem (2.2.1) to the structure theorem to determine the groups we must consider. 20. If R n x GQ i s i n f i n i t e , then one of the following statements i s true: (a) n > 0 ; (b) n = 0 and GQ has an i n f i n i t e compact open subgroup; (c) n = 0 and GQ i s an i n f i n i t e discrete group. It therefore suffices to consider only the group R for case (a); i n case (b) we w i l l be able to construct examples from those known for the i n f i n i t e , compact open subgroup of GQ (see. ( 2 . 3 ) ) . In case (c) the discrete group GQ may have an element of i n f i n i t e order, i n which case i t contains a copy of Z . Otherwise GQ i s a torsion group. In a torsion group, either we have elements of a r b i t r a r i l y large order or there i s a bound on the orders of a l l elements. CO* In t h i s l a t t e r case, we know that GQ contains a copy of Z(r) , the direct sum of countably many copies of Z(r) , and r > 2 (see [12, p. 4 4 9 ] ) . We now give a further reduction for the discrete case. (.2.2.2) Proposition. Let G be an i n f i n i t e discrete abelian group containing a subgroup isomorphic to a discrete group H for which -£p(H) does not have the UMP. Then I (G) does not have the UMP. Proof. We note that S(G) = -^(G) when G i s discrete. Let $: H •+ G be an embedding of H i n G . For any positive constant D there exist f , get-(H) which s a t i s f y |f| < g and | | f | | > D||g|| . For any function h on H , we define h'' on G by 21. , 1 , v v i f x e cf>(H) , otherwise . I t i s clear that i f h e £ (H) , then h"' E I (G) and | |h'| | = | |h| for any r > 1 . Defining f' and g' s i m i l a r l y we immediately obtain | f " " l l > D||g''"ll We need only show that |^| < "g^ . If h s ^ ( H ) , then h'L E l^G) i s supported by <f> (H) and so & i s constant on the cosets of tj)(H) We can i d e n t i f y h"' with a function on G/<j>(H) = <i> (H) = H . By (.1.4.3) we may i d e n t i f y h""1 with h . Hence we have 1^1 - I f I < ? - - ? . and t h i s produces the required example. I t thus suffices i n dealing with discrete groups that have elements of i n f i n i t e order, to consider only the group Z ; likewise for i n f i n i t e discrete groups i n which there i s a bound on the orders of the elements, i t i s enough to consider only the groups Z(r) , for r>2 . We now use t h i s r e s u l t to summarize the groups or classes of groups for which we must prove the main theorem. They are: a) R ; b) Z ; c) Z(r) , for r > 2 ; 22. d) G a nondiscrete, noncompact LCAG with a compact open subgroup; e) G a discrete abelian torsion group with elements of a r b i t r a r i l y large order. We s h a l l see that c) and d) may be derived from the compact case. The integer group Z i s treated separately, and the f i n a l two cases may be derived from the case of the integers. (2.3) Examples derived from the compact case In t h i s section we prove the main theorem for LCAG's that CO* have an i n f i n i t e compact open subgroup or those of the form Z(r) for some integer r > 2 . '(2.3.1) Theorem. Let G be a nondiscrete noncompact LCAG containing a compact open subgroup. I f p i s f i n i t e and not an even integer, then L (G) does not have the UMP. P Proof. We r e c a l l that we must show that for any positive constant D there are f , g e S(G) which s a t i s f y |f| < g and ||f||p > D l | g | | Let GQ be an ( i n f i n i t e ) compact open subgroup of G . Haar oneasure on G can be chosen to assign mass 1 to G Q j since GQ i s open, i t s Haar measure i s the r e s t r i c t i o n to GQ of the Haar measure on G . Let D be any pos i t i v e constant./ From the compact case (see 17.]) , there are trigonometric polynomials f^ , g^ s a t i s f y i n g , | f 1 l i . g 1 on GQ and | | f 1 | | p > D||g1|| . In pa r t i c u l a r ^ and g n Belong to A(Gn). . 23. By (1.4.5), i f h^ e A(GQ) there i s an h e A(G) whose r e s t r i c t i o n to G i s h . Since G A i s compact and open, 1 U 1 U GQ belongs to A(G) (see [13, (31.7(1)).]) and so we can extend h^ to G by defining i t to be zero off G^ . In p a r t i c u l a r , we can extend a function Mi^in S(G^) to a function i n S(G) by l e t t i n g i t s extension h be zero off G_. ; moreover, i t i s clear that h n = h 0 1 p 'P Let f and g be such extensions to G of f^ and g^ respectively. Then i t i s immediate that | | f | | p > D||g||p and so we need only show that |f| < g on G . Since f and g are supported by G^ , so their transforms are constant on the cosets of G^* and hence can be i d e n t i f i e d with functions on G/GQ = G^ . But by (1.4.3) these l a t t e r functions are just f^ and g^ respectively. Thus we have f = f t 0 TT < g 1 O T T = g where TT denot es the canonical projection of G on G.. . This concludes the proof. We now turn our attention to the groups G = Z(r) , r > 2 . In t h i s case we w i l l be able to derive our examples from those of the compact group X = Z ( r ) W , the direct product of countably many copies of Z(r)_ . We note that for each, positive integer N , each of X and G contains a subgroup isomorphic to Z ( r ) ^ which are denoted by X^ and G^ respectively. Thus, 24. 2L. = { (x.) e -X I -x. = 0 i f j > N+l} N 3 1 3 -G N = {(y ) e G | y - 0 i f j > N+l} These groups are self-dual. CO* (2.3.2) Theorem. If G = Z(r) (r>2) and p i s f i n i t e and not an even integer, then L^(G) does not have the UMP. Proof. Let D be a positive constant. From the known results for X we can find trigonometric polynomials f , g on X which s a t i s f y |f| < g on G and | | f | | p > D | | g | | . As f and g are f i n i t e l y supported, there i s a positive integer N for which supp (f) (J supp (g) C G N , where supp(f) = {yeG | f(y) =f 0} . Thus the transforms of f and g , and hence also their r e f l e c t i o n s f and g respectively, must be constant on the cosets of G^ " i n X . We can now i d e n t i f y f and g with functions JL on X/Gjj = G N = G N ' I t ''"S t^1^s Pa:*-r °f functions on G^ (that i s on G,, but supported by G ) that w i l l provide our example. J. IN Before proceeding, we should summarize the main i d e n t i f i c a t i o n s used above. We start with a pair of functions f , g on X , constant on the cosets of Gj^ . This gives r i s e to a pair of functions F , G on X/G^ j = G^ . F i n a l l y , v i a a (topological) isomorphism, we obtain a 1 1 ''i \ pair F"' , G'' on G^ . I t i s F' , G" which w i l l provide our example, and so we w i l l need to check that the following two conditions are s a t i s f i e d : 25. a) on X , and b) llF'Mp > D||G'||P . Since G^ i s compact, F' , G' belong to Z^(G) = S(G) . To obtain a), note f i r s t that the f i n a l paragraph of the proof of (2.3.1) shows that |F| < G since |f| < g . Passing to F' and G' involves a p o s i t i v i t y preserving linear map and so we also have |1F^ | < 'G** . Hence a) holds. To show that b) holds, we trace the behaviour of the L p 0 0 norm of a function h on X which- i s constant on the cosets of G^-. N In the f i r s t stag§ we obtain an H on X/G^" for which h = H o ir , TT being the canonical projection of X on X/G^" . From (1«4.4) we have h(x)dx = r N H(x)dx , since [ X : G^J = r N . But we also have |h-IP = [ H I | P O TT ; thus I f . h- e L P C X ) we have H £ L P ( X / G * ) with I M I p - r~P ||H||p . CD In the second stage, H^ ' i s obtained from H. v i a a topological isomorphism of the underlying groups. As the image of a Haar measure under such, a map i s also a Haar measure, i t follows that there i s a posi t i v e constant c for which-[ H l l | | p C 1 / P | | t t | | p , C2) 26. for every H e L (G.T) . p N Using (1) and (2) we now see that b) holds: ||F'||P - c||F|| P P P = c r N | | f | | P P > c r V | | g | | P = DP||G'||P . P P Henee D||G'|| < ||F'| | p and th i s completes the proof. (2.4) The case of the integers As i n the previous cases, our goal i s to show i f p i s not an even integer or 0 0 , then for any positive D there are functions f , g belonging to ^ ( Z ) which s a t i s f y |f| < g on T and D||g||p < ||f Our method, though different i n d e t a i l , i s ess e n t i a l l y the same as that used by Bachelis [1] and Fournier [ 7 ] . For a discussion of the origins of t h i s method see Shapiro [ 1 8 ] . F i r s t note that the constant D in our de f i n i t i o n s of the UMP (see (1.1.1) and (2.1.3)) must be at least 1, since S(G) contains n o n t r i v i a l functions with nonnegative transforms For the group T , Bachelis proves (1.1.2)(b) by using a suggestion of Y. Katznelson to show, by an i t e r a t i o n method, that i f the UMP f a i l s to hold with D = 1 , then i t f a i l s to hold at a l l (see [1, p.121]). We now give the i t e r a t i o n method; i t i s the dualized version of a special case of Fejer's Lemma (see [1, p.121], and [21, p.49]). 27. C2.4.1) Proposition. Let 1 < p < 0 0 and suppose a i s a f i n i t e l y supported sequence on Z . For a p o s i t i v e integer n we define a by 1° i f m e nZ otherwise. Then for large n we have i n I i i i 2 a*a = a i n i i p p and i n part i c u l a r . 2 lim IIa*a„11= II a I I ^ II £ i l l p II l i p Proof. Note that a e c (Z) and thus a*a e c (Z) . We use induction n c n c on the number of elements £ i n the support of a . For £ g 1 , the 2 result i s obvious. Suppose now that £ > 2 and that ||e*0n|| = [|0|| for large n whenever 0 e C CCZ) and the support of 6 has t-1 members. Write 6 for l r i • Then (6 ) =6 for every n , z e Z . z {z} z n nz Eet a e c (Z) have £ elements i n i t s support and write c £ a = ) a. &, J=l J J £-1 If 6 = Y a.6, , then we have 28. a % = (6 + *l \ > * <Bn + al 6nk £> since 6 * 6 = 6 , whenever z , u e z . z u z+u The supports of these four terms are pairwise d i s j o i n t when n i s s u f f i c i e n t l y large, and so since 6 = 1 for every z . z p 3 Now £ - 1 £-i kl n j = l J k £ n k j j = l J n k j + k £ since a l l the nk.. + k£(l<j<£-l) are d i s t i n c t , we have £ - 1 P = V L IP „ I I 31 IP l i p I K *M| = I |«.| £ p j = l J (2) S i m i l a r l y , 1 ^ * 3 1 | P = ||3|| P . Combining ( 1 ) and (2) we obtain l«*%llp- He*eJ|P + 2|c.^ ||6||P+ C | « £ | P ) 2 , for large n 2 l « M ? -Taking pth roots we obtain the r e s u l t . (2.4.2) Corollary. Let 1 < p < °° be an exponent for which there i s a pair of f i n i t e l y supported functions f , g on Z sa t i s f y i n g |f| < g and ||g|| p < ll fllp • T h e n £ pt z) f a i l s to have the UMP. Proof. Since | | g | I ^ < | | f | | n > there i s a constant C > 1 for which Form f and g as defined i n the statement of (2.4.1). We note that n n f (x) = f(nx) since n I (x) = If (m) e - i m X = J f ( £ ) e - ± n £ x = f(nx) . m-L. n £=-«> Simi l a r l y for g R . Since |f| < g , we have | f S f n G O | 1 | f ( x ) | | f n ( x ) | = | f ( x ) | |f(nx)| < g(x) g(nx) = g^g^Cx) . Thus g*g n majorizes f * f when g majorizes f , for a l l p o sitive integers n . Applying (2.4.1) to each sequence g*g and f * f , n n 2 2 2 and noting that C ||g|1^ < | | f | | ^ , i t follows that there i s an n for 30. which C 2| |g*g | | < |.|f*f | | . For any positive constant D , there i s a positive integer £ I for which D < C . Iteration of the above procedure shows that there i s a pair of f i n i t e l y supported functions d , e on Z with e majorizing d and C^||e|| < ||d)| . Hence D||e|| < l|d|| and thus £ (Z) does not have the UMP . To complete the proof of the main theorem for the integer group, we must, for a given f i n i t e p which i s not an even integer, f i n d a pair of f i n i t e l y supported functions on Z which s a t i s f y the hypotheses of Corollary (2.4>.2). We require some preliminaries about the functions which operate on P^CZ), the real-valued members of P(Z) . Recall that i f A i s a subset of the complex plane C and F : A C , then we say F operates on P(Z) i f F o <j> e P(Z) whenever ((>. e P(Z) and rangeUl<b(T~ A . If A = (=1,1) and F i s real-valued, then Rudin [16] has shown that F must be of the form OO F(x) = I c x n , for |x| < 1 n=0 n and c > 0 for n = 0, 1, 2 n = A fore-runner of t h i s r e s u l t can be found i n Schoenberg Our next res u l t i s stated and proved only for hand. I t w i l l be clear that a more general formulation [17]. the case at i s possible. Its 31. proof requires an elementary application of the convergence theorem for sequences of positive d e f i n i t e functions (see [2, p. 17]). (2.4.3) Lemma. Let F : (-1,1) •> R be continuous and have the property that F o e P r(Z) whenever t|> i s a f i n i t e l y supported member of P r(Z) with range W.i>(T^ (-1,1) • Then F operates on P r(Z) . Proof. Let i|> e P r(Z) have range ( ^ ) > ( T ^ (-1,1) and l e t (B^) be the Fejer kernel i n L^T) . It i s well known that belongs to P r(Z) and i s f i n i t e l y supported for every N . Moreover, we have 0 < < 1 and so range(1^4(3 (-1,1) i f range($M L T (-1,1) • Thus F o (K^ |>) belongs to P r(Z) . Since the (K^) are an approximate i d e n t i t y for L i ( T ) , (1.5.4) shows that ip = lim K^i> pointwise and so F o ip = lim N N F o (K^) belongs to P r(Z) . Thus F operates on P r(Z) . (2.4.4) Proposition. If p i s not an even integer or °° f there exists a function (J> belonging to P r(Z) which i s f i n i t e l y supported and for which |<I>|P does not belong to P(Z) . Proof. Let H : (-1,1) •*• R be defined by H(x) = |x| P . Then i t i s easy to see that H i s not of the form (3) (unless p i s an even integer) and so H does not operate on P r(Z) . A s H i s continuous, we can apply (2.4.3) to conclude that there i s a f i n i t e l y supported $ e P r(Z) for which rangeQ>W<T~~ (-1,1) and H o <j> J: P(Z) ; that i s , |<j>|P \. P(Z) . We can now present the necessary examples. The germ of th i s method i s to be found, i n a disguised form, i n [7, p. 163]. 32. (2.4.5) Theorem. Suppose that p > 1 i s not an even integer or 0 0 . Then £ (Z) does not have the UMP with constant 1 . P Proof. We want to find f i n i t e l y supported functions f , g on Z which s a t i s f y a) |f| < g and b) ||g|| p < | | f | | p . Let <f> be as i n (2.4.4). Positive d e f i n i t e functions are self-adjoint and the r e a l ones are symmetric (<(>(-n) = <t> (n) for every n E Z) .• This implies that |<j)|P i s self-adjoint and so has a r e a l -valued Fourier transform. As |<f>|P £ P r(Z) , there i s an x £ T for which (|<J>|p) (x) < 0 , that i s , I \HD\V e " i £ x < 0 £=-oo (see (1.5.5)). Define a r e a l parameter f t = (1+ty) <f> , where 1 stands for the constant function with value..1. Note that each f f c i s f i n i t e l y supported and f^ £ P(Z) i f t > 0 . I t i s clear that i f t i s nonnegative, we have Y £ Z by Y C O = e , for every I e Z . Then for t we define functions f on Z by I t thus suffices to find a positive t n for which i g i p < n s i i p In other words, for positive t we set f = f , g = f and show that b) holds for some positive t ^ ; To t h i s end we define a function F on R by Ht) - | | f t | . | J = I |i+t Yc-e)| p . Notice that, for every £ , the function t h | l + t y C £ ) | P i s d i f f e r e n t i a b l e at 0 . Since <)> i s compactly supported, the sum defining F(t) i s r e a l l y f i n i t e ; hence we have: F ' ; CO) = I " f F i | + c e ) | p | i + t Y c £ M p ] t : = 0 £=-oo - P I \*U)\*izL\i-*yM\\m0 £=-°o = P I Uce> | p R e c T i m £=-oo = P R e ^ I |<f>(£)|P Y C £ ) ^ = p (*) . Thus F"' CO) •> 0 , and thus there must be a positive t ^ for which. 34. F ( t Q ) < FC-t Q) . Taking pth roots we obtain and b) follows. This completes the proof. (2.4.6) In t h i s section we sketch an alternative proof of (2.4.5) for p a r a t i o n a l number but not an even integer. This method, except for a minor modification due to Z not being compact, i s d i r e c t l y analogous to the example f i r s t produced by Hardy and Littlewood for ^ ( T ) (see {11, p. 305] and [3, p. 255]). Suppose that 1 < p < 0 0 and that p i s not an even integer. For each positive integer n , l e t c n be the binomial c o e f f i c i e n t f Cf - 1) (f - n+1) . Let k be the least positive integer n n! for which c < 0 . We need a function <fe i n I-(Z) with the following n • x properties: ^ 00 (a) (j> = g , for some n o n t r i v i a l , nonnegative C function g on T ; (b) for an integer L , to be specified l a t e r , the functions {cf> |l<£<L} are mutually orthogonal i n t^(l) . Note that these products are defined by pointwise m u l t i p l i c a t i o n . To obtain such a function <f> we need a n o n t r i v i a l , nonnegative oo *t n C function g on T for which the convolution powers g (1<-£.<L) are mutually orthogonal i n T,^(T) •• Since these convolution powers are nonnegative they are orthogonal i f and only i f t h e i r supports are d i s -j o i n t . Suppose g i s supported by a small i n t e r v a l [a,b] with 0 < a < b < 2TT . Then the convolution powers g (1<-£<L) w i l l have d i s j o i n t supports provided that b i s s u f f i c i e n t l y small and a i s s u f f i c i e n t l y close to t^.c. 2 p °° Now l e t i|> = <J> . Since <J> i s the Fourier transform of a C function, the same i s true of ip ; i n p a r t i c u l a r , ip belongs to £^(Z) . Let A be a re a l number with |A| < 1 and l e t t be a positive parameter. We define a function f by A f x = \i>(i + t$ + xt\k) where 1 i s the constant function whose sole value i s 1 . Then f A belongs to -L^(Z) . We f i r s t examine the t (Z)-norm of f, . We have p X | | f ^ | |p = | | f p ^ 2 | 12 and, i t t i s small enough, we can apply the k k p/2 binomial theorem to ( 1 + t<|> + At <|> ) . When we do t h i s we obtain an absolutely convergent double series ( in Z^(Z)) and thus f p / 2 = * P / 2 ( 1 + t * + A t V ) p / 2 = * I Mt<j>r , 1=0 *-where 3^ = for 0 < I < k - 1 and 6 k = C^ A + Cfc . If we write 36. 2k f J / 2 - * { I e W + o ( t 2 k + 1 ) } , • 1=0 L then using (b), with L = 2k+l , we obtain I M S " H'J'2H2= I l ^ l 2 I l / V P • 2 1=0 L 2 Our choice of k shows that 1*1 I | P < f l * • i M P i i i i p I I _ i i i p i f t i s s u f f i c i e n t l y small. This takes care of the norm inequality. We would l i k e to know i f f^ majorizes . This would cer t a i n l y be the case i f ip were positive d e f i n i t e . For general p the author has had no luck i n determining i f we can assume that ip = <j>2^P i s p o s i t i v e d e f i n i t e i n addition to conditions (a) and Xb) . However, i t i s quite easily done when p i s r a t i o n a l . For suppose that p = ; then — = — . I f we l e t ip = g and <j> = g , then ip = <j> , ip i s p o s i t i v e d e f i n i t e , and ip belongs to t^(Z) . Hence f^ i s a majorant of f_^ . The only change required for the norm computation i s to replace L by mL i n condition (b). (2.5) Examples derived from the integer group This section gives the proof of the main theorem for the re-maining two classes of groups, namely R and discrete torsion groups with elements of a r b i t r a r i l y large orders. In both cases we derive these 37. results from the corresponding results for the integers. We f i r s t deal with R . A s l i g h t l y modified form of a device due to de Leeuw (see [14, p. 375]) i s required. The proof i s the same as his and so i s omitted. Before stating t h i s lemma, we remark that f i n i t e l y supported functions on Z can be i d e n t i f i e d with f i n i t e sums of point masses on R v i a E 2 a £ ^ , where 6 £ i s the unit point mass at Z on R . (2.5.1) Lemma. Let 1 < p < «> . Let <j> e A £(R) have the following properties: (1) <{> > 0 ; (2) * > 0 ; (3) / <j>p(x)dx = 1 ; (4) supp(<j>) = {xeR|<f>(x) ^ °} C [a, a+1] for some r e a l number a . Then for any f i n i t e l y supported a on Z ( i d e n t i f i e d with the corres-ponding discrete measure on R) , we have a * <j> e S(R) and l l * * * l l p " l | a | | p • Since any i n t e r v a l of length one w i l l work i n (4), ant example of such a cj) i s , except for the normalizing factor required for (3), <J>(x) = [ l - 2|x| |x| < 1/2 |x| > 1/2 . 38. (2.5.2) Theorem. Suppose that p > 1 i s not an even integer or 0 0 Then L (R) does not have the UMP. P Proof. Let D be any positive constant. In the proof of (2.4.5) we found f i n i t e l y supported X , y on Z which s a t i s f y (a) |X| < y on T , hence on R and '(b) D| |y I I < I Ixl I . I I i i p I I i i p Let <f> be as i n (2.5.1) and set f = X*$ , g = y*<f) . Then f and g belong to S(R) since S(R) i s an id e a l i n M(R) (see (1.5.1)). By (a) we obtain | f | = \x\l < y$ = I , and from (2.5.1) and (b) we have l l f l l p = l.|X**|| p = I M I p > D | M l p - D||y**||p - D||g||p . This completes the proof. (2.5.3) Theorem. Let G be a discrete abelian group containing elements of a r b i t r a r i l y large order. Suppose that p > 1 i s not an even integer or » . Then I (G) does not have the UMP. P Proof. Let D denote an arb i t r a r y positive constant. We know that there are f i n i t e l y supported functions f and g' on Z which s a t i s f y | f * | <1> on T and D||g'||p < | | f ' | | p . 39. For a positive integer n , l e t S r = {-n, -n+l,..., -1,0,1,..., n-1, n} . Then there i s a positive integer n for which supp(f')U supp(g')C S n . has 2n + 1 members and by hypothesis G contains a c y c l i c group H with at least 2n+l members. Suppose that H has r elements, 2 r-1 r > 2n+l , and i d e n t i f y H with the subgroup {1, 5o> £ } of T , where £ = exp (r~) • Define a map p: Z -> G by p(k) = £ ; then p i s a continuous homomorphism with image contained i n H . I t i s easy to see that p| n i s i n f e c t i v e . We define functions on G as follows: set f(x) = f f ' ( j ) i f x = p(j) where -"jeS n - C i f x £ p(S n) ; define s i m i l a r l y g i n terms of g' . As f , g are f i n i t e l y supported, they belong to S(G); since pi S i s m i e c t i v e we have 1 SQ n I I f ' I I = I IfI I and I I g ' I I I I l i p I I I I p I 1 6 I | p Thus D||g||p < | | f | | p . Since f , g are supported by H , we can i d e n t i f y f and g with functions on G/H = H = H regarded as a subgroup of T . As i n prevxous cases, we are r e a l l y i d e n t i f y i n g f with f \U , since i f y £ H 40. and y corresponds to £ (0<k<r-l) , then f ( Y ) = I f(x) Y ( x ) = I f'(j)exp(-2Tri^) xep(S ) . . I r A similar r e l a t i o n holds for g and ^ , so we have If I • l^|'H| < ^ f H - i This completes the proof. We note now that by combining (2.3.1), (2.3.2), (2.4.5), (2.5.2) and (2.5.3), we obtain the main theorem (2.1.5). (2.6) Miscellany As mentioned i n (2.1), other dense subspaces of L p ( G ) (p f i n i t e ) have equal claims to consideration i n the d e f i n i t i o n of the UMP. Obvious candidates are 1^(G) O ^ ( G ) , L±(G) f\ ^(G) , A £(G) , and [A c(G)j = {feL^(G)/f has compact support-}- . The l a t t e r space w i l l be denoted by S c(G) . Our f i r s t aim i s to show that neither the positive results (2.1.4) nor the negative results (2.1.5) are affected by using any of the above four test spaces instead of S(G) i n the d e f i n i t i o n of the UMP. Before doing t h i s , we should note that L^ /*^ L m cannot give us anything new. For i f f , g e I ^ f l with |f{ < g , we have g > 0 and so g z I^CG), 41. by [13, (31.42)]. Hence also f belongs to L^(G) . The Inversion theorem now says that f and g are equal a.e. to functions i n S(G). (2.6.1) Lemma. Let 1 < p < °° and suppose that L p(G) has the UMP as defined i n (2.1.3). Then L (G) also has the UMP when either P S (G) , A (G) , L 1 H L or L.O L i s used instead of S(G) i n the c c 1 <» 1 p d e f i n i t i o n of the UMP. Proof. This i s obvious for S (G) and A (G) , since each i s contained c c in S(G) . Suppose that L p(G) has the UMP as defined i n (2.1.3) and l e t f , g E ^ A l p • Let (u*a) be an approximate ide n t i t y for L^(G) as i n (1.5.4). Then u * f , u * g e S(G) for every a and we have a a I u f [ = u I f I < u g ; 1 a 1 a 1 1 = a thus I |u * f I I < D||U *g|I 1 1 a M p = 1 1 a 0 1 'p D being the constant of the d e f i n i t i o n . Taking l i m i t s , we obtain H f | l p 5 D | | g | | p . When p < 0 0 a similar argument shows that we can also replace SCG) by L x n • 42. (2.6.2) Lemma: Suppose that 1 < p < 0 0 and that L p(G) f a i l s to have the UMP as defined i n (2.1.3). Then i t also f a i l s to have the UMP when S(G) i s replaced by either L^ , L - ] A L p , A c(G) , or S £(G) i n the d e f i n i t i o n . Proof. Since L, f\ L and L^f\ L contain S(G) , the statement for 1' oo 1 p these two cases follows immediately from (2.1.5). In the other two cases i t i s enough to show that our examples can always be assumed either to be compactly supported or to have compactly supported transforms. We f i r s t deal with S (G) . Let (ii' ) be an approximate iden t i t y for L^(G) as i n (1.5.4). Each u^ i s compactly supported. Now suppose that D i s an arbitrary positive constant and l e t f , g e S(G) s a t i s f y | f | < g and D||g|| < | | f | | . We obviously have |u'f| < u g *"*' P p 0(r Ct for every a and there i s an index B for which D||u^*g|| < ||u^*f|| , by (1.5.4)(e). Replacing f and g by uB * f and uB * g proves the B 3 lemma for S (G) . c We now consider the case of ^ ( G ) . As i s to be expected, we simply dualize the argument just given for ^ c(G) . Let (v^) be an approximate i d e n t i t y for L^(G) , with the properties l i s t e d i n (1.5.4). Now v_^ 1 uniformly on compact sets so that | |v J i - h| | •> 0 (l<p<°°) for every h e L p(G) . In pa r t i c u l a r we have | |v^h| | \ |h| | . Now l e t D be an a r b i t r a r y positive constant and l e t f , g E S(G) s a t i s f y |f | < g and D||g|| p < ||f||p • Then there i s an index j for which D l l v - g | I < IIv.fII ] "P J P 43. We also have l e v a r i = | V j * f | < v. * i n < V j * i = (Vjg)' Thus i f we replace f and g by v^.f and y.g , respectively, we have examples a r i s i n g from A £(G) . This completes the proof. 00 (2.6.3) Remark. When G = R , we can use G C ( R ) » t n e space of compactly supported i n f i n i t e l y d i f f e r e n t i a b l e functions on R , as a test space i n our d e f i n i t i o n of the UMP. F i r s t , note that L^(R) has a bounded oo approximate identity of nonnegative functions belonging to C^QR) with 00 nonnegative transforms. For l e t <J> e ^ ( R ) be nonnegative and s a t i s f y CO —]_ 2£ f <f>(x)dx = 1 . I f , for n > 0 , we set ^ ( x ) ~ n ^ ^ ' t n e n i s an approximate identity for L^(R) (see [18, p. 10]). Let ip = (f^* Oj)^ ) , where h (x) = h(-x) . Then i - s a n approximate i d e n t i t y for L^(R) consisting of nonnegative functions i n C c(R) and 0 < < 1 f° r every n . The proof now follows those of (2.6.1) and (2.6.2). mrr>. proof now follows those of (2.6.1) and (2.6.2). I t i s possible to give a d e f i n i t i o n of majorant which i s v a l i d for every pair of functions belonging to L p ( G ) • T o do t h i s we require the notion of d i s t r i b u t i o n as described i n (1.5). * (2.6.4) D e f i n i t i o n . Let G be any LCAG and l e t L,M e S (G) . By J_ | L | < M we mean |<L,u>| < <M,u> for every u e S +(G) , the set of nonnegative real-valued members of S(G) . 44. If L v denotes the inverse transform of an L e S*(G) , we say that M v majorizes L v when | L | < M Note from the d e f i n i t i o n that M must be a positive linear functional on S(G) , that i s , <M,u> > 0 for every u e S +(G) . This leads to an extension of a result from the Schwartz theory which was pointed out to the author by M. Cowling. Although t h i s result must be well known, we could find no reference, so a proof i s included here. A (2.6.5) Lemma. Let M e S (G) be positive. Then M i s a measure (though not necessarily a bounded measure when G i s noncompact). Proof. By the density of A £(G) i n C Q(G) (see [13, (33.13)]), i t suffices to prove that the r e s t r i c t i o n of M to A(G) frj C (G;K) i s continuous on C(G;K) , where C(G;K) denotes the space of compactly supported members of G^(G) whose support i s contained i n the compact set K . We have to fi n d a positive constant B , which may be K-dependent, such that |<M,u>| < B | | u. | | ^ for every u e A(G)f\C(G;K) . Let u E A(G )Ac(G;K) and l e t v E A (G) be such that c v(K) = {1} and v ( G ) C [0,1] . Such functions are guaranteed by [13, (31.37)]. If u i s real-valued we have - W M L ± u „ < " v l l u l loo > and fey. the p o s i t i v i t y of M we obtain |<M,u>| < <M,v> ||u|1^ . 45. It i s easy.to show now that, i f u i s complex-valued and belongs to C(G;K) , then |<M,u>| < <M,v> J |u[|^ (use the method of [12, proof of (11.5)]). Hence M i s a measure. If G i s not compact, then Haar measure i s an unbounded measure which defines a positive member of S*(G) . In the compact case i t was quite clear how to produce a majorant for a trigonometric polynomial. For examples of majorants i n the non-compact case, see [8, pp. 272, 273]. We now give an example to show that elements of S*(G) need not have any majorant at a l l . (2.6.6) Example. We w i l l show how to regard the Hilbert transform as a member of S*(R) and then show that i t ' s inverse transform has no majorant i n S*(R). A l l that we use here concerning the Hilbert transform may be found i n [19, chapter 6]. In what follows, S(R) denotes L 1 ( R ) O A ( R ) CO and i s not to be confused with the Schwartz class of C functions which, along with a l l t h e i r derivatives, are of rapid decrease. For u e S(R) we define a linear functional H on S(R) by <H,u> = lim e+0 x X We now show that for u e S(R) t h i s expression i s sensible. Let u denote the usual Hilbert transform. By the L 2 theory of t h i s transform, we have C u V (5) = - i sgn(£) u ( 0 , 46. since u e L 2 (R) . But u e L^R) f\L (R) and so we also have (u) e L ^ ( R ) O l 2 ^ R ^ • B y t n e inversion formula, we may regard u as a continuous function on R for which u(x) = - i sgn(S) u(?) e 1 5 x dC and i n p a r t i c u l a r , u(0) = [- i s g n C O J u ( ? ) d? . (1) R But u(0) i s just <H,u> . From (1) we obtain |<H,u>| < ||u|\ 1 < ||u s , and so H e S (R) . Furthermore, H i s the Fourier transform of the L function H v(?) = - i sgn(£) . We now show that H has no majorant i n S*(R) . By (2.6.5) i t i s enough to show that there i s no positive measure u sa t i s f y i n g <H,u>| < <u,u> for every u e S (R) . 47. Let u (x) = n nx 1 -nx+n+1 0 < x < -- - n — < x < 1 n -^0 1 < x < 1 + -- - n otherwise . Then each u i s a trapezoidal function and i t i s well known that n u n e S +(R) . Since I l u nl !„, = ^ f° r e v e r v n and the support of every u i s contained i n [0,2] i t follows that {u } i s a bounded n n subset of C(R; [0,2]) . If p ±s anytpositiive meisureiwe then have sup <y,u > < y([0,2]) < °° . n = n However, simple computations show that <H,un> = log n + (n+l) l o g ( l + |) , and t h i s sequence i s unbounded. Thus H has no majorant i n S*(R) . We consider next a d e f i n i t i o n of upper majorant property based on (2.6.4). (2.6.7) D e f i n i t i o n . We say that L p ( G ) has the UMPD i f there i s a positive constant D such that whenever f, g e Lp(G) and |f| < g i n the d i s t r i b u t i o n a l sense of (2.6.4), then l l f l l p S D l | g | l p • It i s immediate that i f L (G) has the UMPD, then L (G) P P has the UMP, since for f f g e S(G) , |f| < g pointwiseeimplies |f| < g 48. d i s t r i b u t i o n a l l y . Hence L p(G) can have the UMPU only..when..p i s an even integer or 0 0 . We now prove the converse i s also true. (2.6.8) Lemma. If f, g e L^(G) s a t i s f y |<f,u>| < <g,u> for every u e S +(G) , then |f| < g pointwise. If f, g e S(G) s a t i s f y |<f,u>| < <g,u> for every u e S +(G) with u l y i n g i n some sphere i n Lq( G) » then [f| < g pointwise. Proof. We prove the f i r s t statement only, since the proof of the second i s es s e n t i a l l y the same. It i s easy to see that g > 0 pointwise. Suppose that the conclusion i s false and l e t y^ e G be such that |f(YQ)I > S^YQ) • We f i r s t consider the case where f i s real-valued. ~ ^ ~ ^ Then either *(YQ) > B(YQ) o r f(YQ) < _ §(YQ) • Suppose, for definiteness, that the f i r s t case occurs. By continuity there i s an open neighbourhood V of YQ such that g(Y) < f ( Y ) f° r every y e V and by [13, (31.34)] there exists w e S (G) such that 0 < w < 1 and the support of w i s c = = /\ z\ contained i n V . In p a r t i c u l a r , w e S +(G) . Then <g,w> = g(Y)w(Y)dy < f(Y)w(y)dy = <f,w> , V and <g,w> < |<f,w>| , which i s a contradiction. Suppose that f i s complex-valued and j f CYQ) I > S^YQ) ' ^ • * T E R m u l t i p l i c a t i o n by a complex number of absolute value one we can assume that Re f (YQ) > &(YQ) ' Now Re f i s the transform of an L^-function, and I<Re f,u>| < <g,u> for a l l u i n S +(G) . By the previous analysis 49. Re f(YQ) < g ( Y Q ) • This contradiction completes the proof of the lemma. (2.6.9) Proposition. Let l<p<«> . If L (G) has the UMP (as defined _ p in (2.1.3)) then L (G) has the UMPD with at most the same constant. P Proof. Let f, g e L p ( G ) s a t i s f y | f | < g d i s t r i b u t i o n a l l y and l e t D be the constant of the d e f i n i t i o n of the UMP. Let (u ) , (v ) be n m sequences i n S(G) such that u -»• f and v -> g i n L (G) . Then n m ° p we also have |<un,u>| -> |<f,u>| and <vm,u> -> <g,u> for every u e S(G) . We can assume that v i s real-valued, and thus that <v ,u> i s r e a l -m m valued for every u e S,(G) . For i f the v are not real-valued, + m 1 ~ * ^ replace v by w = TT(V + v ) . As <g,u> > 0 when u E S,(G) i t m m z m m = + i s easy to show that <(g)",u> = <g,u> whenever u belongs to S +(G) . Hence <wm,u> -*• <g,u> for every u e S +(G) . Furthermore, I I w m I Ip **" I |^(g +g) Mp 5. I |g| Ip • We now assume that the are r e a l -valued . Let 6 > 0 and, for r > 0 , l e t S r = (ueS +(G) | ||u|| < r} Then there i s a positive integer m^ such that for every m > m^ , we have |<f,u>| < -r^T <v ,u> for a l l u e S and I lv I I < i I g(| .1 + 6 . 1 1 = 1-6 m r 1 1 m1'p 1 1 0 1 ' p Si m i l a r l y , there i s a positive integer n^ such that for every n > ng , feu ,u>| < X ' T H F ) < V ,U> for a l l u e S and | | f | | - 6 < I lu I I . 1 no 1 = 1-6 m ' r 1 1 1 'p 1 1 n1'p i ~ i 1+6 In p a r t i c u l a r , we have <u ,u> < (-—r) <v ,u> for every u e S . By ' n = mQ r Lsnsa (2.6.8), | < C Lemma (-27678) ^ |u n | < Oj^r) v m pointwise; since Lp.(G) has the UMP, we also have | |u^ | | p < (g|) D| | V | | p . Hence | |f | | - 6 < (g|) n i l . . II 11.11 . . /1+6S P (inequality continues over) 50. D U | g | L ' + 5 ] and, as 6 > 0 i s a r b i t r a r y , | | f | | <D||g|| . This P p p completes the proof. (2.7) The Lower Majorant Property In t h i s section we discuss analogues for noncompact LCAG's of the LMP for the compact case. (2.7.1) Proposition. Let 1 < p < °° and consider the following state-ments: (1) There i s a positive constant C^ such that i f f e S(G) , there exists g e S(G) such that |f| < g pointwise and l | g | | p 5 C i l | f | l p (2) There i s a positive constant such that i f f e S(G) , there exists g e L (G) such that |f| < g d i s t r i b u t i o n a l l y and (3) There i s a positive constant C„ such that i f f e L (G) , 3 P there exists g§e'L'I(G.) sa t i s f y i n g |f| < g d i s t r i b u t i o n a l l y and llsllp£ c3ll fllp P- P Then (1) implies (2), and i f 1 < p < » , (2) implies (3) . Proof. (1) => (2) i s obvious, since SCG)ClL CG) and since | f | _ g pointwise implies | f | < g d i s t r i b u t i o n a l l y . The constant i s at most C- . (2) => C3). Let f e L CG) and l e t (u ) be a 1 p n sequence i n S(G) such, that u^ -»• f i n LpCG) . By hypothesis, for each n , there exists g e L (G) such, that |u I 5" g d i s t r i b u t i o n a l l y ' e n • p 1 n 1 = n * and IIg M < C _ l l u || . Thus {g } i s a norm-bounded subset of the 1 1 °n1 1 p = 2 11 n 1 ' p n ref l e x i v e Banach space LpCG) and so there i s a weak * convergent s 51. subsequence, s t i l l denoted by (g^) , with weak l i m i t g , say, i n L (G) . Then P and by weak convergence (and (1.5.3)), i f u e S +(G) , ]<f,u>| = lim |<u ,u>| < lim ^ g ,u> = <g,u> . n n Hence (3) follows, with being at most C^ • Note that each of statements (1), (2), and (3) i n t h i s proposition makes sense for 1 < p < 0 0 . (2.7.2) D e f i n i t i o n . Let 1 < p < °° . We s h a l l say that L p(G) has the LMP(j) (j=l,2,3) i f statement (j) of (2.71.1) holds for L p(G) . The content of (2.7.1) i s that i f 1 < p < °° and L p(G) has LMP(l), then i t has LMP(2); s i m i l a r l y , i f L p(G) has LMP(2), then i t has LMP(3) when 1 < p < ~ . Note that i n statements (1) and (2) of (2.7.1), the condition "f £ S(G)" could equally be replaced by "f e S (G)" or "f e A (G)" c c and the conclusions would s t i l l hold. This leads to a new d e f i n i t i o n of LMPtj) when j = 1,2. (2.7.3) Lemma. (a) . L (G) has LMP(3), with constant 1. (b) L 2(G) has LMP(2), with constant 1. 52. Proof. (a) By Lemma (2.6.7), i f f, g e L^G) , then | f | < g d i s t r i b u t i o n a l l y i f and only i f |f| < g pointwise. We proceed exactly as i n Hardy and Littlewood [11, p. 305]. For f e L^(G) , write f = f .j f „ , where f. e L 0 (G) and l l f . l L = I I f I I •, , for i = 1,2 . 1 2 l 2 1 1 i 1 12 1 1 "1 Let g± e L 2(G) be such that g ^ = | r ^ | , i = l , 2 . If g = g ^ , then g e L^(G) and |f| < g pointwise, since |f| = IV^I - 1 !l' * ^ = % * 8 2 = & ([13, (31.29)]). Furthermore, \1<- w , 2 , 2 ' 2 ,2 • i This proves (a) . (b) If f e S(G) , l e t g e L 2(G) be such that g = |f| a.e. Then |f| < g d i s t r i b u t i o n a l l y and | | g j | ^ = | | dff | 1 ^ ' H e n c e 00 follows. (2.7.4) Remark. There are other senses, not covered by (2.7.2), i n which L^(G) and L 2(G) have a LMP. Two examples are: ( i ) L^(G) has the LMP i n the sense that there i s a positive constant C^ such that i f f e ^(G) , there exists g e S^(G) for which |f| < g pointwise and ||g|11 < | |f | | . 53. ( i i ) (G) has the LMP i n the sense that there i s a positive constant C,_ such that i f f e L^iG) , there exists g e sat i s f y i n g |f| < g a.e. and ||g|| 2 < C,. | |f | | 2 . Each i s easy to prove; statement ( i i ) i s essentially (2.7.3)(b) and ( i ) can be derived from (2.7.1)(a) by using the method of (2.1.2)(b). We come now to the duality theorem. In the compact case, one proves that a space L p ( G ) n a s t n e LMP by using the KHardy-Littlewood 2k duality theorem (see (1.1)). For p = 2]n-i » a direct proof that L p(G) has the LMP i s not known, even for the c i r c l e group. Unfortunately, we have found neither a generalization of the Hardy-Littlewood duality theorem nor a direct proof that L p(G) has LMP(j) , for some j , when 2kk p = 2k-± (KeN) . However, i t i s an easy matter to generalize the Boas duality theorem (see (1.1)), and from t h i s we can at least conclude that L p(G) can only have LMP(j) for some j (j = 1,2, or 3) i f p i s 1, 2k 2, or of the form 2k~l k e N . (2.7.5) Lemma. Suppose that L^ M e S*(G) and |L| < M . If <J> , ip e S(G) sa t i s f y |<J>| < ip pointwise, then |<L,<j>>| < 2<M,ip> . Proof. F i r s t suppose that L i s re a l - l i n e a r and that c|> i s real-valued. Then we have - ip < <j> < ip , so that 0 < <J> + i/> and 0 < ^ — cp . Since ]L| < M , we have | <L,<j>-hJj> | < <M,<$>+ip> and |<L,ip-<j>>| < <M,ip7<()> . This i s equivalent to - <M,ip> - <M,<}>> < - <L,<()> - <L,\p> < <M,<j)> + <M,\p> and - <M,ip> + <M,<f>> < - <L,(j>> + <L,ip> < <M,ip> -f <M,<|)> . 54. Adding, we obtain -2<M,)JJ> < -2 <L,<f>> < 2<M,ip> , hence |<L,<j>>| < <M,ip> . Suppose now that L i s general and cf) i s real-valued. Then <L,<j>> = <L^<J<J>> - i<L^, icj)> where i s a r e a l - l i n e a r functional on S(G) . If a i s a complex number of absolute value one such that a<L,<j>> = |<L,c|>>| , then |<L,4>>| = <L,acf>> = <L15a(f)> But i t i s easy to check that < M and |a<f>| < ^ , so that |<L1, a<j>>| < <M,ip> . Thus |<L,<f>>| < <M,IJJ> i n t h i s case. For general L and complex-valued <f> , we have, writing 4> = <j)^ + i(f>2 with <J>^ and cf>2 real-valued, \<L,q»\ = |<L,<J>1+i<j)2>| < |<L,(j)1>| + |<L,<j>2>| < 2<M,ip> . (2.7.6) Theorem. Let 1 < p < <=° . (a) Suppose that L (G) has _ _ p LMP(j) (j = 2 or 3) with constant C . Then L (G) has the UMP with q constant at most 2C . Cb) If L (G) has LMP(1) with constant C , then L (G) has P q 55. the UMP with constant at most C Proof. See Boas [O, p. 256]. 4a) The case j = 2 i s similar to the case j = 3 , so we prove only the l a t t e r . Let f, F e S(G) s a t i s f y | f | < F pointwise. I f g e L p ( G ) > there exists h e L p(G) sa t i s f y i n g ||| < h d i s t r i b u t i o n a l l y and ||h|| p<c||g||. From (1.5.3) we have f(x) g(-x)dx| = |<g,f> < 2 <h,F> (by (2.7.5)) = 2 h(x) F(-x)dx < 2||h||p ||F||q < (2C||F||q) ||g|| p Thus the map g - f(x) g(-x)dx defines a continuous linear functional on L (G) , with norm at most 2C||F|I . However, th i s p 1 1 1 1 q functional i s defined by the L (G) - function f and so has norm q | | f | | q . Consequently we have | | f | | q < 2 C | | F | | , and t h i s proves (a), (b) F i r s t suppose 1 < p < 0 0 . We proceed as i n (a) , using the Parseval formula instead of (2.7.5), and using the density of S(G) in Lp( G) . When p = 0 0 , we again proceed as i n (a) to obtain |J f(x) g(-x)dx| < I | F | | x I|g| G (1) 56. It i s an easy exercise to show that S(G) i s weak-*-dense i n L (G) , and consequently M f l ^ - sup{| J f(x)g(x)dxf |geS(G) and M g M ^ 1 } G Combined with (1), t h i s shows that ||f||-j_ < C | | F j j . This completes the proof. AC (2.7.7) Corollary. Suppose p i s not 1, 2, or of the form 2k~T ^ o r some k e N . Then L p(G) does not have LMP(j) for j =1, 2, or 3 . Proof. If p i s not of the given form,-then q , the index conjugate to p , i s not an even integer or 0 0 and so L (G) does not have the q UMP (see (2.1.5)). The result follows from (2.7.6). 5 7 . CHAPTER 3 MAJORANTS AND FUNCTIONS WITH NONNEGATIVE TRANSFORMS This chapter i s concerned with obtaining generalizations of the results outlined i n (1.1.3) to a l l i n f i n i t e compact abelian groups. (3.1) Even integers again Throughout t h i s section, G i s a compact abelian group. We establish analogues of the positive r e s u l t s for the c i r c l e group described i n (1.1.3). (3.1.1) Theorem. Let A be a symmetric neighbourhood of the iden t i t y i n G . Suppose that f e L^(G) and f > 0 . If f e L 2(A) , then f e L 2(G) and there i s a positive constant C , dependent on A but independent of f , for which i f | lL 2CG) S C H f H L 2 ( A ) Proof. Let m be normalized Haar measure on G . Let K be a compact symmetric neighbourhood of the i d e n t i t y i n G with K + K C A . Set h = rc^jQ 1 R and define g = h * h , where h(x) = h(-x) . Then we have g = |h| 2 > 0 and £(0) = 1 . Moreover, |g| < 1 A , and so gf e L0'(G) , since l A f e L„(G) A 2 = ffi(K) A ' B 2 If 3 E G , we have gfCB) = g * f ( 6 ) = ^ £( Y) f ( 3 - y ) > g(0) f(0) = f ( 3 ) YEG Applying the Plancherel theorem, we obtain 58. H f i i L 2 ( G ) - H ' l i i ^ g ) s I I ^ H L 2 C G ) - I I ^ £ I I L 2( G ) S=W H £ H L 2(A ) . This proves the theorem, with C = ,\ T S . m(K) It i s now an easy matter to derive the corresponding r e s u l t for L p(G) when p i s an even integer or °» . (3.1.2) C o r o l l a r y . Let A be a symmetric neighbourhood of the i d e n t i t y , and suppose f e L^(G) with f > 0 . If p i s an even integer or 0 0 and i f f e L p(A) , then f e Lp(G) . Moreover, there i s a p o s i t i v e constrant C , dependent on A and p but not on f , for which l f M L ( G ) < C||f|| L CA) • P P Proof. F i r s t suppose that p = 2£ , where £ i s a p o s i t i v e integer, £ ^ - -We have f e L 2(A) and f > 0 , since f" = f * f * * f > 0 £ £ (there are £ r ' s h e r e ) . By (3.1.1), f e L.(G) and ||f |L ^2 ' 1 £ < W T r S I If I L ... , where K i s as i n the proof of (3.1.1). Thus = m(K) 1 1 1'L (A) f belongs to L p ( G ) and we have i f i i L (G, • t i i ^ n l 9 ( G ) i w s W W ] " 1 " M f ' l l i V -p L I = i m w r ^ i i f i ^ ( A ) a) -l/£ p Hence, C = ft(K) • 59. Suppose now that p = °° and f e L^CA) . Then f e L^CA) for every positive integer L . Since lim ||h|| =1^11^ i f h e L for some r , p-*» ^ r i t follows from (1) that H ' N L . C G ) i H £ I I L J « • CO CO In p a r t i c u l a r , f E an& C = 1 i n t h i s case. (3.1.3) Remark. If f E L^CG) and f > 0 , then f e Z^(G) and thus i s equal a.e. to a function i n P(G) , the space of continuous positive d e f i n i t e functions of G (see [13, (34.12)] or use the UMP as i n [3, p. 256]). The norm inequality of (3.1.2) t e l l s us nothing new, for i f we assume f e P(G) , we have f(0) = | | f | | L ( G ) < | | f | | L ( A ) < | | f | | L ( G ) = f(0) . However, when p = 0 0 , (3.1.2) i s an alternative proof of a c l a s s i c a l result (('see [6, p. 144]). (3.2) Failure of "good" behaviour when p i s not an even integer In t h i s section we present extensions of the results of Wainger 120] and Shapiro [18]. Throughout, G w i l l be an i n f i n i t e compact abelian group. One chara c t e r i s t i c of the positive results i s that we 60. obtain an inequality of the form N f l l L (G) = c H f N L (A) • p P We refer to t h i s inequality as (W) . We s h a l l show that i f p > 1 i s f i n i t e and not an even integer, then no inequality of the form (W) can hold. Of course, we assume that f > 0 i n (W) . (3.2.1) Lemma. Let B a nonempty symmetric open subset of G . Then we can find u e A(G) with the following properties: (1) u and u are real-valued; (2) u i s supported i n B ; (3) u i s bounded away from zero on some symmetric open set contained i n B . Before proceeding with the proof, we need some more notation. For f e L 1(G) , we l e t f(x) = f(-x) ; for y E G , we l e t (^f)(x) = f(y4x). Note that ( x f ) ~ = -x(f) . x -xx Proof of (3.2.1). Let x^ e B and suppose that V i s a symmetric neighbourhood of x^ contained i n B . There i s a compact symmetric neighbourhood K of the i d e n t i t y such that K + K C - X Q + V . Set w = 1^ * 1„ . Then w i s supported by K + K , w e A(G) and w(0) = m(K) >' 0 . Thus there i s a 6 > 0 and a symmetric neighbour-hood Y of the i d e n t i t y such that Y C K + K and w(x) > 6 for every x i n Y . 61. Let u^ = w ; then supp(u^) = X Q + supp(w) CT V C B . xO , Since may not be real-valued, consider u = " j ^ i + ui) • Then both u and u are real-valued and u e A(G) . Furthermore, supp(u) C supp(u 1) U supp(u 1) = suppC^) U k- supp(u1>} CZ B . F i n a l l y , we note that u i s bounded away from zero on x^ + Y , since i f y e Y , uCxQ+y) = |(wC-x0+j£0+y) + w C x ^ - y ) ) , as (-xQW) ~ = x Q(w) = |(w(y) + w(-y)) > 6 This completes the proof. Lemma (3.2.1) and the f a i l u r e of the UMP when p i s not an even integer provide the key to the next two theorems. Theorem (3.2.3) i s a direct generalization of Shapiro's result [18, p. 16], and the method i s the same. Theorem (3.2.3) i s a s l i g h t variant, the difference being that i n (3.2.3) we have p > 2 and we can conclude that our example belongs to L^tG) and hence also belongs to L^(G) . We can not do t h i s when 1 < p < 2 . 62. (3.2.2) Theorem Suppose p > 1 i s f i n i t e and not an even integer. Let A be a closed symmetric neighbourhood of the i d e n t i t y which i s not of f u l l measure. Then no inequality of the form (W) can hold. Proof. We know that the UMP f a i l s i n L p(G) for these p ; see (1.1.2)(b) Thus, for each positive integer n we can find trigonometric polynomials f , F with the following properties: F majorizes f , I I F I|P < 1 n n n n 1 1 n1'p = and ||f ||P > 2 n . 1 1 n M p = Apply Lemma (3.2.1) to the set B = A' , the complement of A , to obtain u belonging to A(G) , supported by A' , and bounded below by 6 > 0 on a nonempty open set I contained i n A' . Let u* = E ^ | U ( Y ) | Y • yeG Then u* belongs to A(G) and u* majorizes u . We f i r s t show that (1) f can be assumed to have r e a l c o e f f i c i e n t s n and s t i l l be majorized by F , and (2) after a possible replacement of n f as i n (1), we can arrange that the sequence {||f I | P i s unbounded, n n L ( .1) P Let t denote the least number of translates of I required to & cover G , and suppose that G = V"^ (y_. + I) . Then 2° < where z i s that - y (Kj<£) for which | U f | | P , T , = m a X J I f I | P / T N , J 1 'z n M L (I) l<j<x' '-y. n 1 'L (I) P - ~ J p Thus C2) w i l l s t i l l hold when we replace by ; note that z may 63. vary with n . We now show how to make the necessary adjustments for (1) and s t i l l preserve (2). I f h l " K fn + a n d h2 = ^ Z f n ' <zfn>~> > t h e n b o t h h^ and h have real-valued Fourier transforms and at l e a s t one of h^ n—1 and has the pth power of i t s L p(I)-norm > 2 It . We thus replace zf by whichever of h. and h„ has I Ih. l| P > 2 It . Hence we z n J 1 2 1 1 j 1 ' L (I) = P can assume (1) and (2) hold. Denote the replacement by f once again. We now show that (W) cannot hold. Let g = F u* + f u ; °n n n then g n > 0 since F^ majorizes f , u* majorizes u , and the c o e f f i c i e n t s of u and f are real-valued. Now sup I |g I | P , . s < 0 0 , n 1 1 n 1 1 L_ (A) P since g (x)| Pdx = "|F ( x ) | P |u*(x)| Pdx (u i s supported by A') A n < u* P F I l P = M u I l » I l ^ n l l L (G) P < l l u * | | P However, sup ||g || P , . = °° ; for i f t h i s sequence were bounded, the sequence {I If u l | P would also be bounded since f u = g - F u* . 1 1 n 1'L (G) n °n n P But f n U H L (G) * [ l f n ( x ) ! P ! u ( X ) | P dx > 6 P 2n"1/£ P I 64. by (2). This completes the proof. We can be much more d i r e c t when p > 2 . (3.2.3) Theorem. Let p > 2 not be an even integer or 0 0 . Suppose that A i s a closed symmetric neighbourhood of the i d e n t i t y which i s not of f u l l measure. Then there e x i s t s g belonging to L^(G) with the following properties: Ca) g > 0 ; Cb) g E L p(A) ; Cc) g i L p(G) . Proof. By 17, p. 165], there i s a complex-valued function c on G with |c| E l L p ( G ) ] " " and c ^ l L p ( G ) ] " . Let F e L p(G) be such that F = |c| ; since p > 2 , we can f i n d f^ belonging to L^(G) such that = c . Note that f^ does not belong to L p ( G ) • Let u be obtained from Lemma (3.2.1), using B = A' . Let I be a symmetric neighbourhood contained i n A' on which i u i s bounded below by a p o s i t i v e number, <5 say. Since f^ £ ^p^) a n d since only a f i n i t e number of translates of I are required to cover G , there must be a y e G for which v f ^ £ •kpCl) • Since I i s symmetric, (y^ i ^ does not belong to L p ( I ) , and so not both of = + ( y ^ ) ' ) a n d h 2 = ~ | ^ y f i " ^ ' y f l ^ ^ c a n belong to L p ( I ) . As both h^ and have r e a l c o e f f i c i e n t s , we replace vf^ with whichever of h^ and does not belong to L p ( I ) . C a l l t h i s replacement f ; then f i s majorized by F . 65. Let u* = £ J U - ( Y ) | Y , and l e t g = Fu* + fu . Then g belongs to L (G) , and g > 0 . Furthermore, g e L (A) since u i s supported z. p by A' , and so 'L p(A) |F(x)|P |u*(x)| P dx < ||u*|| P ||F||P . P However, g does not belong to L (G): for otherwise we would have P fu = g - Fu* e L p(G) , and yet | f ( x ) | P |u(x)| Pdx > 6 P j |f(x)| Pdx I This completes the proof. For the case 1 < p < 2 , we can give a modified version of (3.2.3) .- This result should be regarded as an extension of Wainger's result [20]. (3.2.4) Proposition. Suppose 1 < p < 2 and l e t A be a closed symmetric neighbourhood of the identity which i s not of f u l l measure. Then there i s a trigonometric series with nonnegative co e f f i c i e n t s which converges to a function i n L p ( A ) , but whose coe f f i c i e n t s are not the Fourier c o e f f i c i e n t s of any function belonging to L p(G) . Proof. Since the UMP f a i l s i n L p(G) , for each positive integer n there exist trigonometric polynomials F^ , f^ with F^ majorizing f^ , 66. I|Fj | p < 2 n , and | | f j | > 2 n . As i n the proof of Theorem (3.2.3) we can assume that f has r e a l c o e f f i c i e n t s and that ||f |L > c 2 n n 1 1 n M L (I) = P for some positive constant c , where I i s as i n the proof of Theorem (3.2.3). Let u be obtained from Lemma (3.2.1) with B = A' , and l e t If g = F u* + f u , then g > 0 and g belongs to A(G) e n n n °n = 6 n 00 for every n . Let g = £ g Then g i s a trigonometric series n=l n with nonnegative c o e f f i c i e n t s . Note that g e L (A) and n p I ^ n I II (A) = 2 n I Iu* I loo ' Hence the series defining g converges i n P L (A) . Since the sequence {||g-|| . .}. i s unbounded, g cannot belong P n p ( , to L (G) . P (3.2.5) Remark. We should note that the series defining g can be assumed to converge a.e. (with respect to Haar measure) by using a subsequence i f necessary. For th i s subsequence, g s t i l l cannot belong to L n(G) (for the same reason as for the o r i g i n a l g) . 67. BIBLIOGRAPHY [1] G.F. Bachelis, On the upper and lower majorant properties i n L (G), Quart. J. Math. Oxford (2) 24(1973), 119-128. p [2] C. Berg and G. Forst, Potential Theory on l o c a l l y compact abelian groups, Springer-Verlag, New York, 1975. [3] R.P. Boas, Majorant Problems for Fourier Series, J. Analyse Math. 10(1962-3), 253-271. [4] , Entire functions, Academic Press, New York, 1954. [5] P. C i v i n , Fourier Coefficients of dominant functions, Duke Math. J. 13(1946), 1-7. [6] R.E. Edwards, Fourier Series: a modern introduction, volume I, Holt, Reinhart and Winston, New York, 1967. [7] J. Fournier, Majorants and L norms, Israel J. Math. 18(1974), 157-166. P [8] , Local complements to the Hausdorff-Young theorem, Michigan Math. J. 20(1973), 263-276. 19J G. Gaudry, Topicss i n Harmonic Analysis, Lecture notes, Yale U n i v e r s i t y , 1969. [10] L. Gluck, Characterization of Fourier transforms of L (G)-functions, Ph.D. Thesis, I l l i n o i s I n s t i t u t e of TeEhnology, 1971. [11] G. Hardy and J. Littlewood, Notes on the theory of series (XIX): a problem concerning majorants of Fourier series, Quart. J. Math. Oxford (1) 6(1935), 304-315. [12] E. Hewitt and K. Ross, Abstract Harmonic Analysis, Volume I, Springer-Verlag, New York 1963. [13] , Abstract Harmonic Analysis, volume I I , Springer-Verlag, New York, 1970. [14] K. de Leeuw, On L m u l t i p l i e r s , Ann. of Math. (2) 81(1965), 364-379. P [15] H. Reiter, C l a s s i c a l Harmonic Analysis and l o c a l l y compact groups, Oxford Mathematical Monographs, Oxford University Press^ Oxford, 1968. [16] W. Rudin, Positive d e f i n i t e sequences and absolutely monotonic -functions, Duke Math. J. 26(1959), 617-622. 68. [17] I. J. Schoenberg, Positive d e f i n i t e functions on spheres, Duke Math. J. 9(1942), 96-108. [18] H. Shapiro, Majorant problems for Fourier c o e f f i c i e n t s , Quart. J. Math. Oxford (2) 26(1975), 9-18. {19] E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, 1971. [20] S. Wainger, A problem of Wiener and the f a i l u r e of a p r i n c i p l e for Fourier series with positive c o e f f i c i e n t s , Proc. A.M.S. 20 (1969), 16-18. [21] A. Zygmund, Trigonometric series, Volume I, Cambridge University Press, Cambridge, 1959.
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Majorant problems in harmonic analysis Rains, Michael Anthony 1976
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Title | Majorant problems in harmonic analysis |
Creator |
Rains, Michael Anthony |
Date Issued | 1976 |
Description | In various questions of Harmonic analysis we encounter the problem of deriving a norm inequality between a pair of functions when we know a (point wise) inequality between the transforms of these functions. Such problems are known as majorant problems. In this thesis we consider two related problems. First, in Chapter two, we extend the known results on the upper majorant property on compact abelian groups to noncompact locally compact abelian groups. We show, using various test spaces and two notions of majorant, that a Lebesgue space has the upper majorant property exactly when its index is an even integer or infinity. Furthermore, if a Lebesgue space has the lower majorant property, then the Lebesgue space with conjugate index has the upper majorant property. In the final chapter we consider the second problem. Here-, we are concerned with deriving global integrability conditions from local integrability conditions for functions which have nonnegative transforms. Such a property holds only in Lebesgue spaces whose index is an even integer or infinity. For Lebesgue spaces whose index is not an even integer or infinity the proof of the failure of this property is based on the failure of the majorant property in these spaces. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0080137 |
URI | http://hdl.handle.net/2429/20159 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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