Convolution operators on groups

Lecture Notes of the Unione Matematica Italiana For further volumes: http://www.springer.com/series/7172 11 Editori...

Author: Derighetti A.

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Lecture Notes of the Unione Matematica Italiana

For further volumes: http://www.springer.com/series/7172

11

Editorial Board

Franco Brezzi (Editor in Chief) IMATI-CNR Via Ferrata 5a 27100 Pavia, Italy e-mail: [email protected] John M. Ball Mathematical Institute 24-29 St Giles’ Oxford OX1 3LB United Kingdom e-mail: [email protected] Alberto Bressan Department of Mathematics Penn State University University Park State College PA. 16802, USA e-mail: [email protected] Fabrizio Catanese Mathematisches Institut Universitatstraße 30 95447 Bayreuth, Germany e-mail: [email protected] Carlo Cercignani Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milano, Italy e-mail: [email protected] Corrado De Concini Dipartimento di Matematica Universitá di Roma “La Sapienza” Piazzale Aldo Moro 2 00185 Roma, Italy e-mail: [email protected]

Persi Diaconis Department of Statistics Stanford University 450 Serra Mall Stanford, CA 94305-4065, USA e-mail: [email protected], [email protected] Nicola Fusco Dipartimento di Matematica e Applicazioni Universitá di Napoli “Federico II”, via Cintia Complesso Universitario di Monte S. Angelo 80126 Napoli, Italy e-mail: [email protected] Carlos E. Kenig Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL 60637-1514 USA e-mail: [email protected] Fulvio Ricci Scuola Normale Superiore di Pisa Piazza dei Cavalieri 7 56126 Pisa, Italy e-mail: [email protected] Gerard Van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam, The Netherlands e-mail: [email protected] Cédric Villani Institut Henri Poincaré 11 rue Pierre et Marie Curie 75230 Paris Cedex 05 France e-mail: [email protected]

The Editorial Policy can be found at the back of the volume.

Antoine Derighetti

Convolution Operators on Groups

123

Antoine Derighetti Ecole polytechnique fédérale de Lausanne EPFL SB-DO MA A1 354 (Bâtiment MA) Station 8 Lausanne Switzerland [email protected]

ISSN 1862-9113 ISBN 978-3-642-20655-9 e-ISBN 978-3-642-20656-6 DOI 10.1007/978-3-642-20656-6 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011932164 Mathematics Subject Classification (2000): 22DXX; 43A10; 43A20; 43A22; 43A40; 43A45; 43A15 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Roughly speaking a convolution operator T on a group G is a linear operator on complex functions ' W G ! C that commutes with left translations g .Tf

/ D T . g f /:

Typically convolution by fixed functions gives rise to convolution operators. To be more precise, one has to specify G and the underlying function space for T . One may suppose that G is a locally compact group with Haar measure m; and choose T to be a continuous linear endomorphism of Lp .G/ D Lp .GI m/; where p > 1 is some fixed real number. It’s these convolution operators that will be the subject of this book, individual cases of them as well as, for given p and G; the space C Vp .G/ of all of them. The set C Vp .G/ is a sub Banach algebra of the Banach algebra of all continuous linear endomorphisms of Lp .G/. If G is abelian, it is possible to define the Fourier transform of every T in C V2 .G/. The Fourier transform is a Banach algebra b Here, G b denotes the Pontrjagin dual of G. isometry of C V2 .G/ onto L1 .G/. Moreover, C Vp .G/ C V2 .G/; this permits to define the Fourier transform of every T in C Vp .G/. The case of G D Rn involves results of classical Fourier analysis. For instance, the fact that the Heaviside function is the Fourier transform of some T 2 C Vp .R/ implies Marcel Riesz’s famous theorem on the convergence in Lp of Fourier series. This convergence still holds in two variables for square summation, but not for circular summation if p 6D 2. This reflects the fact that the indicator function of any square is the Fourier transform of some T 2 C Vp .R2 / but not the indicator function of the disk except if p D 2. In this book, we will be mainly concerned with the investigation of C Vp .G/ for noncommutative groups. 0 L 1 kkkp klkp0 . If k 2 Lp .G/ and l 2 Lp .G/, then k lL 2 C0 .G/ with kk lk Forming series of such functions leads to the very important Figà-Talamanca space Ap .G/ contained in C0 .G/. Ap .G/ is an algebra for the pointwise product. v

vi

Preface

If it is given a norm based on kkkp klkp0 , it becomes a Banach algebra. There is a natural duality between C Vp .G/ and Ap .G/ for a large class of locally compact groups. This duality holds for all locally compact groups if p D 2. It is conjectured that it holds even for all p. If G is abelian, then A2 .G/ turns out to be the space of b Here, again the Fourier transform is a Banach algebra Fourier transforms of L1 .G/. 1 b isometry of L .G/ onto A2 .G/. To every integrable function on G; and more generally to every bounded measure on G; there corresponds by convolution an operator in C Vp .G/. For finite groups all of C Vp .G/ is obtained in this manner. It is not the case for infinite groups like Z; R; T and probably for all infinite groups. Then we may ask whether every convolution operator may be approximated by operators associated to bounded measures, and in which topology. For p D 2 the answer is yes under the weak operator topology. This result was obtained by Murray and von Neumann for discrete groups, by Segal for unimodular groups and finally by Dixmier for general locally compact groups. The duality between C Vp .G/ and Ap .G/ permits to answer positively for p 6D 2 for all amenable groups. Let I be an ideal of the algebra Ap .G/. The set of points of G where all functions in I vanish will be called the cospectrum of I . An elegant formulation of the celebrated tauberian theorem of Wiener is: if G is an abelian group every ideal of A2 .G/ with empty cospectrum is necessarily dense in A2 .G/. In this book, we will show that this statement holds for every group and also every p > 1. The fact that the theorem of Wiener is verified on arbitrary groups is highly surprising: there are papers suggesting the impossibility of such an extension for the group of two by two invertible matrices of complex numbers! There is a hudge amount of literature concerning the non-commutative version of the Plancherel theorem and the inversion formula for C 1 functions with compact support on Lie groups. Such questions are, for commutative groups, very simple. An achievement of this book is the extension to non-commutative groups of theorems which are deep and difficult even for Z; T or R. An important part of this monograph deals with the relation between C Vp .H / and C Vp .G/, where H is a closed subgroup of G. Let i be the inclusion map of H into G. Then i induces a canonical map, also denoted i; of C Vp .H / into C Vp .G/. For G D R and H D Z; this is a famous result due to Karel de Leeuw (1965), and to Saeki (1970) for G abelian and H arbitrary closed subgroup. It is also possible to characterise the image of i in C Vp .G/ and to obtain in this way noncommutative analogs of a result of Reiter (1963) concerning the relations between b and L1 .H b / and also to the fact that H is a set of synthesis in G (1956). L1 .G/ The characterisation in C Vp .G/ of the image of C Vp .H / under the map i , is a deep result due to Lohoué (1980). A large part of Chap. 7 is devoted to a detailed proof of Lohoué’s result. As a consequence we obtain the extension of the Kaplansky– Helson theorem to non-abelian groups and to p 6D 2: for x in a arbitrary locally compact group G; every ideal of Ap .G/ having the cospectrum fxg is dense in the set of all functions vanishing in x.

Preface

vii

In the last chapter, we prove that for amenable groups C Vp .G/ is contained in C V2 .G/: this statement, compared to the commutative case, requires an entirely new approach. The development of harmonic analysis on non-commutative groups is not just a straightforward generalization of the commutative case. It requires new ideas but it also gives rise to new problems which are far from being solved. For instance, the approximation theorem for non-amenable groups and for p 6D 2 is still out of reach. The investigation of the noncommutative case gives a better understanding of the b commutative case! For example, instead of studying the relations between L1 .G/ 1 b and L .H /; it is more conceptual and more fruitfull to investigate the relations between the algebras C V2 .G/ and C V2 .H /. A large part of the results presented appeared here for the first time in a book’s form. The presentation is selfcontained and complete proofs are given. The prerequisities consists mostly with a familiarity with the books of Hewitt and Ross [66, 67]. (Chaps. 4, 6, 8 and 10), Reiter and Stegeman [105] and Rudin [107]. Notes at the end of the volume contain additional information about results of the text. We wish to acknowledge our indebtedness to Professor Henri Joris, who read the proofs and helped to remove some errors and obscurities. His comments have stimulated us to improve the text in several places. Those errors which do appear in the text are, of course, my own responsibility. Thanks are also due to Professor Noël Lohoué and many colleagues for encouragement and help. We would like to thank especially Professor Gerhard Racher for improvements and suggestions in relation with chapter height.

•

Contents

1

Elementary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Basic Notations and Basic Definitions . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Radon Measures and Integration Theory . .. . . . . . . . . . . . . . . . . . . . 1.1.2 Locally Compact Groups .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.3 Convolution of Measures and Functions.. .. . . . . . . . . . . . . . . . . . . . 1.1.4 Amenable Groups.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Convolution Operators .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . b ..................... 1.3 For G Abelian CV 2 .G/ is Isomorphic to L1 .G/ 1.4 For G Abelian CVp .G/ is Isomorphic to CV p0 .G/ . . . . . . . . . . . . . . . . . . . 1.5 CVp .G/ as a Subspace of CV 2 .G/ . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 The Fourier Transform of a Convolution Operator .. . . . . . . . . . . . . . . . . . .

1 1 1 3 5 6 7 12 16 18 20

2 The Commutation’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . p 2.1 The Convolution Operator T G .f / . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 A Commutation Property of CV 2 .G/ . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 An Approximation Theorem for CV 2 .G/. . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

25 25 28 31

3 The Figa–Talamanca Herz Algebra . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Definition of Ap .G/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 A2 .G/ for G Abelian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Ap .G/ is a Banach Algebra . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

33 33 39 41

4 The Dual of Ap .G / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 The Dual of Ap .G/: The Notion of Pseudomeasure . . . . . . . . . . . . . . . . . . 4.2 Applications to Abelian Groups .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Holomorphic Functions Operating on Ap .G/ . . . . .. . . . . . . . . . . . . . . . . . . .

45 45 52 56

5 CV p .G / as a Module on Ap .G / . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . L . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 The Convolution Operator .k l/T 5.2 The Convolution Operator uT . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

65 65 68 ix

x

Contents

5.3 CV p .G/ as a Module on Ap .G/ . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . L for an Amenable G . . . . . . . . . . . . . . . . 5.4 Approximation of T by .k l/T

75 80

6 The Support of a Convolution Operator . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Definition of the Support of a Convolution Operator.. . . . . . . . . . . . . . . . . 6.2 Characterization of the Support of a Pseudomeasure . . . . . . . . . . . . . . . . . 6.3 A Generalization of Wiener’s Theorem .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 An Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Application to Amenable Groups . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

85 85 88 91 94 96

7 Convolution Operators Supported by Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 The Image of a Convolution Operator . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 On the Operator i.T / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 A Canonical Isometry of C Vp .H / into C Vp .G/ .. . . . . . . . . . . . . . . . . . . . 7.4 The Support of i.T / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5 Theorems of de Leeuw and Saeki . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.6 The Image of the Map i . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.7 First Applications: The Theorem of Kaplansky–Helson . . . . . . . . . . . . . . 7.8 A Restriction Property for Ap . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.9 Subgroups as Sets of Synthesis .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

101 101 107 112 118 122 124 133 136 141

8 C Vp .G / as a Subspace of C V2 .G / . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . p 8.1 A Canonical Map of L.Lp .X; // into L.LH .X; // . . . . . . . . . . . . . . . . p 8.2 An Integral Formula for T G .˛/ L2 .G/ . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 C V2 .G/ and C Vp .G/ in Amenable Groups .. . . . . .. . . . . . . . . . . . . . . . . . . .

145 145 150 156

Notes . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 161 References . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 167 List of Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 171

List of Symbols

We list here the symbols which are systematically used throughout the book. The numbers in parentheses refer to the paragraphs where the symbols are defined. Ap .G/; Ap .G/ Œf , fP

.3:1/

.1:1:1/

', L a ', 'a , ', ', Q ' , p '

(1.1.2)

Œf L; p Œf ; a Œf ; Œf a

(1.1.2)

, L , Q

(1.1.2)

F2

(1.1.4)

˛p

(1.5)

ˇ

(7.1)

' (1.1.3) C Vp .G/ b T

(1.2)

(1.6)

C.X I Y /, C00 .X I Y /, C.X /, C00 .X /, C0 .X / p

p

p

p

L .X I /, L .X I / , L .G/, L .G/, p LV .X I /, p

p LV .X I /

L.L .X I // (1.1.1)

p G ./,

G .f /, G .Œf /

ƒGO

(3.3)

p

(1.2)

(1.3)

1 1 M .X; /, M1 00 .X; /, M .G/, M00 .G/ 1

(1.1.1), (1.1.2)

(1.1.1)

jjjT jjjp

p

(1.1.1)

(1.1.1), (1.1.2) xi

xii

List of Symbols

M 1 .X /, kk Ap ;PMp ˆGO

(4.1)

(1.3)

PMp .G/ p ‰G

q

(1.1.1)

(4.1)

(4.1) (7.1)

L .k l/T

(5.1)

uT

(5.2)

spu

(6.1)

supp T TH TH;q

(6.1)

(7.1) (7.1)

Chapter 1

Elementary Results

We give the basic properties of the Banach algebra CVp .G/. For a locally compact b and define the abelian group G we show that CV 2 .G/ is isomorphic to L1 .G/ Fourier transform of every element of CVp .G/.

1.1 Basic Notations and Basic Definitions 1.1.1 Radon Measures and Integration Theory Let X be a topological space and Y a topological vector space. We denote by C.X I Y / the vector space of all continuous maps of X into Y and by C00 .X I Y / the subspace of all maps having compact support. We put C.X / D C.X I C/ and C00 .X / D C00 .X I C/. We denote by C0 .X / the subspace of all elements of C.X / vanishing at infinity. Suppose that X is a locally compact Hausdorff space and that is a complex Radon measure on X . For ' an arbitrary map of X into Œ0; 1 Z '.x/d jj.x/ X

denotes the upper integral in the sense of Bourbaki ([6], p. 112, Chap. IV, Sect. 4.1, no. 3, Définition 3.) We write L1 .X; / for the C-vector space of all ' 2 CX which are -integrable. For ' 2 L1 .X; / the integral of ' with respect to is denoted .'/ or Z '.x/d.x/: X

A. Derighetti, Convolution Operators on Groups, Lecture Notes of the Unione Matematica Italiana 11, DOI 10.1007/978-3-642-20656-6 1, © Springer-Verlag Berlin Heidelberg 2011

1

2

1 Elementary Results

For f 2 CX or Œ1; 1X locally -integrable we denote by f the Radon measure defined by .f/.'/ D .f '/ for every ' 2 C00 .X /. If 1 < p < 1 Lp .X; / is the C-vector space of all ' 2 CG such that ' is -measurable and j'jp is -integrable. If f 2 CX we denote by Œf the set of all g 2 CX with g.x/ D f .x/ -almost everywhere and by fP the set of all g 2 CX with g.x/ D f .x/ locally -almost everywhere. Suppose 1 p < 1 . For f 2 CX or for an arbitrary map of X into Œ1; 1 we put 0 11=p Z Np .f / D @ j'.x/jp d jj.x/A : X p

Np is a semi-norm on L .X; /. With respect to this semi-norm Lp .X; / is complete. For f 2 Lp .X; / we set kŒf kp D Np .f /

n ˇ o ˇ and Lp .X; / D Œf ˇf 2 Lp .X; /

which is a Banach space for the norm kkp . For an arbitrary map f of X into Œ1; 1 we put ˇ n ˇ M1 .f / D inf ˛ 2 Œ1; 1ˇf .x/ ˛

o locally -almost everywhere :

For f 2 CX we set N1 .f / D M1 .jf j/. For f a bounded complex function we set ˇ n o ˇ kf ku D sup jf .x/jˇx 2 X : Let M1 .X; / be the C-subspace of CX of all functions which are -measurable and bounded and L1 .X; / the C-subspace of CX of all functions which are locally -almost everywhere equal to a function of M1 .X; /. Then N1 is a semi-norm on L1 .X; /, with respect to this semi-norm L1 .X; / is complete. By definition n ˇ o ˇ L1 .X; / D fPˇf 2 M1 .X; / : With the norm

kfPk1 D N1 .f /;

L1 .X; / is a Banach space. We denote by M1 00 .X; / the subspace of all f 2 M1 .X; / with compact support. Finally let M 1 .X / be the space of all complex bounded Radon measures on X . For 2 M 1 .X / we put

1.1 Basic Notations and Basic Definitions

3

ˇ n o ˇ kk D sup j.'/jˇ' 2 C00 .X /; k'ku 1 : Then kk is a norm on M 1 .X /. Let 1 p < 1 we put p 0 D p=.p 1/ if p > 1 and p 0 D 1 if p D 1. For 0 f 2 Lp .X; / and g 2 Lp .X; / we set Z ˝ ˛ Œf ; Œg D f .x/g.x/d.x/ X

if p > 1, and if p D 1 ˝

˛

Z f .x/g.x/d.x/:

Œf ; gP D X

˝ ˛ 0 The function ; is a sesquilinear form on Lp .X; / Lp .X; /. Let L.Lp .X; // be the linear space of all continuous endomorphisms of p L .X; /. For T 2 L.Lp .X; //, jjjT jjjp is the bound of the operator T : ˇ n o ˇ jjjT jjjp D sup kTf kp ˇf 2 Lp .X; /; kf kp 1 : For the composition of the operators, L.Lp .X; // is a Banach algebra. For V a topological vector space, V 0 denotes the dual of V . If .V; kkV / is a normed vector space, and if F 2 V 0 we put ˇ n o ˇ kF kV 0 D sup jF .v/jˇv 2 V; kvkV 1 : This norm makes V 0 into a Banach space.

1.1.2 Locally Compact Groups Let G be a group. For a non-empty set Y , ' a map of G into Y , a and x 2 G we put '.x/ L D '.x 1 /; a'.x/ D '.ax/

and 'a .x/ D '.xa/:

Let now be G a locally compact group. We always suppose that the topology of G is Hausdorff. We recall that there is a nonzero positive Radon measure mG on G such that LG mG .'/ D mG .a'/ D mG .'a /G .a/ D mG 'L for every ' 2 C00 .G/ and every a 2 G. Here G is a continuous homomorphism of G into the multiplicative group .0; 1/. Up to a multiplicative real number, the measure mG is unique. The measure mG is called a left invariant Haar measure of G.

4


The function G does not depend of the choice of the measure mG . This function is called the modular function of G. If G is compact we suppose that mG .1G / D 1. Let us present some basic examples. (a) If G D T 1 mT .'/ D 2

Z2

'.e i /d

0

for ' 2 C.T /. (b) For G D R we may choose Z1 mR .'/ D

'.x/dx 1

for every ' 2 C00 .R/. (c) Take now an arbitrary group G. Consider on G the discrete topology. This locally compact group is denoted Gd . Suppose at first that G is finite. Then C00 .Gd / D CG . We have mGd .'/ D

1 X '.x/ jGj x2G

for every ' 2 C.Gd /. If G is infinite then C00 .Gd / is the subspace of CG of all functions having a finite support and we may choose X '.x/ mGd .'/ D x2G

for every ' 2 C00 .Gd /. In all these examples G D 1. (d) Let G be the group of matrices x y 0 x 1 where x; y 2 R; x 6D 0, with the topology induced by R2 . Then we may choose Z1 Z1 mG .'/ D 1 1

One has G

'.x; y/ dxdy: x2

! 1 x y D 2: 1 0x x

In the examples (a), (b) and (d), the integral on the right hand side is the Riemann integral.

1.1 Basic Notations and Basic Definitions

5

Let again G be any locally compact group. For ' 2 CG we set '.x/ Q D '.x 1 /

'.x/ D '.x/;

and ' .x/ D '.x 1 /G .x 1 /:

Let be a complex Radon measure on G. Then we define the three Radon measures , L and Q by .'/ L D .'/; L

.'/ D .'/

and .'/ Q D .'/ Q

where ' 2 C00 .G/. For f 2 CG and 1 p < 1 we also put p .f /.x/ D f .x 1 /G .x 1 /1=p : For f mG -integrable we set Z mG .f / D m.f / D

f .x/dx; Lp .G/ D Lp .G; mG /; Lp .G/ D Lp .G; mG /

G

.1 p 1/ and

1 M1 00 .G/ D M00 .G; mG /:

For f 2 CG we put: Œf L D ŒfL and for 1 p < 1

p Œf D Œp f ;

for a 2 G we also put a Œf

D Œa f and Œf a D Œfa :

Clearly p is an isometric involution of the Banach space Lp .G/ for 1 p < 1.

1.1.3 Convolution of Measures and Functions Formally the convolution of the two Radon measures and on the locally compact group G is defined by Z . /.f / D

f .xy/d.x/ d.y/ GG

whenever the double integral converges absolutely for all f 2 C00 .G/. This is the case for example if one of the two given measures has compact support, or if both

6


of them are bounded. For D gm we have Z

Z f .xy/g.x/dxd.y/ D

.gm /.f / D GG

Z D

0

G .y 1 / @

G

G

Z

1 0 Z @ f .xy/g.x/dx A d.y/ G

1

f .x/g.xy 1 /dx A d.y/ D ..g /m/.f /

G

where we define Z .g /.x/ D

g.xy 1 /G .y 1 /d.y/:

G

Similarly we get hm D . h/m if we define Z . h/.y/ D

h.x 1 y/d.x/:

G

Putting here D gm we set g h D gm h and thus Z g.yx/h.x

.g h/.y/ D G

1

Z /dx D

g.yx 1 /h.x/G .x 1 /dx:

G

We refer to [10] Chapter 8 for a detailed exposition of these questions.

1.1.4 Amenable Groups A locally compact group G is said to be amenable if there is a linear functional M on the vector space C b .G/ of all continuous bounded complex valued functions on G such that M.'/ > 0 if ' > 0, M.1G / D 1 and M.a'/ D M.'/ for every a 2 G. We only recall that compact, abelian or solvable groups are amenable. But SL2 .R/, the group of two by two real matrices with determinant one, and the free group F2 of two generators are not amenable. Every closed subgroup of an amenable group is amenable. If G is a locally compact group and H a closed normal subgroup, and if H and G=H are amenable, then so is G. They are many properties equivalent to the amenability. We will use the following one: for every " > 0 and for every compact subset K of G there is s 2 C00 .G/ with s > 0, N1 .s/ D 1 and N1 .k s s/ < " for every k 2 K. We refer to Chap. 8 of [105] for detailed proofs of all these assertions.

1.2 Convolution Operators

7

1.2 Convolution Operators Theorem 1. Let G be a locally compact group, 1 < p < 1, 2 M 1 .G/ and ' 2 C00 .G/. For x 2 G we then have Z 1=p0 ' G L .x/ D '.xy/G .y/1=p d.y/: G

Moreover 1=p0 1=p0 ' G L 2 Lp .G/ \ C.G/ and Np ' G L kkNp .'/: Proof. We have Z 1=p0 1=p0 L ' G L .x/ D '.xy 1 /G .y 1 /d.G /.y/ G

Z

D

0

'.xy/G .y/G .y/1=p d.y/

G

Z

D

'.xy/G .y/1=p d.y/:

G 1=p0

Next we prove the continuity of ' G . L Let x0 2 G and " > 0. Choose U0 a compact neighborhood of x0 and > 0 such that " : < 1 C kk supy2U 1 supp ' G .y/1=p 0

There is an open neighborhood U1 of e with j'.a/ '.b/j < for every a; b 2 G with ab 1 2 U1 . Let U2 be an open neighborhood of e with U2 U1 and U2 x0 U0 . Let x 2 U2 x0 then for every y 2 G we have j'.xy/ '.x0 y/j 1U 1 supp ' .y/: 0

Therefore ˇ ˇ Z ˇ ˇ 1=p0 1=p 0 L G /.x L 0 /ˇ j'.xy/'.x0 y/jG .y/1=p d jj.y/ < ": ˇ.' G /.x/.' G

8


be any function in C00 .G/. We then have 1 0 Z Z 1=p 0 1=p .x/.' G /.x/dx L D @ '.xy/G .y/ .x/dx A d.y/

To end the proof let Z G

G

G

and therefore ˇ Z ˇZ ˇ ˇ ˇ ˇ 1=p0 .x/.' G /.x/dx L ˇ Np0 /Np 'y G .y/1=p d jj.y/ ˇ ˇ ˇ G

G

D Np .'/Np0 . /kk: Remark. The inequality Np .' / kkNp .'/ is not verified in general. There is a missprint at .3:5:15/ of page 108 of [105]. Corollary 2. Let G and p as in Theorem 1. For every 2 M 1 .G/ there is an unique S 2 L.Lp .G// such that h i 1=p0 S Œ' D ' G L for every ' 2 C00 .G/. For f 2 Lp .G/ and a 2 G we have a .Sf

Moreover

/ D S.a f /:

jjjS jjjp kk:

Proof. For a and ' 2 C00 .G/ we have a

1=p0 1=p0 ' G L D a'/ G L

and therefore a .S Œ'/

D S.a Œ'/: p

Definition 1. The operator S of Corollary 2 is denoted G ./. p

p

p

p

Definition 2. For f 2 L1 .G/ we put G .f / D G .f m/ and G .Œf / D G .f /. In the following we will study a class of continuous operators of Lp .G/ which p have the same essential property as the operator G ./. Definition 3. Let G be a locally compact group and 1 < p < 1. An operator T 2 L.Lp .G// is said to be a p-convolution operator of G if T .a'/ D a .T .'//

1.2 Convolution Operators

9

for every a 2 G and every ' 2 Lp .G/. The set of all p-convolution operators of G is denoted CVp .G/. Proposition 3. Let G be a locally compact group and 1 < p < 1. Then CVp .G/ is a Banach subalgebra of L.Lp .G//. Proposition 4. Let G be a locally compact group and 1 < p < 1. Then: p

1: G is a linear injective contraction of the Banach space M 1 .G/ into the Banach space CVp .G/, 2: for every a 2 G and ' 2 Lp .G/ we have

pG .ıa /' D 'a G .a/1=p ˇˇˇ p ˇˇˇ and ˇˇˇ G .ıa /ˇˇˇp D 1 where ıa is the Dirac measure in a, p p p 3: G .ıab / D G .ıa / G .ıb / for every a; b 2 G, p 4: for f 2 L1 .G/ and ' 2 C00 .G/ we have G .f /Œ' D Œ' p f . p

Remarks. 1: The map x 7! G .ıx / is an isometric representation of the locally compact group G into the Banach space Lp .G/. For p D 2 this map is called the right regular representation of G. p p p 2: In Sect. 4.1 we shall show that G .˛ ˇ/ D G .˛/ G .ˇ/ for ˛; ˇ 2 M 1 .G/. Theorem 5. Let G be a locally compact group 1 < p < 1 and T 2 L.Lp .G//. Then T 2 CVp .G/ if and only if T .f '/ D f T ' for every f 2 L1 .G/ and every ' 2 Lp .G/. Proof. We suppose that T 2 CVp .G/. Let f 2 C00 .G/, ' 2 Lp .G/ and '1 2 T Œ'. We have Œf T Œ' D Œf '1 where for every x 2 G Z f '1 .x/ D

f .xy/'1 .y 1 /dy:

G

Let

0

2 Lp .G/. From Z

0 jf .y/j @

G

Z

1 j'1 .y 1 x/jj .x/jdx A dy Np .'1 /Np0 . /N1 .f / < 1;

G

we obtain ˝

˛ Œf T Œ'; Œ D

Z G

For every y 2 G we have

0 f .y/ @

Z

G

1 '1 .y 1 x/ .x/dx A dy:

10


Z

'1 .y 1 x/ .x/dx D

G

D

˝ ˝

y 1 T Œ'; Œ

y 1

˛ ˝ D T

˛ Œ'; T Œ D

y 1 Œ'; Œ

Z

˛

'.y 1 x/ n.x/dx

G

where T is the adjoint of T and where n 2 T Œ . Therefore ˝ ˛ Œf T Œ'; Œ D

Z

0 f .y/ @

G

Z

1 '.y 1 x/ n.x/dx A dy:

G

We also have Z

0 1 Z jf .y/j @ j'.y 1 x/jjn.x/jdx A dy Np .'/ kT Œ kp0 N1 .f / < 1;

G

G

and consequently Z

0 f .y/ @

G

Z

1 '.y 1 x/ n.x/dx A dy D

G

Z G

0 1 Z @ f .y/'.y 1 x/dy A n.x/dx: G

This implies ˝

˛ ˝ ˛ ˛ ˝ Œf T Œ'; Œ D Œf Œ'; T Œ D T .Œf Œ'/; Œ

and therefore T .Œf Œ'/ D Œf T Œ': If f 2 L1 .G/, choose a sequence .fn / of C00 .G/ with N1 .f fn / ! 0. By the inequality T .Œf Œ'/ Œf T Œ' jjjT jjj N1 .f fn /Np .'/ C N1 .fn f /T Œ' p p p we obtain T .Œf Œ'/ D Œf T Œ': Now suppose that T .f '/ D f T ' for every f 2 L1 .G/ and every ' 2 Lp .G/. Let ' 2 Lp .G/, a 2 G and " > 0. Choose f 2 L1 .G/ such that kf ' 'kp
k'k1 : ˇ ˇ GO

But

Z

jr./jjv./jd N2 .jrj1=2 /N2 .vjrj1=2 / jjjT jjj2

GO

and thus k'k1 < jjjT jjj2 C : Next we verify that T 2 CV 2 .G/. For a 2 G and f 2 L2 .G/ we have F .a f / D "G .a/F .f / and therefore T .a f / D F 1 "G .a/ .'F .f // D a F 1 .'F .f // Da T .f /:

14


2. ƒGO .' / D ƒGO .'/ƒGO . / for '; For f 2 L2 .G/ we have

b 2 L1 .G/.

ƒGO .' /.f / D F 1 '. F .f // D F 1 ' F .ƒGO . /f / D ƒGO .'/.ƒGO . /.f //: b 3. ƒGO .'/ D ƒGO .'/ for ' 2 L1 .G/. 2 Let f; g 2 L .G/. We have ˛ ˝ ˛ ˝ ˛ ˝ ˛ ˝ g; ƒGO .'/ D F g; 'F .f / D 'F .g/; F .f / D ƒGO .'/g; f ˛ ˝ D g; ƒGO .'/ f : b D CV 2 .G/ and for T 2 CV 2 .G/ we have 4. ƒGO .L1 .G// ƒ1 .T /f D F T .F 1 .f // GO b for every f 2 L2 .G/. b Let T 2 CV 2 .G/. We put for f 2 L2 .G/ .f / D F T .F 1 .f // ; b For g 2 L1 .G/ we have is a continuous linear operator of L2 .G/. O / D g F 1 .f / F 1 .Œgf (see [67] (31.27) Theorem p. 230) and therefore .Œgf O / D Œg .f O /: We now show that .Œrf / D Œr .f / b for every r 2 C00 .G/. b with f1 2 f , f2 2 .f / and > 0. There is g 2 L1 .G/ Let f1 ; f2 2 L2 .G/ (see [105], Chap. 5, Proposition 5.4.4, p. 161) with : kgO rku < 2 1 C jjj jjj2 1 C N2 .f1 / 1 C N2 .f2 / From k .Œrf / Œr .f /k2 k .Œrf / .Œgf O 1 /k2 C k .Œgf O 1 / Œg .Œf O O 1 /k2 CkŒg .Œf 1 / Œr .f /k2

1.3 For G Abelian CV 2 .G/ is Isomorphic to L1 .b G/

15

we deduce O 1 / C N2 .gf O 2 rf2 / k .Œrf / Œr .f /k2 jjj jjj2 N2 .rf1 gf and therefore k .Œrf / Œr .f /k2 < : b According to [8] (Chap. II, Sect. 2.3, no. 3, Lemme 3, p. 145), there is ' 2 L1 .G/ with .f / D 'f b For f 2 L2 .G/ we finally get for every f 2 L2 .G/. ƒGO .'/f D F 1 .F .f // D Tf: Remarks. 1. For T 2 CV 2 .G/, ' and

2 L2 .G/ we have

E ˝ ˛ D 1 T '; D ƒGO .T /F '; F : 2. Theorem 2 is already found in Larsen [73] .Chap. 4, p: 92, Theorem 4:1:1:/. . 2G .// D ..b /L/P. 3. For 2 M 1 .G/ we have ƒ1 O G

We also obtain an integral formula for the function Tf for many T 2 CV 2 .G/. Corollary 3. Let G be a locally compact abelian group and T 2 CV 2 .G/ with b Then for f 2 L2 .G/ ƒ1 .T / 2 L1 .G/. O G

h i f: .T / Tf D ˆGO ƒ1 O G Example. Theorem 2 implies that for f 2 L1 .T /, n 2 N and x1 ; : : : ; xn 2 C ˇ ˇ n n ˇ ˇ X X ˇ ˇ b xj xk f .j k/ˇ N1 .f / jxj j2 : ˇ ˇ ˇ j D1

j;kD1

A special case of this result is due to Toeplitz .[116], p. 500, Satz 7/. By Corollary 3 we also obtain for T 2 CV 2 .Z/ and x 2 l 2 .Z/ Z 1 X x.n/ i i.mn/ ƒ1 d .T x/.m/ D O .T /.e /e Z 2 nD1 2

0

.[121], Zygmund, Vol. I, Chap. IV, Sect. 9, p. 168, (9.18) Theorem/.

16


1.4 For G Abelian CVp .G / is Isomorphic to CV p0 .G / Theorem 1. Let G be a locally compact abelian group, 1 < p < 1 and 0 0 T 2 CVp .G/. Then T ' 2 Lp .G/ for ' 2 Lp .G/ \ Lp .G/ \ L1 .G/ and kT 'kp0 jjjT jjjp k'kp0 . 0

Proof. Let ' 2 Lp .G/\Lp .G/\L1 .G/ and 2 C00 .G/. Let r 2 ' and s 2 T .'/. Then Z s.x/ .x/dx D .s L /.e/ D ..T '/ Œ L/.e/: G

The group G being abelian, we have .T '/ Œ L D Œ L .T '/. But according to Theorem 5 of Sect. 1.2, Œ L .T '/ D T .Œ L '/ D T .' Œ L / D ' T .Œ L /; and in particular .T .'/ Œ L/.e/ D .' T .Œ L //.e/ i.e. Z

Z s.x/ .x/dx D G

r.x/t.x 1 /dx

G

where t 2 T .Œ L /. Therefore we get ˇ ˇZ ˇ ˇ ˇ ˇ ˇ s.x/ .x/dx ˇ k'kp0 kŒ kp jjjT jjjp ˇ ˇ G

0

and consequently T .'/ 2 Lp .G/ with kT .'/kp0 jjjT jjjp k'kp0 . Corollary 2. Let G be a locally compact abelian group, 1 < p < 1 and 0 T 2 CVp .G/. There is an unique continuous linear operator S of Lp .G/ with 0 S ' D T ' for every ' 2 L1 .G/ \ Lp .G/ \ Lp .G/. We have jjjS jjjp0 jjjT jjjp . 0

Definition 1. The unique continuous linear operator of the Banach space Lp .G/ of Corollary 2 is denoted jp .T /. Proposition 3. Let X be a locally compact Hausdorff space, a positive Radon measure on X , .Y; k k/ a normed space and f a -moderated and -measurable map of X into Y . Then there exists a sequence .gn / of -measurable step functions with values in Y with following properties: 1: kgn .x/k kf .x/k for every x 2 X and for every n 2 N, 2: lim gn .x/ D f .x/ -almost everywhere, 3: supp gn is compact.

1.4 For G Abelian CVp .G/ is Isomorphic to CV p0 .G/

17

Proof. Suppose f 6D 0. There nis ˇ.Kn /1 sequence nD1 a o compact subsets S of disjoint ˇ 1 of X and N -negligible with x ˇf .x/ 6D 0 D nD1 Kn [ N . Moreover for every n 2 N Kn 6D ; and ResKn f 2 C.Kn I Y /. Let i; n 2 N with i 6 n. For every x 2 Ki , there is V , open neighborhood of x in the subspace Ki , such that kf .x/ f .x 0 /k
ˇ = ˇ ˇ ˇ ˇ ˇ A ˇ '.xy/d.y/ˇ dx ˇ' 2 C00 .G/; Np .'/ 1 ˇ > ˇ ˇ ; G

9 ˇp0 11=p0 ˇ ˇZ > = ˇ ˇ ˇ ˇ ˇ ˇ A ˇ '.xy/d.y/ˇ dx ˇ' 2 C00 .G/; Np0 .'/ 1 : > ˇ ˇ ˇ ; G

Remark. In 1976, Herz proved that for every finite group G and for ˇˇˇ nonabelian ˇˇˇ ˇˇˇ p ˇˇˇ ˇˇˇ p0 ˇˇˇ every p 6D 2, there is f 2 CG with ˇˇˇ G .f /ˇˇˇp 6D ˇˇˇ G .f /ˇˇˇ 0 i.e. p

ˇ ˇ o n o n ˇ ˇ sup kf gkp ˇkgkp 1 6D sup kf gkp0 ˇkgkp0 1 .[63], Corollary 1, p. 12/. See also the notes to Chap. 1.

1.5 CVp .G / as a Subspace of CV 2 .G / Theorem 1 (Riesz–Thorin). Let be a positive Radon measure on a locally compact Hausdorff space X , and E the space of step functions in L1 .X I /. Let 0 < ˛ 1 and let the map T W E ! L1=˛ .X I / \ L1= .X I / be linear and satisfy for ' 2 E the inequalities

1.5 CVp .G/ as a Subspace of CV 2 .G/

19

kT 'k1=˛ M1 k'k1=˛ and kT 'k1= M2 k'k1= : Then T W E ! L1=ˇ .X I / for all ˇ 2 Œ˛; , and jjjT jjj1=ˇ M11t M2t where ˇ D .1 t/˛ C t ; 0 t 1: Proof. Cf [67], Appendix E (E:18), p. 722 and (E:16), p. 719, Example (a). Theorem 2. Let G be a locally compact abelian group, 1 < p < 1 and T 2 CVp .G/. Then we have T ' 2 L2 .G/ and kT 'k2 jjjT jjjp k'k2 for every step function in L1 .G/. 0

Proof. Clearly ' 2 L1 .G/ \ Lp .G/ \ Lp .G/. By Theorem 1 of Sect. 1.4 this 0 implies T ' 2 Lp .G/ and kT 'kp0 jjjT jjjp k'kp0 . We may suppose that p 2 p 0 . By Theorem 1, with M1 D M2 D jjjT jjjp , we obtain immediately T ' 2 L2 .G/ and kT 'k2 jjjT jjjp k'k2 . Corollary 3. Let G be a locally compact abelian group, 1 < p < 1 and T 2 CVp .G/. There is an unique S 2 L.L2 .G// with S Œr D T Œr for every integrable step function r. Moreover jjjS jjj2 jjjT jjjp . Proof. It follows immediately from Theorem 2 and from the fact that the integrable step functions are dense in Lp .G/. Definition 1. The unique operator of Corollary 3 is denoted ˛p .T /. Theorem 4. Let G be a locally compact abelian group and 1 < p < 1. Then ˛p is a contractive algebra monomorphism of the Banach algebra CVp .G/ into the Banach algebra CV 2 .G/. For T 2 CVp .G/ and for ' 2 Lp .G/ \ L2 .G/ we have p ˛p .T /' D T '. Moreover for 2 M 1 .G/ we have ˛p . G .// D 2G ./. Proof. Let T 2 CVp .G/ and ' 2 Lp .G/ \ L2 .G/ we show that ˛p .T /' D T '. Let '1 2 ', there is a sequence of mG -measurable complex valued step functions .rn / with lim rn .x/ D '1 .x/ mG -almost everywhere and jrn .x/j j'1 .x/j for every n 2 N and for every x 2 G. Then lim k˛p .T /Œrn ˛p .T /'k2 D lim kT Œrn T 'kp D 0; consequently ˛p .T /' D T '. We obtain the following improvement of Theorem 2. Corollary 5. Let G be a locally compact abelian group, 1 < p < 1 and T 2 CVp .G/. For every ' 2 Lp .G/ \ L2 .G/ we have T ' 2 L2 .G/ and kT 'k2 jjjT jjjp k'k2 .

20


Corollary 6. Let G be a locally compact abelian group, 1 = ˇ ˇ ˇ ˇ ˇ ˇ A ' 2 C '.xy/d.y/ dx .G/; N .'/ 1 ˇ ˇ ˇ 00 2 ˇ ˇ > ˇ ; G

9 ˇp 11=p ˇ ˇZ > ˇ = ˇ ˇ ˇ ˇ ˇ ˇ '.xy/d.y/ˇ dx A ˇ' 2 C00 .G/; Np .'/ 1 : ˇ > ˇ ˇ ; G

Remark. The proof of Corollary 5 .and also of Corollary 6/ uses in a very strong way the commutativity of G. One the main result of this book is that Corollary 5 .and also Corollary 6/ extends to the class of amenable groups.

1.6 The Fourier Transform of a Convolution Operator Using the results and notations of Sects. 1.3 and 1.5, we introduce the Fourier transform of a p-convolution operator for arbitrary p > 1. Definition 1. Let G be a locally compact abelian group and 1 < p < 1. The b 2 L1 .G/ b of T 2 CVp .G/is defined by T b D ƒ1 ˛p .T / . Fourier transform T O G

Theorem 1. Let G be a locally compact abelian group and 1 < p < 1. 1: 2: 3: 4: 5:

b for S; T 2 CVp .G/. .S C T /bD b S CT b .˛S /bD ˛ S for ˛ 2 C and T 2 CVp .G/. b for S; T 2 CVp .G/. .S T /bD b ST kb S k1 jjjS jjjp for S 2 CVp .G/. For S 2 CVp .G/ we have ˇ (ˇ Eˇˇˇ ˇD bF .'/; F . / ˇˇˇ' 2 L2 .G/ \ Lp .G/; jjjS jjjp D sup ˇˇ T ˇˇ

0

2 L2 .G/ \ Lp .G/;

) k'kp 1; k kp0 1 : 6: 7: 8: 9:

For S 2 CVp .G/ b S D 0 if and only if S D 0. p . G .//bD ..b /L/Pfor every 2 M 1 .G/. p . G .ıa //bD "G .a/Pfor every a 2 G. b Then for ' 2 L2 .G/ \ Lp .G/ we have S 2 L1 .G/. Let S 2 CVp .G/ such that b for S ' the integral formula S ' D ŒˆGO .b S / '.

1.6 The Fourier Transform of a Convolution Operator

21

Proof. This Theorem is a consequence of Theorem 2 and Corollary 3 of Sect. 1.3 and Theorem 4 of Sect. 1.5. b may be considered as a tempered distribution. For Remark. For T 2 CVp .Rn / T 1 n b/ ' F 1 .T b/ being ' 2 C .R / rapidly decreasing we then have T ' D F 1 .T b interpreted as the inverse Fourier transform of the distribution T . Corollary 2. Let 1 < p < 1 and S 2 CVp .Z/. If p < 2 suppose that x 2 lp .Z/, and x 2 l2 .Z/ if p 2. We then have Z 1 X x.n/ b i i.mn/ S .e /e d; m 2 Z: .S x/.m/ D 2 nD1 2

0

The following proposition gives a characterization of the Fourier transform of a p-convolution operator. Proposition 3. Let G be a locally compact abelian group, 1 < p < 1, E a 0 subspace of L2 .G/ \ Lp .G/ and G a subspace in L2 .G/ \ Lp .G/. We suppose both spaces E and G invariant by translations .fa 2 E and ga 2 G for f 2 E and g 2 G and a 2 G/. We also suppose E dense in L2 .G/ and in Lp .G/ and 0 b similarly F dense in L2 .G/ and in Lp .G/. Let also u be an element of L1 .G/. The following statements are equivalent: 1: There is S 2 CVp .G/ such that b S D u. 2: There is K 2 R with K > 0 and ˇ˝ ˛ˇˇ ˇ ˇ u F '; F ˇ Kk'kp k kp0 for every ' 2 E and every 2 G. Moreover for T 2 CV p .G/ we have ˇ ˇ ˛ˇˇˇ ˇ˝ b D sup T F '; F jjjT jjjp ˇˇˇ' 2 E; ˇ

2 G; k'kp 6 1; k kp0 6 1 :

Proof. We show at first that .1/ implies .2/. Let ' 2 L2 .G/ \ Lp .G/ and 0 2 L2 .G/ \ Lp .G/. We have ˇ˝ ˛ˇˇ ˇˇ˝ ˛ˇˇ ˇ ˇ u F '; F ˇ D ˇ T '; ˇ jjjT jjjp k'kp k kp0 : It remains to verify that .2/ implies .1/. There is a unique continuous sesquilinear 0 map L of Lp .G/ Lp .G/ into C with D E L.'; / D u F .'/; F . /

22


2 G. There is therefore a unique S 2 L.Lp .G// with ˝ ˛ L.'; / D S ';

for every ' 2 E and every

0

for every ' 2 Lp .G/ and every 2 Lp .G/. It remains to verify that S 2 CVp .G/. Let ' 2 E; 2 G and a 2 G. We have E D E ˝ ˛ ˝ ˛ D S.a'/; D u F . a'/; F . / D u ."G .a//PF .'/; F . / D u F .'/; F .a1 / ˝ D S ';

a1

˛

D

˝

a .S '/;

˛

:

Corollary 4. Let G be a locally compact abelian group, 1 < p < 1, a net .T˛ / of b Suppose that: CVp .G/, K 2 .0; 1/ and u 2 L1 .G/. 1: jjjT˛ jjjp K for every ˛, b˛ D u for the weak topology .L1 .G/; b L1 .G//. b 2: lim T b Then there is a T 2 CVp .G/ such that T D u. We have jjjT jjjp K. Proof. Let ' 2 L2 .G/ \ Lp .G/ and ˝

But

T˛ ';

˛

0

2 L2 .G/ \ Lp .G/. For every ˛ we have

D F .'/F Q ; Tb˛ :

D E lim F .'/F Q ; Tb˛ D F .'/F Q ; u :

Consequently

ˇ˝ ˛ˇ ˇ u F '; F ˇ Kk'kp k kp0 :

Proposition 3 permits to finish the proof. For m D .m1 ; : : : ; mn / 2 Zn we put m .e i1 ; : : : ; e in / D e i m1 1 CCi mn n . Then cn . Let K be a compact neighborhood of 0 in Rn . For f 2 L1 .T n / and > 0 m 2 T we set X .K/ b.m/m : s f D f m2Zn \ K

Example. For f 2 L1 .T /,

.Œ1;1/ sN f

is the Fourier sum of f .

The following theorem (see [111], p. 74, Teorema 4.1.) relates the Lp theory of Fourier series to CVp .Rn /. Theorem 5. Let K be a compact convex neighborhood of 0 in Rn and 1 < p < 1. The following statements are equivalent. .K/

1: lim !1 kf s f kp D 0 for every f 2 Lp .T n /, b D Œ1K . 2: There is T 2 CVp .Rn / such that T

1.6 The Fourier Transform of a Convolution Operator .Œ1;1/

23

According to Marcel Riesz, lim !1 kf s

f kp D 0 for every f 2 Lp .T / and for every 1 1, for every p > 1 and for every closed convex polyedral set C of Rn there is T 2 CVp .Rn / with b T D Œ1C . But for D the unit ball in Rn (n > 1) and for

b D Œ1D . Most of p … 2n=.n C 1/; 2n=.n 1/ there is no T 2 CVp .Rn / with T these results are due to Schwartz ([109], see also Herz [54]). Fefferman [43] proved b D Œ1D . that for every p 6D 2 and n > 1 there is no T 2 CVp .Rn / with T

•

Chapter 2

The Commutation’s Theorem

We show that for a locally compact unimodular group G, every T 2 CV 2 .G/ is the limit of convolution operators associated to bounded measures. p

2.1 The Convolution Operator T G .f / Theorem 1. Let G be a locally compact group, 1 < p < 1, T 2 CV p .G/, 0 p 2 Lp .G/. Then: f 2 M1 00 .G/, r 2 T Œf , ' 2 L .G/ and 0

rQ 2 Lp .G/,

1:

2: Np0 . 3:

D

Z

rQ / jjjT jjjp Np0 . /

p T G .p f

jf .x/jG .x/

E Z G /Œ'; Œ D '.x/.

p10

dx,

rQ /.x/dx.

G

Proof. To begin with suppose ' 2 C00 .G/. We have D E ˝ ˛ p T G .p f /Œ'; Œ D Œ' r; Œ : From .j'j jrj/j j 2 L1 .G/ we get Z Z .' r/.x/ .x/dx D '.x/. G

rQ /.x/dx:

G

The inequalities ˇZ ˇ ˇ ˇ '.x/. ˇ G

ˇ Z ˇ ˇ 1 rQ/.x/dx ˇ Np .' r/Np0 . / jjjT jjjp Np .'/Np0 . / f .x/jG .x/ p0 dx ˇ G

prove .1/ and .2/. A. Derighetti, Convolution Operators on Groups, Lecture Notes of the Unione Matematica Italiana 11, DOI 10.1007/978-3-642-20656-6 2, © Springer-Verlag Berlin Heidelberg 2011

25

26

2 The Commutation’s Theorem

Suppose now that ' 2 Lp .G/. There is a sequence .'n / of C00 .G/ with Np .'n '/ ! 0. We have Z

D E p r/.x/dx Q D T G .p f /Œ'; Œ

'n .x/.

lim G

and ˇZ ˇ ˇ ˇ '.x/. ˇ

Z rQ /.x/dx

G

'n .x/. G

ˇ ˇ ˇ rQ /.x/dx ˇ Np .'n '/Np0 . ˇ

rQ /:

Consequently Z

D E p r/.x/dx Q D T G .p f /Œ'; Œ :

'.x/. G

0

Remark. Even for p D 2, we are unable to decide whether j j jLr j is in Lp .G/. p

We now show that every T 2 CV p .G/ can be approximated by T G .f /. Proposition 2. Let G be a locally compact group, 1 < p < 1 and I the set of all f 2 C00 .G/ with f .x/ 0 for every x 2 G, f .e/ 6D 0 and Z

0

f .x/G .x/1=p dx D 1:

G

Then: 0 1: on I the relation supp ˇˇˇ pf supp ˇˇˇ f is a filtering partial order, 2: for f 2 I we have ˇˇˇG .p f /ˇˇˇp 1, p 3: for every T 2 CV p .G/ the net T G .p f / converges strongly to T . f 2I

p

Proof. Let T 2 CV p .G/, ' 2 L .G/ and " > 0. Let U be a neighborhood of e in G such that for y 2 U Np ' .'/y 1 G .y 1 /1=p
0 there is ."/ 2 N such that p.un um / < " for every m; n 2 N with m; n > ."/. By induction there is a sequence .nk / of N with nk < nkC1 and 1 nk > k 2 1 for every k 2 N. Then .unk /1 kD1 is a subsequence of the sequence .un /nD1 . For every k 2 N, we have

p.unkC1 unk /
0. There is f1 2 C00 .X I V / with N1 .f f1 /
0, supp en On for every 1 6 n 6 N , en .x/ D 1 on supp f1 and N X

en .x/ 6 1 on X . Then k D

nD1

For every x 2 X we have

N X

nD1

g en f1 .an / belongs to the vectorspace A.X I V /.

nD1

f1 .x/ k.x/ D

N X

g.x/en .x/ f1 .x/ f1 .an / :

nD1

This implies N1 .f k/ < ". Theorem 2. Let X be a locally compact Hausdorff space, a complex Radon measure on X and .V; k kV / a complex Banach space and f 2 L1V .X; /. Then

42

3 The Figa–Talamanca Herz Algebra

Z there is a unique v 2 V such that F .v/ D

F .f .x//d.x/ for every F 2 V 0 . We

X

have kvkV N1 .f /.

Proof. There is a sequence .fn / of A.X I V / with limn!1 N1 .fn f / D 0. For m; n 2 N, we have k.fm / .fn /kV 6 N1 .fm fn /. The sequence ..fn // is a Cauchy sequence in the Banach space V . Then there is v 2 V with lim k.fn / vkV D 0. Let F 2 V 0 . For every n 2 N we have Z ˇ ˇ ˇ ˇ ˇF .v/ F .f .x//d.x/ˇ 6 kF kV 0 kv .fn /kV C kF kV 0 N1 .fn f /: X

Z F .f .x//d.x/. The uniqueness of v is straightforward.

It follows that F .v/ D X

Z

Definition 1. The vector v of Theorem 2, is denoted .f / or

f .x/d.x/. X

L1V .X; /

into V . Remarks. 1: is a continuous linear map of 2: See Bourbaki, [6], Chap. III, Sect. 3, no. 3, Corollaire 2, p. 80 for a more general result. Let ! be a continuous map of a topological space X in a topological space Y . For f 2 C.Y / we put ! .f / D f ı !. Then ! is an algebra homomorphism of C.Y / into C.X /. Lemma 3. Let ! be a continuous homomorphism of a locally compact group G into a locally compact group H , k; l 2 M1 00 .G/ and r; s 2 C00 .H /. We set for h 2 H, j.h/ D k ! .rh / G .l .! .sh //L: Then for 1 < p < 1 we have: Z 1: j 2 C00 .H I Ap .G// and j.h/dh 2 Ap .G/, Z 2:

H

j.h/dh D k G lL! .r H sL /,

H

3: kk G lL ! .r H sL /kAp 6 Np .k/Np0 .l/Np .r/Np0 .s/. Proof. For every h 2 H we put f .h/ D k ! .rh / and g.h/ D l ! .sh /. Then f .h/; g.h/ 2 M1 00 .G/ and j.h/ D f .h/ G g.h/L. But for h; h0 2 H we have

6 krku Np .k/ Np0

kj.h/ j.h0 /kAp g.h/ g.h0 / C krh rh0 ku Np .k/ Np0 g.h0 / :

3.3 Ap .G/ is a Banach Algebra

43

Z This implies j 2 C00 .H I Ap .G//,

j.h/dh 2 Ap .G/ and H

Z j.h/dh

Z

Ap

H

kj.h/kAp dh: G

But Z

Z kj.h/kAp dh 6 H

Z Np .f .h// N .g.h//dh 6

H

p

!1=p Z

Np .f .h// dh

p0

!1=p0 p0

N .g.h// dh p0

H

:

H

We have moreover !1=p

Z

p

Np .f .h// dh

D Np .k/Np .r/

p0

Np 0 .g.h// dh

and

H

and thus

!1=p0

Z

D Np 0 .l/Np0 .s/

H

Z j.h/dh

Ap

H

6 Np .k/Np0 .l/Np .r/Np0 .s/:

Let x be an element of G. We recall that ıx 2 Ap .G/0 , thus 0 1 Z Z @ A ıx j.h/dh D j.h/.x/dh: H

H

But for every h 2 H Z k.xt/ r.!.xt/h/l.t/s.!.t/h/dt

j.h/.x/ D G

and therefore

Z

L j.h/.x/dh D .k G l/.x/.r H sL /.!.x//:

H

Theorem 4. Let ! be a continuous homomorphism of a locally compact group G into a locally compact group H and 1 0. We choose 8
m > 1 we have

n X

1 X

6 kvkAp C 1

! Np .kj /Np0 .lj /

j DmC1

C kukAp C 1

1 X

! Np .ri /Np0 .si / :

i DmC1

There is consequently w 2 Ap .G/ with lim kw un ! .vn /kAp .G/ D 0: For every n 2 N we have kw u ! .v/k1 6 1 C kukAp

1 X

! Np .rj /Np 0 .sj / C kvk1

j DnC1

1 X

! Np .kj /Np 0 .lj /

j D1Cn

and therefore u ! .v/ 2 Ap .G/. We finally have ku ! .v/kAp 6 ku ! .v/ un ! .vn /kAp C kukAp kvkAp C ": Corollary 5. Let G be a locally compact group and 1 < p < 1. For the pointwise product the Banach space Ap .G/ is a commutative Banach algebra. Remarks. 1: Even for G D T this result is not trivial. 2: The fact that Ap .G/ is a Banach algebra is due Herz in ([56], p. 6002, Théorème 1, [57], p. 244, [59], p. 72, Corollary/. Eymard proved earlier that A2 .G/ is a Banach algebra [41]. The proof above is due to Spector .[112], Lemme IV.2.1, p. 52, and Théorème IV.2.3., p. 54/. See the notes to Chap. 3 for Herz’ approach and various generalizations. Corollary 6. Let G be a locally compact group and 1 < p < 1. Let v 2 Ap .G/, for every u 2 Ap .Gd / we have uv 2 Ap .Gd / and kvukAp .Gd / kukAp .Gd / kvkAp .G/ .

Chapter 4

The Dual of Ap .G /

The dual of the Banach space Ap .G/ is the Banach space PM p .G/ of all limits of convolution operators associated to bounded measures. If G is abelian then A2 .G/ Ap .G/. Holomorphic functions operate on Ap .G/.

4.1 The Dual of Ap .G /: The Notion of Pseudomeasure Proposition 1. Let G be an abelian locally compact group. Then for u 2 A2 .G/ and 2 M 1 .G/ we have D E b .u/ D ˆ1 .u/; P GO and j.u/j kb k1 kukA2 : Proof. We have Z

Z

.u/ D G

GO

! ˆ1 .u/./"G .x/./d GO

Z d.x/ D GO

ˆ1 .u/./ GO

!

Z

.x/d.x/ d: G

Moreover

j.u/j kb k1 kˆ1 .u/k1 D kb k1 kukA2 : GO ˇˇˇ ˇˇˇ p Q ˇˇˇ2 D kb k1 . Proposition 2 below According to Theorem 2 of Sect. 1.3 ˇˇˇG ./ extends the inequality of Proposition 1 to every 1 < p < 1 and to every nonabelian locally compact group.


45

4 The Dual of Ap .G/

46

Proposition 2. Let G be a locally compact group and 1 < p < 1. Then for 2 M 1 .G/ and u 2 Ap .G/ we have .u/ D

1 D X

E p Q p kn ; Œp0 ln G ./Œ

nD1 1 X for every .kn /; .ln / 2 Ap .G/ with u D k n lLn . The following inequality holds nD1

ˇˇˇ p ˇˇˇ j.u/j ˇˇˇG ./ Q ˇˇˇp kukAp : Proof. Let ';

2 C00 .G/. For x 2 G we have Z .' L /.x/ D .p '/x 1 .y/G .x 1 /1=p .p0 /.y/dy G

and therefore .' L / D

Z

0 .p0 /.y/ @

G

Z

but

Z

1 .p '/x 1 .y/G .x 1 /1=p d.x/A dy;

G

1=p0

.p '/x 1 .y/G .x 1 /1=p d.x/ D .p ' G /.y/ Q

G

E D p .' L / D G ./Œ Q p '; Œp0 :

hence

0

For k 2 Lp .G/ and l 2 Lp .G/ choose 'n ; n 2 C00 .G/, n D 1; 2; 3; : : :, such 0 that 'n ! k in Lp .G/ and n ! l in Lp .G/. Then D p Q p 'n ; Œp0 lim G ./Œ

n!1

On the other hand

E

n

E D p D G ./Œ Q p k; Œp0 l :

lim 'n Ln D k lL

n!1

in Ap .G/ and thus

D

E p L Q p k; Œp0 l D .k l/: G ./Œ

Finally we have .u/ D

1 X nD1

! kn lLn

D

1 X nD1

.kn lLn / D

1 D X nD1

E p G ./Œ Q p kn ; Œp0 ln

4.1 The Dual of Ap .G/: The Notion of Pseudomeasure

47

and j.u/j

1 1 ˇD Eˇˇ ˇˇˇ X ˇˇˇ X ˇ p p ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ./Œ 0 Q k ; Œ l . / Q Np .kn /Np0 .ln /: p n p n ˇ G ˇ G p nD1

nD1

Remark. From Theorem 2 of Sect. 1.3 and Theorem 2 of Sect. 3.2 it follows, for G abelian, that ˇ n o ˇˇˇ 2 ˇˇˇ ˇˇˇ ./ ˇˇˇ D sup j.u/jˇˇu 2 A2 .G/; kukA 1 : Q 2 G 2 We will prove the following generalization: ˇ n o ˇˇˇ ˇˇˇ p ˇˇˇ D sup j.u/jˇˇu 2 Ap .G/; kukA 1 ˇˇˇ ./ Q p G p for every 1 < p < 1 and for an arbitrary locally compact group G. Scholium 3. Let G be a locally Zcompact group and 1 < p < 1. Then for every p p G .ıx /d.x/ in the following sense: for every 2 M 1 .G/ we have G ./ D ' 2 Lp .G/ and every

0

G

2 Lp .G/

D E Z D E p p G .ıx /Œ'; Œ d.x/: G ./Œ'; Œ D G

This Scholium permits to complete in a very simple way Proposition 4 of Sect. 1.2. p

Corollary 4. Let G be a locally compact group and 1 < p < 1. Then G is a contractive representation of the Banach algebra M 1 .G/ into Lp .G/. p

p

p

Proof. It suffices to verify that G .˛ ˇ/ D G .˛/ ı G .ˇ/ for ˛; ˇ 2 M 1 .G/. Let 0 ' 2 Lp .G/ and 2 Lp .G/. We have ˝ p ˛ G .˛ ˇ/'; D

Z G

Z D

˛ ˝ p G .ıx /'; d.˛ ˇ/.x/ 1 0 Z ˛ ˝ p @ G .ıxy /'; d˛.x/A dˇ.y/

G

G

G

G

1 0 Z Z ˛ ˝ p p D @ G .ıx /G .ıy /'; Lp ; Lp0 d˛.x/A dˇ.y/ Z

D G

˛ ˝ p p G .˛/G .ıy /'; dˇ.y/


48

Z D

˝ p ˛ ˝ p ˛ p0 p0 G .ıy /'; G .˛/ dˇ.y/ D G .ˇ/'; G .˛/

G

˛ ˝ p p D G .˛/G .ˇ/'; : Definition 1. Let G be a locally compact group and 1 < p < 1. The topology on L.Lp .G//, associated to the family of seminorms ˇ 1 ˇ ˇX˝ ˛ˇˇ ˇ T 7! ˇ T Œkn ; Œln ˇ ˇ ˇ nD1

with .kn /; .ln / 2 Ap .G/, is called the ultraweak topology. Remarks. 1: This topology is locally convex and Hausdorff. 2: Let T be a continuous operator of Lp .G/, .S˛ / a net of L.Lp .G// such that 0 sup˛ jjjS˛ jjjp < 1, E a dense subset of Lp .G/ and F a dense subset of Lp .G/. Suppose that lim < S˛ '; >D< T '; > ˛

for ' 2 E and

2 F . Then lim S˛ D T for the ultraweak topology.

Definition 2. Let G be a locally compact group and 1 < p < 1. The closure p of G .M 1 .G// in L.Lp .G// with respect to the ultraweak topology is denoted PM p .G/. Every element of PM p .G/ is called a p-pseudomeasure. Remarks. 1: Clearly PM p .G/ CV p .G/. 2: Theorem 4 of Sect. 2.3 implies that CV 2 .G/ D PM 2 .G/ for every locally compact unimodular group. 3: We will show that PM p .G/ D CV p .G/ for every locally compact amenable group G and every 1 < p < 1. 4: It is unknown whether PM p .G/ D CV p .G/ in general. Lemma 5. Let G be a locally compact group, 1 0. There is 2 M 1 .G/ such that ˇ ˇ 1 1 D ˇXD E X Eˇ " ˇ ˇ p T Œp kn ; Œp0 ln G ./Œp kn ; Œp0 ln ˇ < ˇ ˇ 2 ˇ nD1 nD1


49

ˇ 1 ˇ 1 D ˇXD E X Eˇ " 0 0 0 0 ˇ ˇ p T Œp kn ; Œp0 ln G ./Œp kn ; Œp0 ln ˇ < : ˇ ˇ ˇ 2 nD1 nD1

and

Proposition 2 implies 1 D X

1 D E X E 0 0 p p G ./Œp kn ; Œp0 ln D G ./Œp kn ; Œp0 ln

nD1

nD1

and consequently ˇ 1 ˇ 1 D ˇXD E X Eˇ 0 0 ˇ ˇ T Œp kn ; Œp0 ln T Œp kn ; Œp0 ln ˇ < ": ˇ ˇ ˇ nD1

nD1

Definition 3. Let T be a p-pseudomeasure on the locally compact group G, and p 1 < p < 1. We define the continuous linear form ‰G .T / on Ap .G/ by p

‰G .T /.u/ D

1 D X

1 E X T Œp kn ; Œp0 ln ; u D kn lLn 2 Ap .G/:

nD1

nD1

The following theorem gives a description of the dual of Ap .G/. Theorem 6. Let G be a locally compact group and 1 < p < 1. Then: p

1: ‰G is a conjugate linear isometry of PM p .G/ onto Ap .G/0 I p p 2: ‰G .G .// Q D for every 2 M 1 .G/I p 3: ‰G is a homeomorphism of PM p .G/, with the ultraweak topology, onto Ap .G/0 , with the weak topology .Ap .G/0 ; Ap .G//. 0

Proof. (I) Let F 2 Ap .G/ . We prove that there is a unique operator TF 2L.Lp .G// such that E D L D TF Œp k; Œp0 l F .k l/ 0

for every k 2 Lp .G/ and every l 2 Lp .G/ and that TF 2 CV p .G/ with jjjTF jjjp D kF kAp .G/0 . L is linear in the first variable and conjugate linear The map .Œk; Œl/ 7! F .k l/ in the second variable and ˇ ˇ ˇ L ˇˇ 6 kF k 0 kŒkkp kŒlkp0 : ˇF .k l/ Ap Hence there is an unique operator SF 2 L.Lp .G// with D E L D SF Œk; Œl : F .k l/


50

We have jjjSF jjjp 6 kF kA0p . It is straightforward to verify that SF 2 CV dp .G/. Let TF D p ıSF ıp . Then TF 2 CV p .G/ and E D L D TF Œp k; Œp0 l F .k l/ 0

for every k 2 Lp .G/ and every l 2 Lp .G/. It remains to prove that kF kA0p 6 jjjTF jjjp . 1 X For every u 2 Ap .G/ and every .kn /; .ln / 2 Ap .G/ with u D k n lLn we nD1

have F .u/ D

1 D X

E TF Œp kn ; Œp0 ln :

nD1

This implies jF .u/j 6 jjjTF jjjp

1 X

Np .kn /Np0 .ln /

nD1

and therefore kF kA0p jjjTF jjjp . (II) For every F 2 A0p we put j.F / D TF . Then clearly: i. ii. iii. iv.

j.F1 C F2 / D j.F1 / C j.F2 / for F1 ; F2 2 A0p , j.˛F / D ˛j.F / for ˛ 2 C and F 2 A0p , p Q for 2 M 1 .G/, j./ D G ./ jjjj.F /jjjp D kF kA0p for F 2 A0p . In particular the map j is injective.

We verify only (iii). According to Proposition 2 we have D E p Q p k; Œp0 l k lL D G ./Œ but

D E k lL D T Œp k; Œp0 l : ˇ o n ˇ (III) The set f mG ˇf 2 C00 .G/ is dense in Ap .G/0 with respect to the topology .A0p ; Ap /. Suppose that E; F are two complex vector spaces and ˇ B a bilinear form on n ˇ 0 E F . For M a subset of E we put M D u 2 F ˇB.L; u/ D 0 for every o L 2 M and similarly for a subset of F . We also set M 00 D .M 0 /0 . Suppose now that M is a subspace of E. By the bipolar theorem, the closure of M with respect 00 to the topology B .E; F / coincides E D Ap .G/0 , F D Ap .G/, ˇ with M . Choose o n ˇ B.L; u/ D L.u/ and M D f mG ˇf 2 C00 .G . If u 2 M 0 , then f mG .u/ D 0


51

for every f 2 C00 .G/, and therefore u D 0. We have proved that M 0 D 0 and consequently that M 00 D E. (IV) Consider on Ap .G/0 the topology .A0p ; Ap / and on j.Ap .G/0 / the ultraweak topology. Then j is a homeomorphism of Ap .G/0 onto j.Ap .G/0 /. 1 X p kn p0 ln L. Let Let " > 0, kn ; ln 2 Ap .G/ and u D nD1

ˇ o ˇ U D L 2 Ap .G/0 ˇjL.u/j < " n

and

ˇˇ 1 ˇ ) ˇˇ X D Eˇ ˇˇ ˇ T Œkn ; Œln ˇ < " : V D T 2 j Ap .G/ ˇˇ ˇˇ ˇ (

0

nD1

1

We have j.U / D V and U D j .V /. (V) We have j.Ap .G/0 / PM p .G/. Let F 2 Ap .G/0 . According to (III) there is a net .f˛ / of C00 .G/ with lim f˛ mG D F for .A0p ; Ap / and therefore lim j.f˛ mG / D j.F / for the ultraweak topology. p (VI) j is a bijection of Ap .G/0 onto PM p .G/ and ‰G D j 1 . p

It is straightforward to verify that for T 2 PM p .G/ ‰G .T / 2 Ap .G/0 and 0 p k‰G .T /kA0p jjjT jjjp . But for k 2 Lp .G/ and l 2 Lp .G/ we have D E ˝ ˛ p p ‰G .T / p k .p0 l/L D j ‰G .T / Œk; Œl D T Œk; Œl p and therefore j ‰G .T / D T . Remark. This result is due to Herz [56], [57]. Figà-Talamanca [44] .Theorem 1, p. 496/ proved it for p > 1 for G abelian or compact, and for p D 2 if G is unimodular. For p D 2 and any locally compact group it was proved by Eymard [41] .p. 210 .3.10/ Théorème/. Corollary 7. Let G be a locally compact group and 1 < p < 1. Then: ˇ o n ˇ p 1: G .f /ˇf 2 C00 .G/ is ultraweakly dense in PMp .G/, ˇ o n ˇ p 2: G ./ˇ is a finitely supported complex measure on G is ultraweakly dense in PM p .G/. Proof. The statement .1/ is containedn inˇ the preceding proof. The proof of .2/ is ˇ identical to the proof of .1/ with M D ˇ is a finitely supported complex measure o on G . Remark. For every 2 M 1 .G/ there is a net .˛ / of finitely supported complex p p measures with lim G .˛ / D G ./ ultraweakly.


52

We obtain a converse to Lemma 5. Corollary 8. Let G be a locally compact group, 1 < p < 1 and T 2 L.Lp .G//. Suppose that 1 D 1 D E X E X T Œp kn ; Œp0 ln D T Œp kn0 ; Œp0 ln0 nD1

nD1

1 1 X X 0 0 L k n ln D kn0 lLn0 . Then for every .kn /; .ln / ; .kn /; .ln / 2 Ap .G/ with nD1

T 2 PM p .G/.

nD1

Proof. Let u 2 Ap .G/. We set F .u/ D

1 D X

E T Œp kn ; Œp0 ln

nD1 1 X k n lLn . Clearly F 2 Ap .G/0 . Theorem for every .kn /; .ln / 2 Ap .G/ with u D nD1

6 implies the existence of S 2 PMp .G/ with

E D F .' L / D S Œp '; Œp0 0

for every ' 2 Lp .G/ and every

2 Lp .G/. Then S D T .

Definition 4. Let G be a locally compact group and 1 < p < 1. For every T 2 PMp .G/ and u 2 Ap .G/ we put: ˝

u; T

˛ Ap ;PM p

p

D ‰G .T /.u/:

Remark. Clearly Ap ;PMp is a sesquilinear form on Ap .G/ PM p .G/.

4.2 Applications to Abelian Groups We show that CV 2 .G/ D PM 2 .G/ for G abelian, without using the approximation theorem of Sect. 2.3. Theorem 1. Let G be a locally compact abelian group. Then CV 2 .G/ D PM 2 (G) and we have D E ˝ ˛ b u; T D ˆ1 .u/; T A2 ;PM 2

GO

for every T 2 CV 2 .G/ and for every u 2 A2 .G/. Proof. To prove that T 2 PM 2 .G/ it will be enough (see Corollary 8 Sect. 4.1) to verify the equality

4.2 Applications to Abelian Groups 1 D X nD1

53

E D E T ŒkLn ; ŒlLn D ˆ1 .u/; b T GO

1 X k n lLn . for every .kn /; .ln / with u D nD1

We put rn D F .kn / and sn D F .ln /. By Remark (1) of Theorem 1 of Sect. 1.3 we have, for N 2 N, we have N D X

* N + E X L L b rQn sLn ; T ; T Œkn ; Œln D

nD1

nD1

but by Lemma 1 of Sect. 3.2 we have N X 1 lim rQn sLn ˆGO .u/ D0 nD1

and therefore

1

1 D E D E X b: .u/; T T ŒkLn ; ŒlLn D ˆ1 GO nD1

So T 2 PM2 .G/. But then by the definitions of Sect. 4.1 ˝

u; T

˛ A2 ;PM 2

D E b D ˆ1 .u/; T ; GO

which ends the proof of the theorem. b and Definition 1. Let G be a locally compact abelian group. For every ' 2 L1 .G/ 1 b every f 2 L .G/ we set ˝ ˛ ‚GO .'/.f / D f; ' : b onto L1 .G/ b 0 . Let t ˆ O be The map ‚GO is a conjugate linear isometry of L1 .G/ G the transposed of the map ˆGO . Then 2 t ƒ1 D ‚1 ı ˆGO ı ‰G : GO GO

b and ' 2 L1 .G/ b we put u D ˆ O .f / and T D ƒ O .'/ in Indeed, for f 2 L1 .G/ G G the formula of Theorem 1, we get ˛ ˝ ˛ ˝ f; ' D ˆGO .f /; ƒGO .'/ A2 ;PM 2 D ‰G2 .ƒGO .'//.ˆGO .f // D t ˆGO ‰G2 .ƒGO .'// .f / and thus

t

ˆGO ‰G2 .ƒGO .'// D ‚GO .'/:


54

The following result completes Theorem 2 of Sect. 1.3. Corollary 3. Let G be a locally compact abelian group. Then ƒGO is a homeomorb with the topology .L1 ; L1 /, onto CV 2 .G/ with the ultraweak phism of L1 .G/, topology. b Then Corollary 4. Let G be a locally compact abelian group and u 2 L1 .G/. there is a net .v˛ / of trigonometric polynorms such that: 1: kv˛ k1 kuk1 for every ˛, 2: lim v˛ D u for the topology .L1 ; L1 /. ˛

Corollary 5. Let G be an infinite locally compact abelian group. Then CV 2 .G/ 6D 2G .M 1 .G//: b consider '.f / D Proof. Suppose that CV 2 .G/ D 2G .M 1 .G//. For f 2 L1 .G/ 1 bı"G . Then ' is a linear continuous map of L .G/ b into C0 .G/ with k'.f /ku 6 f b 0. kf k1 . We show that t ' is a surjective map of C0 .G/0 onto L1 .G/ 1 1 1 b 0 1 b Let F 2 L .G/ . Then ‚ O .F / 2 L .G/ and ƒGO .‚ O .F // 2 CV 2 .G/. There G G is 2 M 1 .G/ with ƒGO .‚1 .F // D 2G ./. Consequently ‚1 .F / D ..b /L/P. GO GO t t 1 t Then . '/./ D F . The map ' is injective: let indeed 2 M .G/ with . '/. /D0. b we have For every f 2 L1 .G/ Z b ı"G / D f ./b .f ./d D 0 GO

thus b D 0 and consequently D 0. b D C0 .G/. From 3 Lemma of VI.6. (p. 488) of [38] it follows that '.L1 .G// According [105] Theorem 5.4.5, p. 161 the group G is finite. Remark. 1: One can show that for every 1 < p < 1 and for every infinite locally compact abelian group G one has CV p .G/ 6D CV 2 .G/ .see Larsen [73], Theorem 4.5.2., p. 110:/ 2: Using the results of this book, the following generalization of Corollary 5 can be easily obtained: if a locally compact group G admits an infinite abelian subgroup then CV 2 .G/ 6D 2G .M 1 .G//. Theorem 6. Let V be a Banach space and ! a linear functional of V 0 . Then the following statements are equivalent: 1: There is v 2 V with !.F / D F .v/ for every F 2 V 0 , 2: For every net .Fi / of V 0 and F 2 V 0 with lim Fi D F for .V 0 ; V / and kFi k C for C > 0 we have lim !.Fi / D !.F /. Proof. See 6 Theorem of Chap. V.5., p. 428 of [38].

4.2 Applications to Abelian Groups

55

Theorem 7. Let G be a locally compact abelian group and 1 < p < 1. Then 1: We have A2 .G/ Ap .G/, and kukAp kukA2 for every u 2 A2 .G/. 2: For T 2 PM p .G/ and u 2 A2 .G/ we also have ˝ ˛ ˝ ˛ u; T A ;PM D u; ˛p .T / A ;PM : p

p

2

2

Proof. Let u 2 A2 .G/. There is (Sect. 3.2, Theorem 2) k; l 2 L2 .G/ with u D k lL and kukA2 D N2 .k/N2 .l/. For every F 2 Ap .G/0 we set

p 1 L L !.F / D ˛p .‰G / .F / Œk; Œl : Let .Fi / be a net of Ap .G/0 , F 2 Ap .G/0 and K > 0 such that lim Fi D F for the 0 topology .Ap ; Ap / and with kFi kA0p K for all i . Then ˇˇˇ p 1 ˇˇˇ ˇˇˇ.‰ / .Fi /ˇˇˇ K G p for every i , and

p

p

lim.‰G /1 .Fi / D .‰G /1 .F /

for the ultraweak operator topology on CV p .G/. In particular if r; s 2 M1 00 .G/ we have E D E D p p lim .‰G /1 .Fi /Œr; Œs D .‰G /1 .F /Œr; Œs : From Œr 2 L2 .G/\Lp .G/ we get ˛p .S /Œr D S Œr for S 2 PM p .G/ and therefore p p lim ˛p .‰G /1 .Fi / Œr; Œs D ˛p .‰G /1 .F / Œr; Œs : Moreover we have

ˇˇˇ ˇˇˇ ˇˇˇ ˇˇˇ p ˇˇˇ˛p .‰G /1 .Fi / ˇˇˇ K 2

for every i 2 I . By the second remark to Definition 1 of Sect. 4.1 this implies that lim ˛p .‰Gp /1 .Fi / D ˛p .‰Gp /1 .F / for the ultraweak operator topology on PM2 .G/. In particular L ŒlL D ˛p .‰ p /1 .F / Œk; L ŒlL lim ˛p .‰Gp /1 .Fi / Œk; G and therefore lim !.Fi / D !.F /. According to Theorem 6, there is a v 2 Ap .G/ 0 with !.F / D F .v/ for every F 2 Ap .G/ .


56 p

For F D ‰G .T / with T 2 PMp .G/ we hence obtain D h i h iE p p ‰G .T /.v/ D ! ‰G .T / D ˛p .T / kL ; lL D ‰G2 ˛p .T / .k lL/; and in particular for 2 M 1 .G/: p p p L D ‰ 2 ˛p .p .// .u/: ‰G G ./ .v/ D ‰G2 ˛p .G .// .k l/ G G p p p Q D .u/ Q and From ˛p .G .// D 2G ./ and ‰G G ./ D Q we deduce .v/ therefore u D v. This implies u 2 Ap .G/ and ˝ ˛ ˝ ˛ u; T Ap ;PMp D u; ˛p .T / A2 ;PM2 : Consequently

kukAp 6 kukA2 :

Remarks. 1: For u 2 A2 .G/ and T 2 PM p .G/ we have ˝ ˛ b/ ˆ1 .u/ : u; T Ap ;PM p D ‚GO .T GO 2: Clearly the map ˛p is the adjoint of the inclusion of A2 .G/ in Ap .G/. 3: We will generalize this theorem to the class of amenable groups in Sect. 8.3.

4.3 Holomorphic Functions Operating on Ap .G / In analogy with the case of the Fourier algebra of a locally compact abelian group ([105] Chap. 6), we investigate whether in the case of an arbitrary locally compact group G the composed F ıu of u 2 Ap .G/ and the holomorphic function F 2 CU still belongs to Ap .G/. Observe at first that for G a locally compact group, 1 < p < 1, a 2 G and u 2 Ap .G/ we have a u; ua 2 Ap .G/ and ka ukAp D kua kAp D kukAp . Lemma 1. Let G be a locally compact group, 1 0 and x0 2 G, and let u D

1 X

k n lLn with .kn /; .ln / 2 Ap .G/.

nD1

Choose M 2 N such that 1 X nDM C1

Np .kn /Np0 .ln /
0. Then there is v 2 Ap .G/ \ C00 .G/ and V neighborhood of a such that: 1: 2: 3: 4: 5:

0 v.x/ 1 for every x 2 G, v.x/ D 1 on V , kvkAp < ˛, supp v U , kuv u.a/vkAp < ".

Proof. Without loss of generality we can assume that a D e. First suppose that u.e/ D 0. By Theorem 2, there is v 2 Ap .G/ \ C00 .G/ and an open set U1 neighborhood of e with v.y/ D 0 on U1 and ku vkAp
max ju.x/jjx 2 G :

For n 2 N and z 2 C with jzj < R define fn .z/ D

zn . We have Rn

lim kfn ıukAp D 0 and therefore

1=n

lim kun kAp R:

n!1

Remarks. 1: This proof is directly inspired from Zygmund .[121], Vol. I, Chap. VI .5:8/ Theorem, p. 246/. 2: For p D 2 and G abelian see Hewitt and Ross .[67], Chap. X, Sect. 39, p. 519/.

Chapter 5

CV p .G / as a Module on Ap .G /

By the map .f; g/ 7! f g, L1 .G/ is a left module on L1 .G/. Similarly for u 2 Ap .G/ and T 2 CV p .G/ we define a convolution operator uT . With the map .u; T / 7! uT CV p .G/ is a left module on Ap .G/. For f 2 L1 .G/ and g 2 L1 .G/ the function f g is more regular than the function g. Similarly the convolution operator uT is a smoothing of the convolution operator T . In particular uT 2 PM p .G/. For G amenable we obtain a generalization of the approximation theorem for CV p .G/.

5.1 The Convolution Operator .k lL/T Lemma 1. Let G be a locally compact group, 1 < p < 1, ' 2 M1 00 .G/ and p p L k 2 L .G/. Then the map G ! L .G/, t 7! t 1 .k/' is continuous, and we have 0 11=p Z p L @ Np t 1 .k/' dt A D Np .k/Np .'/: G

Lemma 2. Let G be a locally compact group, 1 < p < 1, k 2 Lp .G/, l 2 0 Lp .G/, '; 2 M1 00 .G/ and T 2 CV p .G/. Then the function D E L F W G ! C; t 7! T Œ t 1 .k/'; Œ t 1 .lL/ belongs to L1C .G/ \ C.G/, and N1 .F / jjjT jjjp Np .k/Np0 .l/Np .'/Np0 . /.


65

5 CV p .G/ as a Module on Ap .G/

66

Proof. By Lemma 1 F is clearly continuous, and Z

Z jF .t/jdt jjjT jjjp

G

Np

t 1

kL ' Np0 1 lL dt t

G

0 jjjT jjjp @

Z Np

11=p 0 11=p0 Z p 0 p L dt A @ Np0 1 lL dt A t 1 k ' t

G

G

D jjjT jjjp Np .k/Np0 .l/Np .'/Np0 . /: Proposition 3. Let G be a locally compact group, 1 < p < 1, k 2 Lp .G/, 0 l 2 Lp .G/ and T 2 CV p .G/. There is an unique bounded operator U of Lp .G/ such that D E Z D E L L dt T Œ t 1 .k/'; U.Œ'/; Œ D Œ t 1 .l/ G

for every ';

2

M1 00 .G/.

The operator U is a p-convolution operator and jjjU jjjp 6 jjjT jjjp Np .k/Np0 .l/:

Proof. We verify only that U is a p-convolution operator. Let '; 2 M1 00 .G/ and a 2 G. For t 2 G we have D E D E L a ' ; t 1 .l/ L L a ' ; a1 t 1 .l/ L T t 1 .k/ D 1 T t 1 .k/ a D E D E L L a1 L ; .at /1 .l/ L a1 D T .at /1 .k/' ; a1 . t 1 .l// D T a1 . t 1 .k//' and therefore E Z D D E L ; .at /1 .l/ L a1 T .at /1 .k/' dt U a Œ' ; Œ D G

Z D E L ; t 1 .l/ L a1 dt D T t 1 .k/' G

D D U Œ';

a1 Œ

E E D D a U.Œ' ; Œ :

L . Definition 1. The convolution operator U of Proposition 3 is denoted .k l/T Proposition 4. Let G be a locally compact group and 1 < p < 1. Then: L C .k 0 l/T L , 1: .k C k 0 / lL T D .k l/T L C .k lL0 /T , 2: k .l C l 0 /L T D .k l/T

L 5.1 The Convolution Operator .k l/T

67

L 3: .˛k/ lL T D ˛ .k l/T D .k .˛l/L T D .k lL/.˛T /, L L for k; k 0 2 Lp .G/, l; l 0 2 Lp0 .G/, 4: .k l/.S C T / D .k lL/S C .k l/T S; T 2 CV p .G/ and ˛ 2 C. L has a very concrete description. For T D G ./ the convolution operator .kl/T p

Proposition 5. Let G be a locally compact group, 1 0. Then there is U open neighborhood of e in G such that

2

ˇ ˇˇ ˇD E ˇ ˇ p ˇ .uT /Œ'; Œ u.T G .˛ // Œ'; Œ ˇ 6 " jjjT jjjp ˇ ˇ Z for every T 2 CV p .G/ and every ˛ 2 C00 .G/ with ˛ 0,

˛.x/dx D 1 and G

supp ˛ U .

1 X Proof. Let .kn /; .ln / 2 Ap .G/ with k n lLn D u. There is N 2 N with nD1 1 X

" : Np .kn /Np0 .ln / < 4 1 C N .'/ 1 C Np 0 . / p nD1CN

5 CV p .G/ as a Module on Ap .G/

76

It suffices (taking in account Lemma 2 of Sect. 5.2) to choose an open neighborhood U of e in G such that: Np .kn y 1 kn /
0. hı"G W we have h; T show that for every h 2 L1 .G/ b such that There is K compact subset of G Z jh./jd < O GnK

bk1 / 3.1 C kT

:

b and b There is rZ2 L1 .G/ with rO 2 C00 .G/ r D 1 on K. There is also ' 2 C00 .G/ with ' > 0,

'.x/dx D 1, supp ' U and G

k' r rk1 < : bk1 3 1 C khk1 1 C kT We therefore obtain for every 2 K jb ' ./ 1j < : bk1 3 1 C khk1 1 C kT This implies ' h < h b b k1 1 1 C kT from

Z

b T ./b ' ./h./d D 0

ˇ˝ ˛ˇ ˇ b ˇˇ < ; and ˇ h b ' h; T ˇ˝ ˇ ˇ b˛ˇ we have ˇ h; T ˇ<

GO

˝ ˛ b D 0. Theorem 2 finally implies that x 62 sp T b. and therefore h; T

6.2 Characterization of the Support of a Pseudomeasure In analogy with the property .4/ of Theorem 2 of Sect. 6.1, we get at first a characterization of the support of a pseudomeasure. Theorem 1. Let G be a locally compact group, 1 < p < 1, T 2 PMp .G/ and x 2 G. Then x 2 supp T if and˝ only˛ if for every neighborhood V of x there is u 2 Ap .G/ with supp u V and u; T Ap .G/;PMp .G/ 6D 0.

6.2 Characterization of the Support of a Pseudomeasure

89

Proof. Suppose that x 2 supp T . There is U1 ; V1 open subsets of G with e 2 U1 , x 2 V1 , and U11 V1 V . There is also '; 2 C00 .G/ with supp ' U1 , supp ˝ ˛ ˝ ˛ V1 and T Œ'; Œ 6D 0. For u D .p '/ .p / we have u; T Ap ;PMp 6D 0 with supp u V . Suppose conversely ˝ ˛that for every neighborhood V of x there is u 2 Ap .G/ with supp u V and u; T Ap ;PMp 6D 0. Let U; V open subsets of G with e 2 U and x 2 V . Let W an open neighborhood of x relatively compact withW V . Choose ˝ ˛ u 2 Ap .G/ with supp u W u; T Ap ;PMp 6D 0 and .kn /; .ln / 2 Ap .G/ with uD

1 X

k n lLn . There is N 2 N such that

nD1 1 X

" Np .kn /Np0 .ln / < 4 nDN C1 Z There is ' 2 C00 .G/ with ' > 0,

where " D

ˇ˝ ˛ ˇ u; T

Ap ;PMp

ˇ ˇ

2 jjjT jjjp

:

'.x/dx D 1, supp ' U 1 and

G

Np ' kn kn
0. There is g 2 C00 .G/ with " Np .' ' g/ < 1 C jjjT jjjp 1 C Np0 .k/ and supp g U . We have h i h i

.T Œ'/mG .k/ D T ' ; k :

6.3 A Generalization of Wiener’s Theorem

93

But ˇ ˇ ˇ ˇ ˇ h i h i ˇ ˇ h i h i ˇ ˇ ˇ ˇ ˇ ˇ T ' ; k ˇ 6 jjjT jjjp Np .' ' g/Np0 .k/ C ˇ T ' g ; k ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ h i h i ˇ ˇ ˇ < " C ˇ T g ; ' k ˇ: ˇ ˇ From supp ' k .supp '/1 W V we get h i h i

T ' ; ' k D 0;

ˇ ˇ ˇ h i h i ˇ ˇ ˇ T Œ' mG .k/ D 0: ˇ T ' ; k ˇ < " and ˇ ˇ

This implies x … supp.T Œ'/mG and finally supp T Œ' mG D ;. Remark. For p D 2 and G abelian, this result is precisely the dual formulation of Wiener’s theorem .see Theorem 3, Sect. 6.1 and [105], Proposition 7.1.9, p. 196/. Proposition 4. Let G be a locally compact group, 1 < p < 1, T 2 CV p .G/ and u 2 Ap .G/. Suppose that u D 0 on a neighborhood of supp T . Then uT D 0. Proof. We have supp T \ supp u D ;, supp uT D ; (Theorem 2 of Sect. 6.2) and therefore uT D 0. Proposition 5. Let G be a locally compact group, 1 < p < 1, T 2 CV p .G/ with compact support and u 2 Ap .G/. Suppose that u D 1 on a neighborhood of supp T . Then uT D T . Proof. Let v be an arbitrary element of Ap .G/. By Proposition 4 .uv v/T D 0 and therefore v.uT T / D 0. Theorem 2 of Sect. 6.2 implies supp.uT T / D ; and thus uT T D 0. Corollary 6. Let G be a locally compact group, 1 < p < 1 and T 2 CV p .G/ with compact support. Then T 2 PMp .G/. Proof. Let U be a compact neighborhood of supp T and u 2 Ap .G/ \ C00 .G/ with u.x/ D 1 for every x 2 U . We have T D uT , but according to Theorem 7 of Sect. 5.2, uT is a p -pseudomeasure. Proposition 7. Let G be a locally compact group, 1 < p 1, T 2 CV p .G/ and x 2 G. The x 2 supp T if and only if for every u 2 Ap .G/ with uT D 0 we have u.x/ D 0. Proof. Let x 2 supp T and u 2 Ap .G/ with uT D 0. Suppose that u.x/ 6D 0. By the Theorem 2 of Sect. 6.2 we have x 2 supp uT and therefore uT 6D 0. Consequently u.x/ D 0. Conversely suppose that for every u 2 Ap .G/ with uT D 0 we have u.x/ D 0. Assume that x … supp T . Let U be an open relatively compact neighborhood of x with U \ supp T D ;. Choose v 2 Ap .G/ with v.x/ D 1 and supp v U . Then supp vT D ; and therefore vT D 0. This implies u.x/ D 0. Remarks. 1: This generalizes the property .2/ of Theorem 2, Sect. 6.1. 2: For G a general locally compact group and p D 2 this result is due to Eymard .see [41], p. 225 (4.4) Proposition/. 3: For 1 < p < 1 see Herz [61], p. 119.

6.4 An Approximation Theorem Lemma 1. Let G be a locally compact group, 1 < p < 1, ' 2 Lp .G/ and T 2 CV p .G/. Suppose that supp ' mG is compact. Then supp.T '/mG supp ' mG supp T . Proof. Let x … supp ' mG supp T . By Lemma 2 of Sect. 6.3 ˝ there ˛is U and V open subsets of G with e 2 U , .supp ' mG /1 x V and T Œr; Œs D 0 for every r; s 2 C00 .G/ with supp r U , supp s V . There is Z open neighborhood of e with .supp ' mG /1 xZ V . Let f 2 C00 .G/ with supp f W where W D xZ. Let " > 0. There is g 2 C00 .G/ with supp g U and " : k' ' Œgkp < 1 C jjjT jjjp 1 C Np0 .f / According to Lemma 1 of Sect. 6.3 we have ' 2 L1 .G/ and therefore

and

h i

h i h i T .' Œg/; f D T g ; ' f

6.4 An Approximation Theorem

95

ˇ ˇ ˇ ˇ ˇ ˇ h i ˇ h i h i ˇ ˇ ˇ ˇ ˇ ˇ T .'/; f ˇ < " C ˇ T .' g /; f ˇ: ˇ ˇ ˇ ˇ h i h i There is r 2 C00 .G/ with r D ' f . We have supp r V and therefore h i h i

T ' g ; f D 0: This implies ˇ ˇ ˇ h i ˇ ˇ ˇ .T '/mG .f / D 0: ˇ T .'/; f ˇ < " and consequently ˇ ˇ We finally obtain that x … supp.T '/mG . Lemma 2. Let G be a locally compact group, 1 < p < 1, f 2 C00 .G/ and T 2 CV p .G/. Suppose that supp T is compact. Then 1: p T Œf 2 L1C .G/,

p p 2: T G .p f / D G p T Œf , ˇˇˇ ˇˇˇ ˇˇˇ ˇˇˇ p 1=p0 3: ˇˇˇG p T ŒG f ˇˇˇ jjjT jjjp N1 .f /, p p 4: supp G p T Œf supp f supp T , 5: T Œf f 2 C00 .G/. Proof. To verify .2/ and .3/ observe that for ' 2 C00 .G/ p 1=p0 1=p0 1=p0 G p T ŒG f Œ' D Œ' T ŒG f D T Œ' ŒG f 1=p0 p p 1=p0 p but Œ' ŒG f D G f Œ' consequently G p T ŒG f D T G f this indeed implies .2/. We also have p 1=p0 1=p0 G p T ŒG f Œ' jjjT jjjp kŒ' ŒG f kp jjjT jjjp Np .'/N1 .f / p

and therefore .3/. As a direct consequence of Proposition 2 of Sect. 2.1, and of the preceding lemma, we obtain the following approximation theorem. Theorem 3. Let G be a locally compact group, 1 0; f .e/ 6D 0; G .x/1=p f .x/dx D 1 : G

96

6 The Support of a Convolution Operator

Consider on I the filtering partial order f f 0 defined by supp f supp f 0 . Then for every, T 2 CV p .G/ with compact support 1: For every f 2 I we have ˇˇˇ ˇˇˇ ˇˇˇ ˇˇˇ p ˇˇˇG p T Œf f ˇˇˇ 6 jjjT jjjp I p

2: The net p G

p T Œf f

! f 2I

converges strongly to T ; 3: For every " > 0, for every ' 2 Lp .G/ and for every neighborhood U of supp T there is f0 2 I such that for every f 2 I with f0 f we have p T ' G p T Œf f ' < " p

p and supp G p T Œf f U . Remark. Theorem 3 is due to Herz .[61], Proposition 9, p. 117/. Corollary 4. Let G be a locally compact group, 1 < p < 1, T 2 CV p .G/ with compact support and U a neighborhood of supp T . Then there is a net .f˛ / of C00 .G/ such that: 1: 2: 3:

p

ˇlim ˇˇ G .fˇ˛ˇˇ/ D T ultraweakly, ˇˇˇp .f˛ /ˇˇˇ jjjT jjj for every ˛, p G p supp f˛ U for every ˛.

6.5 Application to Amenable Groups From Corollary 7 of Sect. 4.1 and from Corollary 3 of Sect. 5.4 we know that every p T 2 CV p .G/ is in the ultraweak closure of G .C00 .G// if G is amenable. In this paragraph we will obtain a more precise result. Lemma 1. Let G be a locally compact group, 1 0 and ' 2 Lp .G/. Then there is V , a neighborhood Z of e contained in U , such 0

f .x/G .x/1=p dx D 1,

that for f 2 C00 .G/ and T 2 CV p .G/ with f 0 G

supp f V one has p

kT ' T G .p f /'kp jjjT jjj ":

6.5 Application to Amenable Groups

97

Proof. A minor change to the proof of Proposition 2 of Sect. 2.1 justifies this lemma. The following lemma is a consequence of Sects. 5.4 and 6.2. Lemma 2. Let G be a locally compact amenable group, 1 0. Then there is k; l; f 2 C00 .G/ with Z 1: Np .k/ D Np0 .l/ D k.x/l.x/dx D 1, k 0, l 0, G

2: Np .x 1 k k/ < "Zand Np0 .x 1 l l/ < " for every x 2 K; 0

f .x/G .x/1=p dx D 1;

3: f 0, f .e/ 6D 0, G

L U. 4: supp f V and .supp f /2 supp.k l/T Definition 1. Let G be a locally compact amenable group, 1 0, k; l; f 2 C00 .G/ with following properties: Z Np .k/ D Np0 .l/ D

k.x/l.x/dx D 1; k 0; l 0 G

Np .x 1 k k/ < "; Np0 .x 1 l l/ < " for every x 2 K; Z 0 L U: f 0; f .e/ 6D 0; f .x/G .x/1=p dx D 1; .supp f /2 supp.k l/T G

Definition 2. Let G be a locally compact amenable group, 1 < p < 1, T 2 CV p .G/, U a neighborhood of supp T and .K; "; k; l; f /; .K 0 ; "0 ; k;0 l 0 ; f 0 / 2 JU . We say that .K; "; k; l; f / .K 0 ; "0 ; k;0 l 0 ; f 0 / if K K 0 , " "0 and supp f supp f 0 . Theorem 3. Let G be a locally compact amenable group, 1 < p < 1, T 2 CV p .G/ and U a neighborhood of supp T . Then 1: is a filtering partial order on the set JU ; 2: For every .K; "; k; l; f / 2 JU we have ˇˇˇ ˇˇˇ ˇˇˇ p L Œf f ˇˇˇˇˇˇ 6 jjjT jjj I ˇˇˇG p .k l/T p p

p L Œf f U for every .K; "; k; l; f / 2 JU ; 3: supp G p .k l/T 4: In the ultraweak topology we have lim

.K;";k;l;f /2JU

p G

L p .k l /T Œf f D T:

98

6 The Support of a Convolution Operator

Proof. The claim .1/ is a consequence of Lemma 2, while .2/ and .3/ result from Theorem 3 of the preceding paragraph. To prove .4/ it’s enough to show that * lim

.K;";k;l;f /2JU

p G

+ ˝ ˛ L p .k l/T Œf f Œ'; Œ D T Œ'; Œ

for '; 2 C00 .G/. By Theorem 2 of Sect. 5.4 for every " > 0 there is K0 compact subset of G and 0 > 0 with the following property: let K be a compact subset of G with K0 K, let 0 < 0 and let k; l 2 C00 .G/ with Z Np .k/ D Np 0 .l/ D

k.x/l.x/dx D 1; k 0; l 0; Np .x 1 k k/ < ;

Np 0 .x 1 l l/ <

G

for every x 2 K, then we have ˇ˝ ˛ ˝ ˛ˇˇ " ˇ ˇ .k lL/T Œ'; Œ T Œ'; Œ ˇ < : 2 By Lemma 1 there is a neighborhood V of e such that for every f 2 C00 .G/ with Z 0

f .x/G .x/1=p dxD1 we have

supp f V , f 0 and G

p L L Œ' .k l/T G .p f /Œ' < k l/T p

" 2.1 C Np0 . // Z

for every k; l 2 C00 .G/ with Np .k/ D N .l/ D 1; k 0; l 0; p0

k.x/l.x/

dx D 1. By Lemma 2 there is k0 ; l0 ; f0 2 C00 .G/ such that .K0 ; 0 ; k0 ; l0 ; f0 / 2 JU . Then for every .K; ; k; l; f / 2 JU with .K0 ; 0 ; k0 ; l0 ; f0 / .K; ; k; l; f / we clearly have ˇ ˇ* + ˇ ˝ ˛ˇˇ ˇ p L Œf f Œ'; Œ T Œ'; Œ ˇ < ": ˇ G p .k l/T ˇ ˇ Corollary 4. Let G be a locally compact amenable group, 1 < p < 1, T 2 CV p .G/ and U a neighborhood of supp T . Then there is a net .f˛ /˛2J of C00 .G/ such that: 1: 2: 3:

p

ˇlim ˇˇ G .fˇ˛ˇˇ/ D T ultraweakly, ˇˇˇp .f˛ /ˇˇˇ jjjT jjj for every ˛, p G p supp f˛ U for every ˛.

6.5 Application to Amenable Groups

99

Proof. Let JU be the set of Definition 1 with the partial order of Definition 2. For L Œf f . According to Lemma ˛ D .K; "; k; l; f / 2 JU we put f˛ D p .k l/T 2 of Sect. 6.4 we have precisely f˛ 2 C00 .G/. Scholium 5. Let G be a discrete amenable group, 1 < p < 1 and T 2 CVp .G/. Then there is a net .f˛ / of C00 .G/ such that: 1: 2: 3:

p

ˇlim ˇˇ G .fˇ˛ˇˇ/ D T ultraweakly, ˇˇˇp .f˛ /ˇˇˇ jjjT jjj for every ˛, p G p supp f˛ supp T for every ˛.

Remarks. 1: Corollary 4 (and Corollary 4 of Sect. 6.4) were obtained by Herz [61], p. 117. 2: There is no hope to extend the Scholium 5 to non-discrete groups! In fact let G be a non-discrete locally compact abelian group, then there is always a T 2 CV 2 .G/ 2 which is not an ultraweak limit of a net G .˛ / with ˛ a measure with finite support contained in supp T [91]. 3: For results on the approximation of T 2 CV p .G/ by finitely supported measures see Lohoué [80], Théorème II, p. 82, [79], Corollaire 2, [81], Théorème 1, Meyer [94], p. 665, Delmonico [25], Corollary 3.3 and [26], Derighetti [33], Theorem 9.

•

Chapter 7

Convolution Operators Supported by Subgroups

Let H be a closed subgroup of G. We investigate the relations between C Vp .H / b/ and C Vp .G/ and obtain noncommutative analogs of the relations between L1 .H 1 b and L .G/. We prove that ResH Ap .G/ D Ap .H /. We also generalize to noncommutative groups the Theorem of Kaplansky–Helson and basic results on sets of synthesis.

7.1 The Image of a Convolution Operator Let G be a locally compact group and H a closed subgroup. For x 2 G we put xP D xH D !.x/ and a xP D .ax/Pfor a 2 G. For the following we refer to Chap. 8 of [105]. There is q 2 C.G/ with q.x/ > 0, q.xh/ D q.x/H .h/G .h/1 for every x 2 G and every h 2 H . Let mH beZ a left invariant Haar measure of the locally compact group H . We set mH .f / D

f .h/dh for f 2 C00 .H /. H

Definition 1. For f 2 C00 .G/ and x 2 G we put Z f .xh/ P D dh TH;q f .x/ q.xh/ H

Z

and TH f .x/ P D

f .xh/dh: H

Theorem 1. There is a unique Radon measure on the locally compact space G=H , such that mG .f / D .TH;q f / for every f 2 C00 .G/. A. Derighetti, Convolution Operators on Groups, Lecture Notes of the Unione Matematica Italiana 11, DOI 10.1007/978-3-642-20656-6 7, © Springer-Verlag Berlin Heidelberg 2011

101

102

7


Remark. The measure depends on the choice of q. Definition 2. The unique measure of Theorem 1 is denoted mG=H . For f 2 C00 .G=H / we put Z mG=H .f / D

f .x/d P q x: P G=H

We also put mG=H D

mG : mH

If q D 1G .for instance if H is normal in G or if H is compact/ we set dq xP D d x. P Theorem 2. Let G be a locally compact group, H a closed subgroup, ! the canonical map of G onto G=H , q 2 C.G/ with q.x/ > 0 q.xh/ D q.x/H .h/G .h/1 for every x 2 G and every h 2 H , mG a left invariant Haar measure on G, mH a left invariant Haar measure on H and mG=H such that mG=H D

mG : mH

Then for every f 2 L1 .G/ there is a unique g 2 L1.G=H I mG=H / such that !.f mG / D gmG=H . If f 2 C00 .G/ we have g D TH;q f . This permits us to complete Definition 1. Definition 3. We put g D TH;q Œf . Theorem 3. 1: TH;q is a contractive epimorphism of the Banach space L1 .G/ onto the Banach space L1 .G=H I mG=H /, 2: if H is normal in G then TH is contractive epimorphism of the involutive Banach algebra L1 .G/ onto the involutive Banach algebra L1 .G=H /. Definition 4. For every x; y 2 G we put .x; y/ P D

q.xy/ . q.y/

Proposition 4. For every f 2 C00 .G=H / and every y 2 G we have Z

Z f .x/d P q xP D G=H

f .y x/.y; P x/d P q x: P G=H

Corollary 5. Let G be a locally compact group and H a closed normal subgroup of G. Then mG=H is a left invariant Haar measure of the locally compact group G=H . Definition 5. Let G be a locally compact group and H a closed subgroup of G. We say that ˇ 2 C.G/ is a Bruhat function for H if:

7.1 The Image of a Convolution Operator

103

i. for every compact subset K of G there is r 2 C00 .G/ with r.x/ 0 for every x 2 G and ResKH ˇ D ResKH r, Z ˇ.xh/dh D 1 for every x 2 G.

ii. H

Theorem 6. Let G be a locally compact group and H a closed subgroup of G. Then there is a Bruhat function for H . Proposition 7. Let G be a locally compact group, H a closed subgroup of G and ˇ a Bruhat function for H . If we put Z q.x/ D

ˇ.xh/G .h/H .h1 /dh; x 2 G

H

we have: 1: q 2 C.G/, 2: q.x/ > 0 for x 2 G, 3: q.xh/ D q.x/H .h/G .h1 / for x 2 G and for h 2 H . Lemma 8. Let G be a locally compact group, H a closed subgroup of G, p 1, f 2 C00 .G/ and " > 0. There is U open neighborhood of e in G such that 0 @

Z

11=p jf .y 1 xh/ f .xh/jp dhA

1, '; 2 C00 .G/ and " > 0. Then there is U open neighborhood of e in G such that ˇ* " # " #+ˇ # " #+ * " ˇ ˇ ' ' ˇ ˇ ˇ T ; T ; ˇ " jjjT jjjp 0 0 ˇ q 1=p x;H q 1=p x;H q 1=p y 1 x;H q 1=p y 1 x;H ˇ

for every y 2 U , for every x 2 G and for every T 2 C Vp .H /. Proposition 10. Let G be a locally compact group, H a closed subgroup of G, p > 1, '; 2 C00 .G/ and 2 M 1 .H /. The function * xP 7!

p H ./

"

' q 1=p

# " ;

x;H

#+

q 1=p0

x;H

is well defined and continuous with compact support on G=H . Let i./ be the image of the measure under the inclusion map i of H in G. Then ˝ p ˛ G .i.//Œ'; Œ D

" # " #+ Z * ' p P H ./ ; dq x: 0 q 1=p x;H q 1=p x;H

G=H

Proof. We have by Sect. 1.2 ˛ ˝ p G .i.//Œ'; Œ D

Z

1=p0 ' G .i.//L .x/ .x/dx

G

1=p0 where ' G .i.//L Z

2 C00 .G/ and therefore

1=p0 ' G .i.//L .x/ .x/dx D

G

Z G=H

0 1 Z 1=p0 ' G .i.//L .xh/ .xh/ @ P dhA dq x: q.xh/ H

But for every x 2 G and every h 2 H we have 1=p0 ' G .i.//L .xh/ D

Z G

'.xhy/G .y/1=p d.i.//.y/ D

Z

'.xh/G ./1=p d./

H

and therefore ! 1=p0 ' G .i.//L .xh/ ' 1=p0 D H L .h/: q.xh/1=p q 1=p x;H

7.1 The Image of a Convolution Operator

105

This finally implies Z Z 1=p 0 ' G .i.//L .xh/ .xh/ dh D q.xh/

H

q 1=p

H

*

D

'

p H ./

x;H

"

'

1=p0 H L

# " ;

q 1=p

!

x;H

.h/

q

0

x;H

#+

q 1=p

.h/dh

1=p0

: x;H

Lemma 11. Let G be a locally compact group, H a closed subgroup of G, p > 1, '; 2 C00 .G/ and T 2 C Vp .H /. Then ˇ Z ˇ ˇ ˇ ' ˇ ˇ T ; dq xP ˇ jjjT jjjp Np .'/Np0 . /: ˇ 0 1=p 1=p ˇ ˇ q q x;H x;H G=H

Proof. We have ˇ ˇ ˇ Z * ˇ + ˇ ˇ ' ˇ ˇ T dq xP ˇ ; ˇ 0 1=p 1=p ˇ ˇ q q x;H x;H ˇG=H ˇ

Z jjjT jjjp Np

B jjjT jjjp @

Z

x;H

Np

G=H

' q 1=p

Np0

q 1=p

G=H

0

!

'

!p

x;H

q 1=p0 11=p0

C dq xPA

!

'

B @

dq xP x;H

Z

G=H

Np0

' q 1=p0

x;H

p 0

11=p0 C dq xPA

:

But

Z Np G=H

' q 1=p

!p

Z dq xP D

x;H

G=H

0 @

Z

H

1 j'.xh/jp dhA dq xP D Np .'/p : q.xh/

Proposition 12. Let G be a locally compact group, H a closed subgroup of G, p > 1, and T 2 C Vp .H /. Then there is an unique bounded operator S of Lp .G/ such that + Z * ˝ ˛ ' S Œ'; Œ D dq xP ; T q 1=p x;H q 1=p0 x;H G=H

for every ';

2 C00 .G/. We have jjjS jjjp jjjT jjjp .

106

7


Definition 7. The operator S of Proposition 12 is called the image of T under the map i . We set S D i.T /. p

p

Remark. According to Proposition 10 we have i.H .// D G .i.// for 2 M 1 .H /. Theorem 13. Let G be a locally compact group, H a closed subgroup of G and p > 1. Then i is a linear contractive map of the Banach space C Vp .H / into C Vp .G/. Proof. It is enough to prove that i.C Vp .H // C Vp .G/. Let T 2 C Vp .H /; '; C00 .G/, a 2 G and * #+ a' ; : f .x/ P D T q 1=p x;H q 1=p0 x;H By Proposition 4 we have ˛ ˝ i.T / a Œ'; Œ D

Z

.a1 ; x/f P .a1 x/d P q x: P

G=H

From f .a

1

a' ; ; x/ P D T q 1=p a1 x;H q 1=p0 a1 x;H 1=p q.x/ ' a' D and q 1=p a1 x;H q.a1 x/ q 1=p x;H 1=p0 q.x/ a1 D q 1=p0 a1 x;H q.a1 x/ q 1=p0 x;H

we get * + ' 1 a T ; f .a1 x/ P D .a1 ; x/ P q 1=p x;H q 1=p0 x;H 1

and therefore D

+ Z * E ' a1 i.T / a Œ'; Œ D dq xP ; T q 1=p x;H q 1=p0 x;H G=H

E D E D D i.T /Œ'; Œ a1 D a i.T /Œ' ; Œ :

2

7.2 On the Operator i.T /

107

7.2 On the Operator i.T / The relation ˝ ˛ i.T /Œ'; Œ D

Z ' T ; dq xP 0 q 1=p x;H q 1=p x;H

G=H

was obtained in the preceding paragraph for '; 0 tion to ' 2 Lp .G/ and 2 Lp .G/.

2 C00 .G/. We need a generaliza-

Definition 1. Let X be a topological space. Then T C .X / denotes the set n

ˇ ˇ f 2 Œ0; 1X ˇf

o is lower semi-continuous on X :

Proposition 1. Let G be a locally compact group, H a closed subgroup of G and f 2 T C .G/. Then: Z

1: xP 7! Z 2: G=H

f .xh/ dh is in T C .G=H /, q.xh/ H ! Z Z f .xh/ dh dq xP D f .x/dx. q.xh/ H

G

C .G/; ' C00

Proof. Let A D f'j' 2 x 2 G we have consequently

f q

' '2A q x;H

D sup x;H

f g. We recall that f D supf'j' 2 Ag, for

and therefore

f q

x;H

ˇ ˇ ' ˇ' 2 A q x;H ˇ

The set

being filtering we have mH

f q

! P D sup.TH;q '/.x/: '2A

x;H

Let

Z F .x/ P D H

f .xh/ dh q.xh/

2 T C .H /:

108

7


we have F D sup TH;q '; F 2 T C .G=H /

and finally mG=H .F / D sup m.'/ D m .f /:

'2A

'2A

Remark. See Reiter and Stegeman .[105]. see p. 230/. Corollary 2. Let G be a locally compact group, H a closed subgroup of G and f an arbitrary map of G into Œ0; 1. Then: Z

mH

G=H

f q

!

x;H

dq xP mG .f /:

Proof. Let g 2 T C .G/ with f g. By Proposition 1 Z

mH

G=H

f q

x;H

! dq xP mG .g/:

This implies Z G=H

mH

f q

!

ˇ n o ˇ dq xP inf mG .g/ˇg 2 T C .G/; f g D mG .f /: x;H

The following theorem is a generalization of Theorem 1 of Sect. 7.1. Theorem 3. Let G be a locally compact group, H a closed subgroup of G and f 2 L1 .G/. Then there is a mG=H -negligible subset A of G=H with the following properties: f 2 L1 .H I mH /, 1: For every x 2 G with xP 62 A we have q x;H 2: The function g, defined by g.x/ P D 0 if xP 2 A and by g.x/ P D mH

f q

!

x;H

if xP 62 A, belongs to L1 .G=H / and we have mG=H .g/ D mG .f /. Proof. There is .kn / a sequence of C00 .G/ with Z jf .x/ kn .x/jdx < G

1 2n

7.2 On the Operator i.T /

109

for every n 2 N. Corollary 2 implies that Z G=H

0 @

Z

H

1 jf .xh/ kn .xh/j A 1 dh dq xP < n q.xh/ 2

for n 2 N. We set for every x 2 G and for every n 2 N Z Hn .x/ P D H

jf .xh/ kn .xh/j dh; q.xh/

we have Hn 2 Œ0; 1G=H . Then 1 X

mG=H

! Hn

nD1

1 X 1 D 1: n 2 nD1

There is A, mG=H -negligible subset of G=H , such that for every xP 62 A 1 X

Hn .x/ P < 1:

nD1

For xP … A we have Z H

jf .xhj dh < 1 q.xh/

this implies

f q

x;H

ˇ Z ˇˇ ˇ kn ˇ ˇ f .h/ .h/ˇdh D 0; lim ˇ n!1 ˇ q x;H q x;H ˇ

and

H

2 L1 .H /.

P D 0 if xP 2 A and by Let g 2 RG=H defined by g.x/ Z g.x/ P D H

f .xh/ dh q.xh/

if xP 62 A. For xP 62 A we have jg.x/ P TH;q kn .x/j P Hn .x/ P and therefore

1 mG=H jg TH;q kn j mG=H .Hn / < n : 2

110

7


This implies that g 2 L1 .G=H /. Finally for n 2 N we have ˇ ˇ ˇ ˇ Z ˇ ˇ ˇ ˇ ˇmG=H .g/ f .x/dx ˇ ˇmG=H .g/ mG=H TH;q kn ˇ ˇ ˇ ˇ ˇ G

ˇ ˇ ˇ ˇ ˇ CˇmG=H TH;q kn mG .kn /ˇˇ C jmG .kn / mG .f /j mG=H jg TH;q kn j C mG .jkn f j/
0. There is k 2 C00 .G/ and f 2 C00 .H / with Np ' q 1=p .k H f /
n0 we have ˇ˝ ˇ ˇ T Œ' r; Œ

˛ˇˇ s ˇ jjjT jjjp k'ku k ku Np .rn r/Np0 .s/ C jjjT jjjp k'ku k ku Np0 .sn s/ Np .r/ C 1 :

Theorem 3. Let G be a locally compact group, H a closed subgroup and p 2 .1; 1/. Then for every T 2 C Vp .H / we have supp i.T / D supp T . Proof. 1. supp i.T / H . Let x0 2 G n H . There is U and V open subsets of G=H with U \ V D ;, !.e/ 2 U , and !.x0 / 2 V . Let U1 D ! 1 .U / and V1 D ! 1 .V /. Then U1 \V1 D ;. Let '; 2 C00 .G/ with supp ' U1 and supp V1 . For x 2 G n V1 we have ! q 1=p

D0 x;H

and therefore ˝

i.T /Œ' ; Œ

˛ D

Z V1

* " # " #+ ' ˇ.x/q.x/ T ; dx: q 1=p x;H q 1=p0 x;H

7.4 The Support of i.T /

121

But if x 2 V1 then x … U1 and therefore !

' q 1=p

D0 x;H

˝ ˛ consequently i.T /Œ' ; Œ D 0. This implies that x0 … supp i.T /. 2. supp i.T / supp T . Let h0 2 H˝ with h0 … ˛ supp T . There is V0 ; V1 open subsets of H with e 2 V0 , h0 2 V1 and T Œ'; Œ D 0 for every '; 2 C00 .H / with supp ' V0 and supp V1 . There is U0 ; U1 open subsets of G with U0 \ H D V0 and U1 \ H D V1 . There is U2 open neighborhood of e in G such that U2 D U21 , U22 U0 and U22 h0 U1 . Choose finally U3 , open neighborhood of e in G, with U 3 U2 and U 3 compact. Let '; 2 C00 .G/ with supp ' U3 and supp U3 h0 . For x 2 G n U3 H we have ! ' D0 q 1=p x;H

and therefore ˝ i.T /Œ' ; Œ

* " # " #+ ' ; dx: ˇ.x/q.x/ T q 1=p x;H q 1=p0 x;H

Z

˛ D

U3 H

Let x 2 U3 H . We have x D uh with u 2 U3 and h 2 H . Then * " # " #+ * " # " #+ ' ' T ; D T ; : q 1=p x;H q 1=p0 x;H q 1=p u;H q 1=p0 u;H !

From supp we obtain * " T

'

q 1=p

u;H

' q 1=p

" ;

! V0

u;H

0 q 1=p

and

supp

q 1=p0

#+ D 0 and therefore

V1 u;H

˝ i.T /Œ' ; Œ

˛ D 0;

u;H

consequently h0 … supp i.T /. 3. supp T i.T /. Let h0 2 H˝ with h0 … supp i.T /. There is U0 ; V0 open subsets of G with e 2 U0 , ˛ h0 2 V0 and i.T /Œa; Œb D 0 for every a; b 2 C00 .G/ with supp a U0 and supp b V0 . There is U1 open neighborhood of e in G, V2 open neighborhood of h0 in G such that U12 U0 and U1 V2 V0 . Let W1 D U1 \ H and W2 D V2 \ H .

122

7


Consider '; 2 C00 .H / with supp ' W1 and supp W2 . We show that ˝ ˛ T Œ'; Œ D 0. Let " > 0. According to Lemma 1 there is V3 open neighborhood of e in G and k 2 C00 .G/ such that: 1: 2: 3:

V3 U 1 , k.x/ 0 for every x 2 G, supp k U1 , Z

4:

k.xh/dh 1 for every x 2 G, H Z

5:

k.h/dh D 1,

H ˇ ˇ " # " #+ 0 6: ˇ* ˝ ˛ˇˇ 1V3 H q 1=p .k H '/ 1V3 H q 1=p .k H / ˇ ; T Œ'; Œ ˇ ": ˇ i.T / 0 ˇ ˇ mG=H .!.V3 //1=p mG=H .!.V3 //1=p

But supp k H ' U0 and supp k H rD

q 1=p 1V3 H \supp kH ' mG=H .!.V3 //1=p

Lemma 2 implies

D

V0 . Let 0

and s D

q 1=p 1V3 H \supp kH

i.T / r.k H '/ ; s.k H

mG=H .!.V3 //1=p

/

E

0

:

D0

and therefore ˇ˝ ˛ ˇˇ ˇ ˇ T Œ'; Œ ˇ < ":

Consequently

˝

˛ T Œ'; Œ D 0:

We obtain that h0 … supp T . Remark. This result is due to Anker .[2], p. 631 and [1], Lemme IV.6, p.109/.

7.5 Theorems of de Leeuw and Saeki In this paragraph, G is a locally compact abelian group, H an arbitrary closed subgroup, mG a Haar measure on G, mH a Haar measure on H , ! the canonical mG . For 2 H ? we set ./.!.x// D .x/ map of G onto G=H and mG=H D mH where x 2 G. The dual map b ! is a topological isomorphism of G=H onto H ? . Let ? b onto G=H b . Then the dual map b is a topological be the canonical map of G ? ?? b isomorphism of G=H onto H . For h 2 H we set ..h/..// D .h/ for b Finally for 2 G b we also set ..// D ResH . By Hewitt and every 2 G.

2

1

7.5 Theorems of de Leeuw and Saeki

123

Ross ([67], p. 242–246) we have 1 .mHO / D

where mH ? D b ! mG=H :

mGO mH ?

b

b

Observe that for 2 M 1 .H / we have i./ D b ı ı . We show that this relation is verified for every T 2 C Vp .H /. Theorem 1. Let G be a locally compact abelian group, 1 < p < 1, H a closed T . Then subgroup of G, T 2 C Vp .H /, f 2 CGO , g 2 CHO with fP D i.T / and gP D b b f ./ D g. ../// mGO -locally almost everywhere on G.

b

2 C00 .G/. We have

Proof. Let ';

E Z ˝ ˛ D ' ./b ./d mGO . /: i.T /Œ'; Œ D .i.T //F Œ'; F Œ D f ./b

1

GO

On the other hand ˝

Z

˛ i.T /Œ'; Œ D

˝ T Œ'x;H ; Œ

˛

x;H

d mG=H .!.x//:

G=H

For every x 2 G we have ˝

˛

T Œ'x;H ; Œ

x;H

Z

b b ./d m

D

g. /'x;H ./

x;H

HO . /;

HO

consequently ˝ ˛ i.T /Œ'; Œ D

Z HO

0 B @

Z

b b ./d m

g. /'x;H ./

x;H

1 C

G=H .!.x//A d mHO . /:

G=H

b we have But for every x 2 G and every 2 H 0 Z Z 'x;H ./ x;H ./ D .h0 / @ 'x;H .h/

b b

H

1 0 0 x;H .hh /d mH .h/A d mH .h /;

H

and thus ˝ ˛ i.T /Œ'; Œ Z Z

Z g./

D HO

G=H H

0 Z .h / @ 'x;H .h/ 0

H

1 x;H

.hh0 /d m

A d mH .h0 /d mG=H .!.x//d m O ./

H .h/

H

124

7 Z

Z

D

.h0 /

g./ HO

H

Z

Z

g./ @

D HO

0 0 A x;H .hh /d mH .h/ d mG=H .!.x//d mH .h /d mHO ./

H

G

0

Z

1

0 1 Z .h0 / @ '.x/ .xh0 /d mG .x/A d mH .h0 /d mHO ./

H

HO

0 Z @ 'x;H .h/

G=H

g./

D

Z


Z

1 .h/.' Q/.h/d mH .h/A d mHO ./ D

H

Z

D

Z

g./ ResH .' Q/ b./d mHO ./

HO

g. ../// ResH .' Q / b. ..///d mG=H ? ..//: O

? O G=H

Taking into account that ResH .' ˝

Z

˛ i.T /Œ'; Œ D

Q/ bı D TH ? .'

g. ../// TH ? .'

Q/bwe get

Q/b ..//d mG=H ? ..// O

? O G=H

Z

D

g. ../// b ' ./b./d mGO ./;

GO

b and therefore f ./ D g. ../// mGO -locally almost everywhere on G. Remarks. 1. For G D R and H D Z this result .together with the isometry of Theorem 2 of Sect. 7.3/ is due to de Leeuw .[74], Theorem 4.5, p. 377 /. 2. Theorem 1 is due to Saeki .[108], Corollary 3.5, p. 417/. 3. For the above proof cf [31] .Proposition 11, p. 60/. Corollary 2. Let G be a locally compact abelian group, H a closed subgroup of G b /. Then sp.gı ı /P D sp gP and k.gı ı /Pk1 D kgk and g 2 L1 .H P 1. b D gP Proof. Let T D ƒGO .g/ P (see Sect. 1.3, Definition 2). We have T 2 C V2 .H /, T 1 b and sp T D .supp T / by Theorem 3 of Sect. 6.1. By Theorem 1 g ı ı 2 i.T /b. The results then follow from supp i.T / D supp T (Theorem 3 of Sect. 7.4), Theorem 1 of Sect. 1.3 and Theorem 2 of Sect. 7.3. Remarks. 1: We have sp.g ı ı /P H . 2: See Reiter and Stegeman [105], Proposition 7.1.20, p. 198. 3: This corollary will be completed in Sect. 7.7.

7.6 The Image of the Map i The essential result of the paragraph and one of the most important in the present book is the Theorem 10 which says that each T 2 C Vp .G/ supported by H is an i.S /.

7.6 The Image of the Map i

125

Proposition 1. Let G be a locally compact group, H a closed subgroup of G, 1 < p p p p < 1 and 2 M 1 .G/ with supp.G .// H . Then G ./ D i.H .ResH //. Proof. We have D 1H D i.ResH /. By the remark after Definition 7 of p p Sect. 7.1, we get G ./ D i.H .ResH //. Remark. For G abelian and p D 2 see Reiter and Stegeman .[105], Proposition 5.4.7, p. 162:/ We intend to generalize Proposition 1 to all convolution operators of G. Definition 1. Let group, we denote by A00 .G/ the linear o n G ˇbe a locally compact ˇ span of the set r s ˇr; s 2 C00 .G/ . Lemma 2. Let G be a locally compact group, 1 < p < 1, T 2 C Vp .G/ and f 2 A00 .G/. Then there is an unique g 2 C.G/ \ Lp .G/ with T Œf D Œg. Proof. It suffices to verify the existence of g. We have f D Pn

T Œsj D

rj sj and therefore

j D1

Pn

'j with 'j 2 T Œsj . Then rj 'j 2 n X C.G/ \ Lp .G/, and so T Œf D Œg with g D rj 'j .

T Œf D

j D1 Œrj

n X

j D1 Œrj

j D1

Definition 2. The function g of Lemma 2 is denoted by aT;f . Lemma 3. Let G be a locally compact group and 1 < p < 1. Then 1: .T; f / 7! aT;f is a bilinear map of C Vp .G/ A00 .G/ into Lp .G/I 2: aT;x f D x .aT;f / for x 2 G. Definition 3. For f 2 L1 .G/, 1 < p < 1 and for every ' 2 Lp .G/ we put Mf Œ' D Œ'f . Proposition 4. Let G be a locally compact group, H a closed subgroup of G, 1 < p < 1, S 2 C Vp .H / and f 2 L1 .G=H; mG=H /. Then i.S /ıMf ı! D Mf ı! ıi.S /. Proof. Let ';

2 C00 .G/. For x 2 G we have

f ı! ' q 1=p

x;H

'x;H D f .!.x// 1=p : q x;H

By Theorem 5 of Sect. 7.2 * " # " #+ Z E D ' ; d mG=H .!.x// f .!.x// S i.S/Mf ı! Œ'; Œ D 0 q 1=p x;H q 1=p x;H G=H

# " #+ Z * " ' D S ; f .!.x// 1=p0 d mG=H .!.x// q 1=p x;H q x;H G=H

126

7


#+ # " Z * " ' f ı! S ; d mG=H .!.x// D 0 q 1=p x;H q 1=p x;H G=H

D D i.S/Œ'; Œf ı!

E D E D E D f ı!i.S/Œ'; Œ D Mf ı! .i.S//Œ'; Œ :

We now need the notion of carrable set. We refer for this notion to Dinculeanu [34], Chap. 5, Sect. 6.1, p. 362. Lemma 5. Let G be a locally compact group, H a closed subgroup of G, 1 < p < 1 and T 2 C Vp .G/ with supp T H . For every open relatively compact and mG=H -carrable neighborhood V of eP we then have T ıM1! 1 .V / D M1! 1 .V / ıT . Proof. Let f D 1! 1 .V / and ' 2 C00 .G/. (I) If supp ' ! 1 .V / then TMf Œ' D Mf T Œ'. We have f ' D '. Let W be an open neighborhood of e in G with .supp '/W ! 1 .V /, and > 0. There is 2 C00 .GI R/ with supp W and Np .' ' We have supp.' supp T .Œ'

: /< 2 1 C jjjT jjjp

/ ! 1 .V /. From Lemma 1 of Sect. 6.4 we get

/mG ! 1 .V /;

and consequently supp aT;' ! 1 .V /:

This implies f aT;' D aT;' From f .'

/D'

and therefore Mf T Œ'

it follows TMf Œ' Mf T Œ'

D T Œ'

D T Œ'

D TMf Œ'

:

and finally

:

Then kTMf .Œ'/ Mf T .Œ'/kp 2 jjjT jjjp Np .' '

/ < :

(II) If .supp '/ \ ! 1 .V / D ; then TMf Œ' D Mf T Œ' D 0. There is Z1 open neighborhood of supp ', with Z1 \ ! 1 .V / D ;. There is Z2 open neighborhood of e in G with .supp '/Z2 Z1 . Let > 0. There is 2 C00 .G/ with supp Z2 and Np .' '

/
0. By Theorem 2 of Sect. 5.4 there is k; l 2 C00 .G/ with Np .k/ D Np0 .l/ D 1 and ˇ ˇ D ˇ Eˇ ˇ ˇ L Œ p '; Œ p0 T Œ p '; Œ p0 ˇ < ": ˇ k l/T ˇ ˇ With the Theorem 1 we obtain ˇD Eˇˇ ˇ ˇ p T p '; Œ ˇ < " C p .k lL/T p ' " C k lL jjjT jjj k'k2 " C jjjT jjj k'k2 : p p ˇ ˇ Ap 2

Definition 1. Let G be a locally compact group and 1 0 such that k p T p 'k2 C k'k2 for every ' 2 Lp .G/ \ L2 .G/. Corollary 3. Let G be a locally compact group and 1 < p < 1. Then: 1: Ep is a subalgebra of C Vp .G/, p 2: G .M 1 .G// Ep , 3: Ap .G/C Vp .G/ Ep .

8 C Vp .G/ as a Subspace of C V2 .G/

158

Lemma 4. Let G be a locally compact group, 1 < p < 1 and T 2 Ep . Then there is an unique operator S 2 L.L2 .G// with S ' D p T p ' for every ' 2 Lp .G/ \ L2 .G/. We have 2 S 2 2 C V2 .G/. Definition 2. Let G be a locally compact group, 1 < p < 1; T 2 Ep and S as in Lemma 4. We put ˛p .T / D 2 S 2 . Theorem 5. Let G be a locally compact group and 1 < p < 1. Then: 1: ˛p is an injective homomorphism of the algebra Ep into C V2 .G/, p 2: ˛p .G .// D 2G ./ for 2 M 1 .G/. In the following two theorems we extend Theorem 4 and Corollary 6 of Sect. 1.5 to the whole class of locally compact amenable groups. Theorem 6. Let G be an amenable locally compact group and 1 < p < 1. Then: homomorphism of the Banach algebra C Vp .G/ into C V2 .G/, 1: ˇ˛ˇˇp is anˇˇinjective ˇ 2: ˇˇˇ˛p .T /ˇˇˇ2 jjjT jjjp for every T 2 C Vp .G/. Theorem 7. Let G be an amenable locally compact group ˇˇˇ ˇˇˇ and ˇˇˇ 1p < p ˇˇˇ < 1. For every bounded Radon measure on G we have ˇˇˇ2G ./ˇˇˇ2 ˇˇˇG ./ˇˇˇp . Theorem 8. Let G be a locally compact group, 1 < p < 1, u 2 A2 .G/ and v 2 Ap .G/. Then: 1: uv 2 Ap .G/ and kuvkAp kukA2 kvkAp , i.e. Ap .G/ is a Banach module on A2 .G/ D E ˝ ˛ for every T 2 PMp .G/. 2: uv; T Ap ;PMp D u; ˛p .vT / A2 ;PM2

Proof. Let .kn /; .ln / 2 A2 .G/ with uD

1 X

kn lLn :

nD1

For every F 2 Ap .G/0 we put !.F / D

1 X

p 1 ˛p v ‰G .F / Œ 2 kn ; Œ 2 ln

nD1 p

where ‰G is defined in Sect. 4.1 (Definition 3). Let F be an element of Ap .G/0 and .Fi /i 2I a net of Ap .G/0 such that lim Fi D F for the topology .A0p ; Ap / and with kFi kA0p C for every i 2 I for some C > 0. We claim that lim !.Fi / D !.F /. By Theorem 6 of Sect. 4.1 we have p 1 p 1 lim ‰G .Fi / D ‰G .F /

8.3 C V2 .G/ and C Vp .G/ in Amenable Groups

159

ˇˇˇ ˇˇˇ ˇˇˇ p 1 ˇˇˇ for the ultraweak topology and moreover ˇˇˇ ‰G .Fi /ˇˇˇ C for every i 2 I . p

This implies that

p 1 p 1 lim v ‰G .Fi / D v ‰G .F /

for the ultraweak topology and that ˇˇˇ ˇˇˇ ˇˇˇ ˇˇˇ ˇˇˇ p 1 ˇˇˇ ˇˇˇ ˇˇˇ p 1 ˇˇˇv ‰G .Fi / ˇˇˇ kvkAp ˇˇˇ ‰G .Fi /ˇˇˇ C kvkAp : p

p

1 .G/ we have For r; s 2 M00

p 1 p 1 lim v ‰G .Fi / Œ p r; Œ p0 s D v ‰G .F / Œ p r; Œ p0 s : p 1 p 1 From Œr 2 Lp .G/ \ L2 .G/, v ‰G .Fi / ; v ‰G .F / 2 Ep we obtain + * + p 1 p 1 lim ˛p v ‰G .Fi / Œ 2 r; Œ 2 s D ˛p v ‰G .F / Œ 2 r; Œ 2 s *

with for every i 2 I ˇˇˇ ˇˇˇ ˇˇˇˇˇˇ ˇˇˇ ˇˇˇ ˇˇˇ˛p v ‰ p 1 .Fi / ˇˇˇ ˇˇˇˇˇˇv ‰ p 1 .Fi / ˇˇˇˇˇˇ C kvkA : p G G ˇˇˇ ˇˇˇ p 2

Consequently p 1 p 1 lim ˛p v ‰G .Fi / D ˛p v ‰G .F / for the ultraweak topology and consequently lim !.Fi / D !.F /. According to Theorem 6 of Sect. 4.2 there is w 2 Ap .G/ with ˝ !.F ˛ / D F .w/ ˝ for every ˛ F 2 Ap .G/0 , for every T 2 PMp .G/ we thus have w; T Ap ;PMp D u; ˛p .vT / A2 ;PM2 .

In particular (taking into account Proposition 5 of Sect. 5.3) for every 2 M 1 .G/ ˛ ˝ p p w; G ./ Ap ;PMp D u; ˛p vG ./

A2 ;PM2

D E D u; 2G .Qv/

A2 ;PM2

and therefore w D uv. This implies indeed uv 2 Ap .G/. Moreover we have for every T 2 PMp .G/ ˇ˝ ˛ ˇ uv; T

Ap ;PMp

ˇ ˇˇ˝ ˛ ˇ D ˇ u; ˛p .vT /

whence kuvkAp kukA2 kvkAp .

ˇ ˇ kukA2 jjj˛.vT /jjj2 kukA2 kvkAp jjjT jjjp ; A2 ;PM2 ˇ

8 C Vp .G/ as a Subspace of C V2 .G/

160

Theorem 9. Let G be an amenable locally compact group and 1 < p < 1. Then: 1: A2 .G/ Ap .G/, 2: For every u 2 A2 .G/ one has kukAp kukA2 . Proof. Let u 2 A2 .G/ and .kn /; .ln / 2 A2 .G/ with uD

1 X

kn lLn :

nD1

For every F 2 Ap .G/0 we put !.F / D

1 X

p 1 ˛p ‰G .F / Œ 2 kn ; Œ 2 ln :

nD1

There is v 2 Ap .G/ with v 2 Ap .G/ with !.F / D F .v/ for every F 2 Ap .G/0 , as in the proof of Theorem 8 u D v and kukAp kukA2 . Remarks. 1: Theorem 9 generalizes Theorem 7 of Sect. 4.2. 2: Theorem 7 is not trivial even for a finite group! 3: For G amenable, the map ˛p is the adjoint of the inclusion of A2 .G/ into Ap .G/. 4: Theorems 6, 7 and 9 are due to Herz .[59], Theorems B and C, p. 72/. Nota bene, the second sentence of Theorem C should be read: “Dually there is a morphism C Vp .G/ ! C Vq .G/, i.e. convolution operators on Lp .GI C/ are convolution operators on Lq .GI C/ with contraction of norms”. The complete proof of Theorem C is given by Herz and Rivière in [65] .Corollary p. 512/. 5: For G compact and p > 2, Theorems 6 and 7 were obtained earlier .1966/ by and Rider .[47], p. 511 line 2 from below/ with the inequalities ˇFig` ˇˇ a-Talamanca ˇˇˇ ˇˇˇ˛p .T /ˇˇˇ Cp jjjT jjj for T 2 CVp .G/ and kukA Cp kukA for u 2 A2 .G/, p p 2 2 with an explicit constant Cp depending only of p (and not of the group G). Figà-Talamanca and Rider’s contribution is not mentioned in [59] and [65]. 6: The author is grateful to professor Gerhard Racher for the communication of unpublished notes and explanations in relation with Sects. 8.1 and 8.2.

Notes

In these notes we give additional comments and supplementary bibliographical references.

Chapter 1 For the older literature and in particular for C Vp .T / we refer to Chap. 16 of Edwards book on Fourier series [39]. For Rn an important paper on the subject is due to Hörmander [68]. It contains in particular various generalizations to Rn of Marcel Riesz’s Theorem. For p D 1 and G, an arbitrary locally compact group, the space of operators which corresponds to C Vp .G/ is precisely M 1 .G/. This a famous result due to Wendel [117]. For p D 1 the question is more subtle. Our Theorem 5 of Sect. 1.2, is no more true. Indeed, according to Stafney [113] there is on L1 .T / an non-zero linear continuous invariant functional M which is zero on all continuous functions, consequently the operator ' 7! M.'/' does not commute with the action of L1 . The author is indebted to the referee for communication of this fact which was unknown to him. For other informations on the case p D 1 see [73], p. 74–78. To justify the Remark 2 of Theorem 8 in Sect. 1.2 we used the following fact: for amenable groups the convolution norm in Lp of a positive measure is exactly its total mass. One should mention that this result fails for non amenable groups: if for someZ1 < p < 1 the convolution norm in Lp of every positive f 2 L1 .G/ coincides with

f .x/dx then G is amenable ([105], Theorem 8.3.19, p. 241). For specific

non amenable groups more precise results have been obtained. In [85] Lohoué constructed, for any 1 < p0 < 1, a positive measure on SL2 .R/ which convolves Lp0 but does not convolve any other Lp ! Similar results were also obtained latter by Nebbia for the group of isometries of a homogenous tree [97]. According to Leinert

A. Derighetti, Convolution Operators on Groups, Lecture Notes of the Unione Matematica Italiana 11, DOI 10.1007/978-3-642-20656-6, © Springer-Verlag Berlin Heidelberg 2011

161

162

Notes

[75] every l 2 function on F2 supported by a (possibly infinite) free set defines a bounded convolution operator on l 2 . The fact that, for G D SL2 .R/ (see Remark 4 of Theorem 8 in Sect. 1.2) every function in Lp .G/ convolves L2 .G/ for 1 < p < 2, requires, besides the Plancherel formula of Harish-Chandra, the study of a new class of infinite dimensional representations obtained by “analytic continuation” of the principal series and “some vigorous” classical Fourier analysis. These representations acts on a fixed Hilbert space, they are indexed by a complex parameter and depend analytically on the parameter, included among them are the complementary series [72]. A locally compact unimodular group G is said to have the “Kunze-Stein property” if L2 .G/ Lp .G/ L2 .G/ for every 1 < p < 2 [77]. Clearly compact groups have this property but noncompact amenable groups don’t. In 1970 Lipsman [78] proved the following result: let G be a noncompact locally compact connected unimodular group. Then G has the Kunze-Stein property if and only if G has a compact normal subgroup H such that (1) G=H is a connected semisimple Lie group with finite center and (2) G=H has the Kunze-Stein property. Finally Cowling proved that connected semisimple Lie groups with finite center have the KunzeStein property [17]. Oberlin [98], with the aid of a computer, proved that for a certain function f , defined on the dihedral group D4 of height elements, ˇˇˇ ˇˇˇ ˇˇˇ 4 ˇˇˇ ˇˇˇ .f /ˇˇˇ 6D ˇˇˇˇˇˇ4=3 .f /ˇˇˇˇˇˇ D4 D4 4

4=3

:

As mentioned in Sect. 1.4 (Remark after the Corollary 7), Herz proved a similar statement for every finite nonabelian group [63]. By the Corollary 3 of Chap. 7, Sect. 7.3 it follows that, for a locally compact group G having a finite ˇˇˇ ˇˇˇ non-abelian p subgroup, and for every p 6D 2 there is a bounded measure with ˇˇˇG ./ˇˇˇp 6D ˇˇˇ 0 ˇˇˇ ˇˇˇ p ˇˇˇ ˇˇˇG ./ˇˇˇ 0 . A consequence of [85], is that for SL2 .R/ for every p 6D 2 there is p

0

a positive measure which convolves Lp but not Lp . In [63] and [64] Herz proved the following theorem: for each non-abelian nilpotent Lie group ˇˇˇ pG andˇˇˇeach p with 2 < p < 1 there exists a sequence .kn / of L1 .G/ such that ˇˇˇG .kn /ˇˇˇp 1 while ˇˇˇ ˇˇˇ 0 ˇˇˇ ˇˇˇ p ˇˇˇG .kn /ˇˇˇ 0 ! 1. See Dooley et al. [37] for refinements (one could require for p

instance that the kn have their support in a given neighborhood of e) and various improvements on these questions.

Chapter 3 0

The sesquilinear map .f; g/ 7! f gL of Lp .G/ Lp .G/ into C0 .G/ fac0 b Lp .G/ into C0 .G/ where torizes through a linear contraction P of Lp .G/˝ 0 0 p p p b L .G/ is the projective tensorproduct of L .G/ with Lp .G/. Then L .G/˝

Notes

163 0

b Lp .G/=KerP . clearly Ap .G/ coincides with the quotient Banach space Lp .G/˝ This was used by Herz for his proof of Corollary 5 in Sect. 3.3. It is possible to repeat the Sects. 3.1 and 4.1 replacing the regular representation in L2 by an arbitrary continuous unitary representation of G. One gets a Banach space, denoted A , of nice uniformly continuous bounded functions on the group G. See Arsac [4]. He proved, among other things, that A is a Banach algebra if and only if the representation ˝ is quasi equivalent to . In relation with the KunzeStein property, Herz considered such an A , choosing for the representation induced by the identity representation from a certain closed subgroup H of a connected semi-simple Lie group of finite centre [58]. The group H is AN , where G D KAN is the Iwasawa decomposition of G. He proved (“principe de majoration”) that for u 2 A2 .G/ and " > 0 there is v 2 A with kvkA < kukA2 C " and u.x/ v.x/ for every x 2 G. This was used in [17]. Let B.G/ be the vector space of the coefficients of all continuous unitary representations of G. Generalizing the fact that A2 .G/ is an algebra, P. Eymard ([41]) proved that each u of B.G/ “multiplies” A2 .G/ i.e. uA2 .G/ A2 .G/. If G is abelian the converse is true: if u multiplies A2 .G/ then u 2 B.G/ (see [107], p. 73). This holds also for amenable G [27]. Losert proved that if G is non amenable then there are multipliers of A2 .G/ which are not in B.G/ [86]. For a better understanding of the multipliers of A2 .G/ (and also of Ap .G/) we need to consider an important class˚ of convolution operators. ˇ We denote by PFp .G/ p the norm closure in C Vp .G/ of G .f / ˇ f 2 L1 .G/ . If G is abelian PF2 .G/ b See [70], p. 45 for T . The Banach algebra PF2 .T / is is isomorphic to C0 .G/. isomorphic to c0 .Z/ (and C V2 .T / to l 1 .Z/). Kahane call the elements of C V2 .T / “pseudomeasures” and the elements of PF2 .T / “pseudofunctions”. For p 6D 2 the algebra PFp .Rn / has been considered by Hörmander in [68], p. 111. Answering a question raised by Hörmander, Figà-Talamanca and Gaudry proved that for every b 2 C0 .Rn / such that T 62 PFp .Rn / ([46]). p 6D 2 there is a T 2 C Vp .Rn / with T They obtained a similar statement for T n . For a general locally compact group PF2 .G/ is called the reduced C - algebra of G. According to Eymard [41] PF2 .G/ coincides with the full C - algebra of G if and only if G is amenable. Recall (see again [41]) that the dual of the full C algebra of G is precisely B.G/. Observe that if G has the Kunze-Stein property then clearly Lp .G/ PF2 .G/ for every 1 2. If G is abelian then, according to Lohoué, the space of all multipliers of Ap .G/ is the dual of the Banach space PFp .G/ for p > 1 ([79], Théorème 1). In [62] Herz proved the following: (1) for an arbitrary locally compact group G and for p > 1 every function in the dual of PFp .G/ multiplies Ap .G/I (2) if G is amenable the dual of PFp .G/ coincides with the space of all multipliers of Ap .G/. To obtain these results, C. Herz introduces two interesting commutative unital Banach algebras denoted respectively Vp .G G/ and Bp .G/. We have Vp .G G/ C b .G G/ and Bp .G/ C b .G/ and in both algebras the product is simply the pointwise product. For every k 2 C00 .G G/ we denote by Tk the operator of Lp .G/ having the kernel k. Then by definition Vp .G G/ D

164

Notes

˚ ˇ u ˇ u 2 C b .G G/, there is C > 0 such that jjjTuk jjjp C jjjTk jjjp for every k 2 C00 .G/ ˝ C00 .G/ . For u 2 Vp .G G/ the ˚ norm of u is the infimum of all C . The definition of Bp .G/ is: Bp .G/ D u ju 2 C.G/ and the function on G G .x; y/ ! u.yx 1 / belongs to Vp .G G/ , the norm of u being the norm in Vp .G G/ of the function .x; y/ ! u.yx 1 /. Then Herz proved the following statements: (1) Bp .G/Ap .G/ Ap .G/I (2) the dual of PFp .G/ is contained in Bp .G/I (3) if G is amenable then Bp .G/ coincides with the set all multipliers of Ap .G/ (and thus with the dual of PFp .G//. The letter V of Vp .G G/ refers to Varopoulos (see [56]). The algebra Bp .G/ has been used by Herz to study the asymmetry of the norms of convolution operators [64]. For G abelian Lohoué was able to relate Bp .G/ with Bp .Gd /: he proved in particular that Bp .G/ D Bp .Gd / \ C.G/ and more subtle results (see Théorème 3, p. 28 in [81] and Théorème 0.2.2, p. 78 in [80]). For a generalization of these results of Lohoué to all locally compact groups, see [62] and [33] (Theorem 7, p. 936). For more on Bp .G/ see [11, 12, 51]. Eymard proved in [41] that the dual of PF2 .G/ is the vector space of the coefficients of the unitary representations weakly contained in the regular representation of G. Cowling and Fendler ([19], Theorem 2) obtained a similar description of the dual of PFp .G/ for every p > 1. They replace the unitary representations by isometric representations in certain reflexive Banach spaces of a very precise type and use the theory of ultraproducts of Banach spaces.

Chapter 4 Like L1 , A2 .G/ is weakly sequentially complete. This follows from the fact that A2 .G/ is the predual of the von Neumann algebra C V2 .G/ ([114], Chap. III, Corollary 5.2, p. 148). If p 6D 2 the question is not completely answered. If G is compact metrizable, then Ap .G/ is weakly sequentially complete (Lust-Piquard [88]). Lust-Piquard’s proof uses the characterization (see the notes to Chap. 3) of b Lp0 .G/. For R the problem seems to be open. Ap .G/ as a quotient of Lp .G/˝ ˚ b jT 2 PFp .Rn / Hörmander proved that holomorphic functions operate on T ([68], Theorem 1.18, p. 112). In [118] Zafran, for G D Rn or G D T n , shows the b and such that the spectrum of T properly existence of T 2 C Vp .G/ with b T 2 C0 .G/ b b contains T .G/ [ f0g. The existence of a bounded measure having this property is the classical theorem of Wiener-Pitt ([107], Theorem 5.3.4.) See also [119].

Chapter 5 For a large class of nonamenable groups G, including SO.1; n/; S U.1; n/ and Sp.1; n/, the algebra A2 .G/ admits approximate units bounded in the norm B2 .G/ [20], [21].

Notes

165

Cowling proved in [16] that C Vp .G/ D PMp .G/ for every p > 1 for G D SL2 .R/. He obtained also this result for the above mentioned groups, proving following general result: PMp .G/ D C Vp .G/ if A2 .G/ has an approximate unit bounded in the norm of B2 .G/ ([18], p. 413).

Chapter 7 Let H be a closed subgroup of a locally compact abelian group G and T 2 C Vp .G/ b then there is S 2 C Vp .G=H / with ResH ? b T Db Sıb with b T 2 C.G/, ! . One has jjjS jjjp jjjT jjjp [74], [108]. Figà-Talamanca and Gaudry, supposing H ? discrete, obtained the following extension theorem: for every S 2 C Vp .G=H / there is T 2 b D Sıb b 2 C.G/, b jjjT jjjp K jjjS jjjp and ResH ? T C Vp .G/ with T ! [45]. For G D n n n R and Z D R see Jodeit [69]. Cowling [15] has been able to suppress the extra S 2 C.G=H / condition on the discreteness of H ? : for every S 2 C Vp .G=H / with b b jjjT jjjp jjjS jjjp and ResH ? b T D Sıb there is T 2 C Vp .G/ with b T 2 C.G/, !. ˚ b one could consider the norm closure Instead of b T jT 2 C Vp .G/; b T 2 C.G/ of the set of all convolution operators having compact support. For this class of convolution operators the preceding restriction and extension results can be generalized to noncommutative groups [2, 29, 60]. Let G be an arbitrary locally compact group, H a closed normal subgroup and F a closed subset of G=H . In analogy with a famous result of Reiter ([105], Theorem 7.3.1, p. 203), Lohoué proved, that F is locally of p -synthesis in G=H if and only if ! 1 .F / is locally of p -synthesis in G [82]. A closed subset F of G is said to be locally p -Ditkin in G if for every u 2 Ap .G/ \ C00 .G/ vanishing on F and for every " > 0 there is v 2 Ap .G/ \ C00 .G/ with supp v \ F D ; and ku uvkAp < ": The closed set F is said to be p -Ditkin in G if for every u 2 Ap .G/ vanishing on F and for every " > 0 there is v 2 Ap .G/ \ C00 .G/ with supp v \ F D ; and kuuvkAp < ". If every element u of Ap .G/ belongs to the closure of uAp .G/ these notions coincide. The preceding theorem of Lohoué is valid also for Ditkin sets. Indeed we proved in [30] that every closed normal subgroup is locally p -Ditkin in G. This result was later extended to neutral subgroups [24] and to amenable subgroups [32]. Finally Ludwig and Turowska proved that every closed subgroup of a second countable locally compact group is locally 2 -Ditkin [87].

1

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List of Definitions We list here only the definitions which are systematically used throughout the book. The numbers in parentheses refer to the paragraphs where the definitions are given Amenable groups (1.1.4) Bruhat function (7.1) Convolution operator (1.2) Convolution operator associated to a bounded measure (1.2) Duality between Ap .G/ and PMp .G/ (4.1) Figà-Talamanca Herz algebra (3.1), (3.3) C Vp .G/ as a module on Ap .G/ (5.2) Pseudomeasure (4.1) Spectrum (6.1) Support of a convolution operator (6.1) Ultraweak topology (4.1)

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Convolution operators on groups

Convolution Operators on Groups

Matrix convolution operators on groups

Matrix Convolution Operators on Groups (Lecture Notes in Mathematics)

Elliptic Operators and Compact Groups

Convolution Equations and Singular Integral Operators: Selected Papers

Approximation by Complex Bernstein and Convolution Type Operators

Lectures on Gaussian Integral Operators and Classical Groups

Approximation of Additive Convolution-Like Operators: Real C*-Algebra Approach

Approximation by complex Bernstein and convolution type operators

The Douglas Convolution

Lectures on Quantum Groups

Seminar on Transformation Groups

Fourier Analysis on Groups

Probability Measures on Groups

Lectures on Lie Groups

Lectures on Quantum Groups

Lectures on quantum groups

M.G.Krein's Lectures on Entire Operators

Seminar on transformation groups

Lectures on Quantum Groups

Lectures on Lie Groups

Probability Measures on Groups

Composition operators on function spaces

Finite collineation groups: With an introduction to the theory of groups of operators and substitution groups

Composition Operators on Function Spaces

Composition Operators on Function Spaces

Lectures on singular integral operators

the douglas convolution

The Convolution Transform

Analysis and geometry on groups

Convolution operators on groups