Analysis: Conference Proceedings

PROCEEDINGS OF THE ANALYSIS CONFERENCE, SINGAPORE 1986 NORTH-HOLLAND MATHEMATICS STUDIES NORTH-HOLLAND -AMSTERDAM N...

Author: Stephen T.L. Choy | etc. | P.Y. Lee

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PROCEEDINGS OF THE ANALYSIS CONFERENCE, SINGAPORE 1986

NORTH-HOLLAND MATHEMATICS STUDIES

NORTH-HOLLAND -AMSTERDAM

NEW YORK OXFORD *TOKYO

150

PROCEEDINGS OF THE ANALYSIS CONFERENCE, SINGAPORE 1986 Edited by

StephenT. L. CHOY Judith P. JESUDASON and

P. Y. LEE Department of Mathematics National University of Singapore

1988

NORTH-HOLLAND -AMSTERDAM

NEW YORK OXFORD *TOKYO

Elsevier Science Publishers B.V., 1988

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 70341 1

Published by: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM THE NETHERLANDS

Sole distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017 U.S.A.

PRINTED IN THE NETHERLANDS

V

PREFACE

It has become a t r a d i t i o n t h a t one

or two mathematical conferences be h e l d

a n n u a l l y i n Singapore, and t h e second such conference o f 1986 was a workshop and Conference on a n a l y s i s ,

h e l d on t h e campus o f t h e N a t i o n a l U n i v e r s i t y o f

Singapore from June 12 t h r o u g h June 21, 1986.

T h i s volume forms t h e

proceedinys o f t h e workshop and conference, which emphasized m a i n l y harmonic and f u n c t i o n a l a n a l y s i s . A.

The i n v i t e d speakers were E. H e w i t t , S.

Igari,

Miyachi, ti. P i s i e r , and J. J. Uhl, J r . , and t h e y have c o n t r i b u t e d a t o t a l o f

f i v e papers t o t h e s e proceedings.

One o t h e r i n v i t e d speaker, S. Gong, was

u n f o r t u n a t e l y unable t o a t t e n d t h e conference, due t o unforeseen c i r c u m s t a n c e s , b u t has k i n d l y sent us a paper based on t h e address t h a t he would have given.

As t h e r e were more papers s u b m i t t e d t h a n c o u l d be i n c l u d e d i n t h i s volume, t h e remainder o f t h e c o n t r i b u t e d papers were s e l e c t e d on t h e b a s i s o f r e f e r e e s ' reports.

A l i s t o f t h e t a l k s y i v e n a t t h e conference, as w e l l as t h e names and

a f f i l i a t i o n s o f p a r t i c i p a n t s and c o n t r i b u t o r s i m m e d i a t e l y f o l l o w t h i s p r e f a c e . We would l i k e t o thank our c o l l e a y u e s , authors, typists,

referees, publisher,

and many o t h e r s who have helped us i n t h e p r e p a r a t i o n , e d i t i n g , and

p r o d u c t i o n o f t h e s e proceedings; i n p a r t i c u l a r we a r e g r a t e f u l t o Madam Luin and Miss Tan f o r t h e i r e x p e r t t y p i n g o f most o f t h e manuscripts.

A l s o , we would

l i k e t o thank a l l o f t h e o r g a n i z a t i o n s who gave f i n a n c i a l support,

including

t h e Depdrtment o f Mathematics, N a t i o n a l U n i v e r s i t y o f Singapore, t h e Singapore Mathematical S o c i e t y , t h e Singapore N a t i o n a l Academy o f Science, and t h e Southeast Asian Mathematical S o c i e t y .

We hope t h a t t h i s volume w i I I serve as a

u s e f u l r e c o r d o f our conference, and t h a t t h e memories o f t e n days spent i n Singapore i t b r i n g s t o t h e p a r t i c i p a n t s and i n t e r e s t e d readers a r e b o t h as fruitful,

and as p l e a s a n t , as those memories t h a t we r e c a l l .

The E d i t o r s , Singapore, August 1987

vi

I n v i t e d Addresses

E. H e w i t t , U n i v e r s i t y o f Washington, Marcel Riesz's theorem on conjugate Fourier transforms : a progress report I - I I I ; Alfred Haar and his

measure. S.

I g a r i , T8hoku U n i v e r s i t y , Application of an interpolation theorem for mixed normed spaces I : An estimate of Riesz-Bochner means of Fourier

transforms; Application of an interpolation theorem for mixed normed spaces 11 : Restriction problem of Fourier transforms. A. r l i y a c h i , H i t o t s u b a s h i U n i v e r s i t y , A factorization theorem in Hardy spaces;

Boundedness of pseudo-differential operators with non-regular symbols; Estimates for pseudo-differential operators with exotic symbols. I;. P i s i e r , U n i v e r s i t g de P a r i s V I , Factorization through weak-Lp and Lpl and

non-commutative generalizations. J. J. Uhl, Jr.,

U n i v e r s i t y o f I l l i n o i s a t Urbana, Differentiation in Banach spaces I , I I ; Geometry and Dunford-Pettis operators on LI.

Short C m u n i c a t i o n s I/.It. Bloom and J . F. F o u r n i e r : Generalized Lipschitz spaces on Vilenkin

groups.

P. S. S u l l e n : On the solution of = f(x,y). T. S . Chew : A Denjoy-type definition of the nonlinear Henstock integral. M. T.

C.

C h i e n : Perturbations of C*-algebras.

H. Chu and L. S. L i u : A localized version of Choquet's theorem.

S. Darmawijaya and P. Y. Lee : The controlled convergence theorem for the approximately continuous integral of Burkill. C. S. D i n g : Absolutely Henstock integrable functions. J. L. Geluk : AsymptoticalZy balanced functions. B. J e f f r i e s : Pettis integral operators. C. H. Kan : Extreme contractions from Lp to Lq, p e 1 S q . C. M. Kim : shift invariant Markov measures. E. P I . L a g a r e : Approximations of integrals of Henstock integrable functions using uniformly regular matrices. H. C. L a i : Translation invariant operators and multipliers of Banach-valued function space. P. Y. Lee : A proof of the generalized dominated convergence theorem for Henstock integrals. D. J. Luo : On limit cycle bifurcations. P. P. Narayanaswarni : The separable quotient problem for Frechet and ( L F )-spaces.

Short CornrnunicationsJWorkshop Lectures

vii

C. W . Unneweer : Weak L -spaces and weighted norm inequalities for the Fourier

P compact Vilenkin groups. transform on locally

[i. Z. Ouyang : Multipliers of Segal algebras. P.

L.

P a p i n i : Norm-one projections onto subspaces of sequence spaces.

S. Pethe W.

: On linear positive operators generated

R i c k e r : Joint spectral subsets of

by power series.

IFf" for comuting families of operators

in Banach spaces. R o s i han Mohained A1 i : Mijbius transformations of convex mappings. S.

Y.

Shaw : Uniform ergodic theorems for semigroups of operators on L" and

similar spaces. I . H. S h e t h : Centroid operators. K. N. S i d d i q i : On density of Fourier coefficients of a function of Wiener's class. S. L. Tan : The successive conjugate spaces of dual C*-algebras. K. T a v i r i : Fixed points of nonexpansive mappings in Banach spaces. H. J . Tu : Some new applications of potential theory to conformal mappings.

H. C. Wang : On the Fourier transforms. S. L. Warig : Weighted norm inequalities for some maximal functions. J . A. Ward : A refZexivity condition for some homogeneous Banach spaces. B. E. \lu : The second dual of Cesaro sequence spaces of a non-absolute type.

D. Yost : There can be no Lipschitz versia of Michael's selection theorem. X . W.

Zhou : O n a conjecture of band limited function extrapolation. Workshop Lectures

S.

I g a r i : Interpolation of linear operators on product measure spaces, I - I v .

A. M i y a c h i : Singular integral operators and pseudo-differential operators, I-Iv

.

G. P i s i e r : Probabilistic and volume methods in the geometry of Banach spaces, I-III.

viii

L i s t of P a r t i c i p a n t s and C o n t r i b u t o r s

I Z Z A H BTE. ABIIULLAH, U n i v e r s i t i Kebanysaan M a l a y s i a , M a l a y s i a ACHMAD AHIFIN, I n s t i t u t e T e k n o l o y i Banduny, I n d o n e s i a RAVI P. AGARWAL, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e AZLINA AHMAD,

U n i v e r s i t i Kebanysaan M a l a y s i a , E l a l a y s i a

SHICHAN ARVDRN, Chiangmai U n i v e r s i t y , T h a i l a n d NACHLC ASPIAK,

C a l i f o r n i a S t a t e U n i v e r s i t y a t Long Beach, U.S.A.

LYN BLOOM, WACAE ( N e d l a n d s Campus), A u s t r a l i a WALTEH

H. BLOUM, I l u r d o c h U n i v e r s i t y , A u s t r a l i a

P. S. HIJLLEN, The U n i v e r s i t y o f B r i t i s h Columbia, Canada

SEHGIU S. CAO, U n i v e r s i t y o f t h e P h i l i p p i n e s ,

Philippines

CHAN CHUN-WAH, Hony Kony P o l y t e c h n i c , Hony Kony

CHAN K A I MENG, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e CHAN WAI-KIT,

Hony Kony U n i v e r s i t y , Hony Kony

SHAO-CHIEN CHANG, B r o c k U n i v e r s i t y , Canada

TSU-KUNG CHANG, N a t i o n a l T s i n y Hua U n i v e r s i t y , Taiwan LOUIS H. Y. CHEN, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e

CHENG K A I NAH, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e CHEW TUAN SENG, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e MAU-TING CHIEN, Soochow U n i v e r s i t y , Taiwan CHUNG C H I TAT, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e STEPHEN T.

L. CHOY, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e

C. H. CHU, U n i v e r s i t y o t London, U n i t e d Kingdom SOEPARNA DARMAWIJAYA, I n s t i t u t T e k n o l o y i Banduny, I n d o n e s i a DING CHUANSUNG, N o r t h w e s t e r n T e a c h e r ' s C o l l e y e ,

People's Republic o f Chind

DANIEL FLATH, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n g a p o r e J. F. FOUHNIEK, U n i v e r s i t y o f B r i t i s h Columbia, Canada

J.

L. GELUK, Erasmus U n i v e r s i t y , N e t h e r l a n d s

S.

GONG, The C h i n e s e U n i v e r s i t y o f S c i e n c e L Technology, China

R. C.

People's Republic o f

GUPTA, N a t i o n a l U n i v e r s i t y ot S i n y a p o r e , S i n g a p o r e

RENATO GUZZARDI,

Universita d e l l a Calabria,

Italy

EDWIN HEWITT, U n i v e r s i t y o f Washinyton, U.S.A HU K A I Y U A N , N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e

S.

I G A H I , T6hoku U n i v e r s i t y , Japan

BRIAN JEFFEHIES, M a c q u a r i e U n i v e r s i t y , A u s t r a i a JUUITH P. JESUOASON, N a t i o n a l U n i v e r s i t y o f S nyapore, S i n g a p o r e

List of Participants and Contributors K A N CYAKN HUEN, N d t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e

CHUO-WHAN KIM, Simon F r a s e r U n i v e r s i t y , Canada Y A T I KKISNANGKUKA, S r i n a k h a r i n w i r o t U n i v e r s i t y , T h a i l a n d EMMANUEL PI. LAGAKE, Mindanao S t a t e U n i v e r s i t y ,

Philippines

HANG-CHIN L A I , N a t i o n a l T s i n y Hua U n i v e r s i t y , Taiwan

LEE PENG YEE,

IVational U n i v e r s i t y o f Sinyapore, Sinyapore

1.1 S H I XIUNG, Anhui U n i v e r s i t y , P e o p l e ' s R e p u b l i c o f C h i n a LIM SUAT KHOH, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e LIlJ LIANG SHEN, Zhonyshan U n i v e r s i t y , P e o p l e ' s K e p u b l i c o f C h i n a L I U YU-OIANG,

S o u t h C h i n a Normal U n i v e r s i t y ,

Peoples's Republic o f China

LOU J I A N N HUA, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e LUO DINtiJUN, N a n j i n y U n i v e r s i t y , P e o p l e ' s K e p u b l i c o f C h i n a SIDNEY S. MITCHELL, C h u l a l o n y k o r n U n i v e r s i t y , T h a i l a n d

A. M I Y A C H I , H i t o t s u b a s h i U n i v e r s i t y , J a p a n TAKA R. NANOA, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e

P. P. NARAYANASWAMI, Memorial U n i v e r s i t y o f Newfound1 and, Canada NG BOON Y I A N , Uni v e r s i t i Ma1 aya, Ma1 a y s i a NG K. F.,

C h i n e s e U n i v e r s i t y , Hony Kony

NG PENG NUNG, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e NUKUL MUCHLISAH, U n i v e r s i t a s Hasanuddin,

Indonesia

ONG BOON HUA, U n i v e r s i t i S a i n s M a l a y s i a , M a l a y s i a C. W.

ONNEWEEK, U n i v e r s i t y o f New Mexico, U. S. A.

OUYANG GUANGZHONG, Fudan U n i v e r s i t y , P e o p l e ' s K e p u b l i c o f C h i n a ROONGNAPA PAKUEESUSUK, Chi angmai U n i v e r s i t y , T h a i 1arid

PIER LUItiI P A P I N I , U n i v e r s i t y o f Boloyna, I t a l y SHARADCHANURA PETHE, U n i v e r s i t i Ma1 aya, Ma1 a y s i a MINUS PETKAKIS, U n i v e r s i t y o f I l l i n o i s a t Urbana, U.S.A GILLES P I S I E R ,

U n i v e r s i t 6 de P a r i s V I ,

ROGER POH KHENG SIONG,

France

N a t i o n a l U n i v e r s i t y o f Sinyapore, Sinyapore

QUEK TONG SENG, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e

W. J . KICKER, M a c q u a r i e U n i v e r s i t y , A u s t r a l i a

I . ROBERTS, D a r w i n I n s t i t u t e o f T e c h n o l o y y , A u s t r a l i a R O S I H A N I'IOHAMED A L I , U n i v e r s i t i S a i n s M a l a y s i a , M a l a y s i a AHAMAD SHABIR S A A R I , U n i v e r s i t i Kebanysaan M a l a y s i a , M a l a y s i a

SEN-YEN SHAW, N a t i o n a l C e n t r a l U n i v e r s i t y , Taiwan ICHHALAL HAKILAL SHETH, G u j a r a t U n i v e r s i t y , R. N. S I D D I I J I ,

India

Kuwait University, Kuwait

BAMBANG SUOIJUNU, U n i v e r s i t a s Gajah Mada, I n d o n e s i a TAN S I N LENG, U n i v e r s i t i r l a l a y a , M a l a y s i a ABU USMAN BIN MD TAP, U n i v e r s i t i Kebanysaan M a l a y s i a , M a l a y s i a HAKA TAVIKI, U n i v e r s i t y o t Papua New Guinea, Papua New Guinea

ix

List of Participants and Contributors

X

TU HUNGJI, Fuzhou U n i v e r s i t y , P e o p l e ' s R e p u b l i c o f C h i n a

PETEK C. T. TUNG, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e J.J.

UHL JR.,

U n i v e r s i t y o f I l l i n o i s a t Urbana, U.S.A.

HWAI-CHIUAN WANG, N a t i o n a l T s i n y Hua U n i v e r s i t y , Taiwan WANG S I L E I , Hanyzhou U n i v e r s i t y , P e o p l e ' s R e p u b l i c o f C h i n a JU WARU, Flurdoch U n i v e r s i t y , A u s t r a l i a

S. J. WILSUN, N a t i o n a l U n i v e r s i t y o f S i n y a p o r e , S i n y a p o r e WU BO-ER,

South China Normal U n i v e r s i t y , P e o p l e ' s R e p u b l i c o f C h i n a

XU FENG, N o r t h e a s t Normal U n i v e r s i t y , P e o p l e ' s R e p u b l i c o f China DAUD YAHAYA, LEONARD Y.

ti.

U n i v e r s i t i Malaya, M a l a y s i a YAP,

National U n i v e r s i t y o f Sinyapore, Sinyapore

D A V I U YOST, A u s t r a l i a n N a t i o n a l U n i v e r s i t y , A u s t r a l i a ZHANG WENYAO,

L i a o n i n y Normal U n i v e r s i t y , P e o p l e ' s K e p u b l i c o f C h i n a

ZHENG XUE AN, Anhui U n i v e r s i t y , P e o p l e ' s R e p u b l i c o f C h i n a ZHOU X I N G WEI, Nankai U n i v e r s i t y , P e o p l e ' s R e p u b l i c o f C h i n a

ZOU CHENZU, J i l i n U n i v e r s i t y , P e o p l e ' s R e p u b l i c o f C h i n a

xi

CONTENTS N a k h l 6 Asmar and Edwin H e w i t t , Marcel Riesz’s theorem on conjugate

Fourier series and its descendants Chew Tuan Seny,

&i

nonlinear integrals

1

57

Soeparna flarrnawijaya and Lee Peng Yee, The controlled convergence

theorem for the approximately continuous integral of Burkill

63

Gony Sheng, L i S h i XiOng and Zheny Xue An, Harmonic analysis on

classical groups

69

SatOrU I g a r i , Interpolation of operators in Lebesgue spaces with

mixed norm and its applications to Fourier analysis

115

Choo-Whan K i m , Shift invariant Markov measures and the entropy

129

map of the shift Hang-Chin L a i and Tsu-Kuny Chang, Translation invariant operators

and multipliers of Banach-valued function spaces

151

Lee Peng Yee, A proof of the generalized dominated convergence

theorem for the Denjoy integral A k i h i k o M i y a c h i , A factorization theorem for the real Hardy spaces

163 167

A k i h i k o M i y a c h i , Estimates for pseudo-differential operators of

class K . F. Ny and L. S.

in Lp

, hp , and

bmo.

177

L i u , A note on a lifting property f o r convex

189

processes C. W. Onneweer, Weak L

spaces and weighted norm inequalities for p- transform on locally compact Vilenkin groups the Fourier

19 1

Ouyang Guanyzhong, MultipZiers of Segal aZgebras

2113

Minos P e t r a k i s and J . J . Uhl, Jr.,

219

Differentiation in Banach spaces

associated with commuting families of linear operators

Werner R i c k e r , “Spectral subsets” of

243

R o s i h a n Mohamed A1 i , The class of MZbius transformations of convex

mappings

249

Sen-Yen Shaw, Uniform ergodic theorems for operator semigroups

26 1

Wang S i l e i , Weighted norm inequalities for some maximal functions

267

Wu Bo-Er,

The second duals of the nonabsolute Cesaro sequence spaces

L i u Yu-Qiang and Lee Peng-Yee,

Xu Feng and Zou Chenzu, Banach reducibility of decomposable operators

285 29 1

Contents

xii

D a v i d Yost, There can be no Lipschitz version of Michael's selection

theorem Zhang Wenyao, A net) smoothness of Banach spaces

29 5

30 1

Proceedings of the Analysis Conference, Singapore 1986 S.T.L. Choy, J.P. Jesudason, P.Y. Lee (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1988

MARCEL RIESZ’S THEOREM ON CONJUGATE FOURIER SERIES AND ITS DESCENDANTS by KakK Asmar and Edwin Hewitt’

Marcel Rie8z centum anno8 ez anno nafalie 8ui dedicatae Opera imperitura reliquit

$0. Notation.

Throughout this paper we adhere to the following notation: N denotes the positive integc .

Z+the nonnegative integers; Z the integers; Q the rational numbers; R the real numbe: Q: the complex numbers; D the set ( 2 E CC : 121 < l}. The symbol G will always denote locally compact Abelian group, with one or another property imposed on it. The symbol X will always denote the character group (dual group) of G. Haar measure (chosen arbitrarily) on G will be denoted by p , All notation and terminology not otherwise explained are as in Hewitt and Ross [1979,197Ola.We suppose that the reader is at home with the basic facts of Fourier transforms on locally compact Abelian groups.

For a real number p > 1, p’ means

L.

P--l

1

N. Asmar and E. Hewitt

2

$1. Marcel Riese’s original theorem.

(1.1) Early history. Conjugate Fourier series, despite their intrinsic interest and many applications, are for many analysts a little mysterious. Some explanation may therefore be appropriate. The earliest study that we have found of conjugate Fourier series is a Jugendarbeit of Alfred Tauber [1891]’. His description is that followed to the present day. Let

c(a,a7

F ( E )=

(1)

ib,)Y

(av1bvreal, b, = 0)

u=o

be a power series convergent in D. (The minus sign in (1)is merely a convenience.) Write z = rexp(it) with 0 5 r < 1 and - H < t 5 H. The definition (1)becomes F ( z ) = F(rexp(it)) w

=

(2)

00

r”(a, cos vt + b, sin v t ) + i

u=o

= p(r, t )

r” ( -b, cos vt

+ a, sin v t )

,=I

+ i@(r,t ) .

Tauber loc. cit. and a host of later authors have been concerned with the relation between the real-valued harmonic function (p and its conjugate real-valued harmonic function qb. For later use, we will write (1) and (2) a bit differently. For n E N, let c, = ;(a, - ib,) and let c - , = ;(a, I t is plain that 00

(3)

C

~ ( = t )F(rexp(it)) =

r h , exp(ivt) +

v=-ro

where sgnv =

for v

# 0 and

+ ib,).

Let c, = (a,

+ ib,)

= a,.

00

C

sgnvrlvlc, exp(ivt),

,=-00

sgnO = 0.

To apply (3)to Fourier series, we now consider a complex-valued function f in C,( -r1 H). The Fourier traneform f of f is the function on Z4 such that

and the Fourier eerier off is the formal infinite series 00

(5) ,=-a7

The conjugate Fourier eeriee off is the formal infinite series sgnvj(v) exp(ivt).

-i ,=-a7

Marcel Riesz 's Theorem on Conjugate Fourier Series

3

-

The function f is real-valued if and only if i(-n) = )(n) for all n E Z+.In this case we go back to the infinite series (1) and make the following special choice of its coefficients:

and

b,, = i(f(n) - ](-n)),

(8)

We now write the two infinite series in (3), quite formally, with r = 1. This gives us the two formal series (5) and (6), and these turn into 00

(9)

fao

+

(a, cos vt

+ b,

sin vt)

u=1

00

C(-b,cosvt+a,sint), respectively. From now on, the coefflcients a,, and 6, in series of the form (9) and (10) will be as they are defined in (7) and (8) respectively, that is, they will be the real forms of the Fourier transform of f. Already in 1891 much was known about the convergence of the Fourier series IS), and it was natural to find conditions under which the conjugate series (10) converges. The 5rst successful effort in this direction was made by Pringsheim [1900]. His results are now only of historical interest, since for example he did not have the powerful instrument provided by the Lebesgue integral. Fatou's thesis Il906] contains the seed of much of the study of conjugate series. Again we go back t o the two infinite series in (3), with c, = )(v), where f is an arbitrary complex-valued function in C1(-a, T). If f is not real-valued, we will have c-, # F for some integer v. This need not prevent us from writing the infinite series in (3), and the Riemann-Lebesgue lemma implies that the radius of convergence of the power series in r appearing on the right side of (3) is not less than 1. We write the first series in (3) as

An easy calculation shows that

4


We write the second series in (3) as w

(13)

f(r, t ) = -i

C

sgnvrlvlj(v) e x p ( i ~ t ) . ~

Y=-m

Another easy calculation yields

=

LIT 2A

-r

f ( t - u)Q(r, u)du.

The integrals appearing in (12) and (14) are very convenient in studying lim f ( r , t ) and rt1

lim i ( r , t ) . However, the generalizations to locally compact Abelian groups that we have in rt 1

mind require the form (13) of j ( r , t ) . So we introduce it here. For continuous f, Fatou [1906], p. 360, showed that

exists for a fixed t if and only if the Cauchy principal value

exists, and then the limits in (15) and (16) are equal. On p. 363, he showed that if f satisfies a Lipschitz condition of order a < 1, then f(t) exists for all t and is also in Lipa. Privalov [1916(’)] extended this result. If f is of class Lipl, then f(t) exists for all t and is of class Lips for all B < 1. He gave an example showing that f need not be of class Lipl. W. H. Young I19111 showed that if f is real-valued and has finite variation on [ - A , A ] , then the series (10) converges a t a given point t if and only if the limit (16) exists for this value o f t , and then (16) is the sum of (10).

W. H. Young and G. C. Young 119131 made a thoroughgoing study of the Riesz-Fischer theorem. One of their theorems is the following (p. 57). Let u n ( f ( t ) be the nth Cesa.rb mean of the partial sums S k f ( t ) of the Fourier series (9) of a real-valued function f in C ~ ( - A , A )For . 1 < p < co, f belongs to 6 p ( - ~ ifl and ~ ) only if the 6, norms of the functions a,f are bounded over all n. (See for example Zygmund [1959, Vol. I], Ch. IV, p. 145,Theorem (5.7).) On page 58, the Youngs ask if this characterization remains true when the means a,f are replaced by the partial sums S, f of the Fourier series. Should this be the case, the authors point out a remarkable consequence. For f E C P ( - a ,A ) and g E Cpt(-~A , ) , the equality

Marcel Riesz s Theorem on Conjugate Fourier Series

5

holds, the series on the right side of (17) being convergent. (We have recast the Youngs’ statement in complex form.) It was already known to the Youngs that the series on the right side of (17) is Cesarb summable to the left side of (17). The convergence of the series on the right side of (17) the Youngs regarded as so unlikely that they rejected the conjecture that the 2, norms of the partial sums S,f are bounded in n. FejCr (19131 used the conjugate Fourier series to compute lim[f(t 610

+

E)

- f ( t - E ) ] . His

result was later completed by Lukicz [1920]. See Zygmund [1959, Vol. I], Ch. 11, p. 60, Theorem (8.13). We come now to a decisive step in the theory of conjugate Fourier series. (1.2) Theorem (Privalov [1917,19191). Let f be in (1.1.13) and (1.1.14). Then the limit

(4 exists and is finite for almost all t in

21(-7r,7r),

and let f”(r,t) be asin

lim f(r, t ) = f(t) rtl [-7r,

7r]. The value of ( i ) where it exists is the expression

(1.1.16).

(13) Subsequent hlstory of Privalov’s theorem. The Comptes Rendus note Privalov (19171 contains a complete statement of the first assertion of Theorem (1.2), as well as a sketch of the proof. It attracted scant attention. The only citation we have found is in a preliminary announcement by Hardy and Littlewood (19241, who describe it as follows. The subject matter of the8e n o t d i o extremely intereeting, but the indication8 of demonetrations ate ineuficient. In view of the great importance of Theorem (1.2), and the efforts that Hardy and Littlewood later lavished on conjugate Fourier series (19251, we think that the two English savants should have exerted their great powers to fill in the trifling gaps in Privalov’s note (19171. Privalov [1919] is a book dealing with boundary values of analytic functions defined in various domains. I t was apparently unknown in the West for some time. The earliest reference we have found is Lichtenstein [1924/1926]. Privalov 119411 revised his book [1919] and added much new material. We have found no non-Soviet references to this work. After Privalov’s untimely death in 1941 at the age of 50, his friends undertook a revision of Privalov [1941], which appeared as Privalov [1950]. This work is widely available and widely cited. A complete proof of Theorem (1.2) appears there; indeed it is merely the overture in a long work of great profundity. Many other proofs of Theorem (1.2) are in existence. See Zygmund (1959, Vol. I], Ch. IV, $3, pp. 131-136 and the notes to this $, p. 377. The proof in Ch. VII, $1, pp. 252-253, is very like Privalov’s. I t was observed early on that the conjugate Fourier series of an integrable function need not be the Fourier series of an integrable function, even if the conjugate series converges everywhere. Kolmogorov showed that something close holds.


6

(1.4) Theorem (Kolmogorov [1925]). Let f be in &(-T’ T ) andlet E be arealnumber such that 0 < E < 1. The function f is in C 1 - s ( - ~ l s),and there is a constant C such that

Privalov kept his interest in special conditions for the convergence of the conjugate Fourier series under various conditions: see Privalov 11923, 19251.

(1.5) Finally we are ready to describe Marcel’s Riesz’s fundamental contribution. If f is in C2(-7rls), the Riesz-Fischer theorem shows that .f exists as an &limit: Privalov’s Theorem (1.2) is not needed. It is clear that “,

(1)

f(v)= -i sgnvj(v), v E zI

and that (2)

Ilr”ll2

5

Ilf

112’

with equality holding in (2) if and only if i ( 0 ) = 0. One may ask if something like this holds for all p > 1: viz., if f is in C,(-T,T), does f also belong to C , ( - A , T ) , does (1) hold, and does something resembling (2) hold? In the early 1920’s, Marcel Riesz provided affirmative answers to all three of these questions. Lars Girding I19701 gives an interesting account of exchanges between Riesz and Hardy. In 1923, Riesz wrote to Hardy that he had affirmative answers. We now quote from Girding. Hardy wrote back ‘eome month8 ago you said y’ai ddmonfrd que 2 eerier trig. conjugudeo eont toujours en mime temps lee edrier de Fourier de fonctione de claoee L,(p > 1)... ’ I want the proof. Both I and my etudent Titchmarsh have tried in vain to prove it . . n And in the nezt letter ‘very many thank8 - you eupply a“ that ie eeeentiol. I have eent on your letter t o Titchmareh. Moet elegant and beaufqul. Of couree p. 2 io the real point. It is amazing that none of ue should have eeen if before (even for p = 4!). ’

.

Directly thereafter, Riesz I19241 published a Comptes Rendus note stating his theorems but giving no proofs. Hilb and Riesz 119241 stated some theorems about ffor f in .C,(-T, T)’ citing a “demnachst erscheinenden Arbeit” of M. Riesz. Actually four years passed before Riesz published the paper. E. W. Hohson 119261, pp. 610-614 and p. 698, cited Riesz’s work without furnishing a proof and drew some interesting consequences. Titchmarsh I19261 published a theorem very like Riesz’s and also - uer6atim- Theorem (1.8) infra. He notes Riesz’s priority but fails to mention that he had seen Riesz’s 1923 letter to Hardy. One may judge this as ungracious.‘ In 1928, Riesz published the whole story.

Marcel RieszS Theorem on Conjugate Fourier Series

7

(1.6) Theorem (M.Ries5 [1928]). Let p be a real number such that 1 < p < M, and let f be a function in C p ( - n , n ) . The function ?is also in C p ( - n , n ) ,and there is a constant M p such that

Furthermore, we have

(1.7) Riesz did not write out (1.6.ii) explicitly, though he indica.ted it plainly. A stronger version of this identity is due to Titchmarsh [1929], uiz.: if f is in CI( -H, n) and if f is also in C ~ ( - Rn), , then (1.63) holds. Something similar appears in Smirnov [1929].

(1.8) As corollaries to Theorem (1.6), Riesz proved that (i)the 2, norms of the partial sums S,f of the Fourier series o f f are bounded over all n,

and (ii)

lim J J-fSnfJlp = 0.

n-oo

From ( i ) he proved that the series on the right side of (1.1.17)converges for all f E Cp(-n,n ) and g E C P t ( - r , n ) , and that its sum is the left side of (1.1.17). He gave an example to show that the series on the right side of (1.1.17) may converge only conditionally if p # 2. This settled decisively the question raised by the Youngs 119131, already described in $1. Though Riesz did not spell it out, one can show from ( i ) that a function ( c U ) F . - , on Z is the Fourier transform of a function in Cp(-n, n ) if and only if the Cp norms of the functions CE=-, c, exp(ivt) are bounded over n. (See Hobson 119261, pp. 610-614.) There is yet more in Riesz [1928].

(1.9) Theorem (Riesa [1928]). Let p be a real number such that 1 < p < 03, and let f be a function in Cp(R). The Cauchy principal value

exists for almost all z E R. The function Hf belongs to (ii)

IlHfllP

&(a), and an inequality

IMPllfllP

obtains, the constant Mp being the same as in (1.6.i).

(1.10) The function Hf is of course the Hilbert tran8forma of f . We shall have constant recourse to it in the sequel. Zygmund soon contributed to the subject.

8


(1.11) Theorem (Zygmund [1929]). Suppose that Ifllog+Ifl is in C ~ ( - T n)'. , Then El( - A , T ) and there are constants B and C such that

f is in

Zygmund cites, and may well have been inspired by, Riesz 119281. Pichorides [1972] comments on the smallests values of B and C in Theorem (1.11)and also on the smallest value of C in (1.4.i).

A great deal more has been published on Hilbert transforms and other singular integrals. This enormous and still expanding theme is not part of our modest endeavor.

Marcel Riesz's Theorem on Conjugate Fourier Series

9

$2. The Bochner-Helson theorem.

Bochner [1939]'O achieved a far-reaching generalization of Riesz's Theorem (1.6), which can be formulated, as Helson 119591 recognized, in terms of orders on groups.

(2.1) Deflnitions. Let X be an Abelian group, written additively. A subset called un order in (or on) X if: (1) P

P

of X is

+ P = {x + rc, : x, sl) E P} = P;

(2) P n (-P)= P n {-x : x E P} = {o};

P u (-P)= x. We write X 5 sl) if sl) - X E P. The relation 5 is a complete order on X I P being the set of nonnegative elements. With P we associate the function sgnp on X. This function is (3)

defined by:

(4)

sgnp(x) =

i

1 if 0 if -1 if

x

E P\{o};

x =O; x

E -P\{O}.

(2.2) Remarks. An Abelian group with an order is ( 0 ) or is torsion-free and infinite.

Every torsion-free Abelian group admits at least two orders, and may admit a great many. For a detailed discussion of orders and their Haar measurability if X has a locally compact topology, see Hewitt and Koshi [1983]. We shall have occasion infra to use some results from this paper. (2.8) Deflnitions. Let X be an Abelian group with an order P, and let G be the (compact and connected] character group of X. Let a be a complex-mlued function on X that vanishes except for a finite number of points in X. For tradition's sake, we write ax for the value of a at the element X of X. A function

is called a trigonometric polynomial on G. The trigonometric polynomial

(2)

-i

C sgnp(X)aXx = f x EX

is called the conjugate trigonometric polynomial to f."

(2.4) Theorem (Bochner [lSSS]). Notation is an in (2.1) and (2.3). Let p be a real number such that 1 < p < 00. There is a constant A, such that

(4

Ilfll, 5 A P l l f l l P l

for all trigonometric polynomials f on G. We will see in the sequel that A, is exactly the constant M, in the original Theorem (1.6) of Marcel Riesz.


10

(2.5) N o t a t i o n and remarks. Let G be a locally compact Abelian group (not necessarily compact) with character group X. Recall that ~1 denotes a Haar measure on G. For a real number p 2 I, we write Cp(G) for the space of all complex-valued Haar-measurable functions g on G for which the norm

is finite. For g E C1(G), its Fourier truneforrn is the function

a on X such that

JG

If G is compact (and only if G is compact), all trigonometric polynomials f are in C1(G). Well-known orthogonality relations show that for a trigonometric polynomial f as in (2.3.1), we have

f ( x ) = ax, x E x.

(3)

If X has an order P, so that we can define the conjugate trigonometric polynomial f as in (2.3.2), it is clear from (3) that

(2.6) Theorem (Bochner [1939], Helson [1968]). Let G be a compact Abelian group with ordered character group X, as in (2.3). Let p be a real number such that 1 < p < 00. Let f be a function in C,(G). There is a function f i n C,(G) such that

(4 The function (ii)

h

= -i sgn,(X)j(x),

x E X.

f satisfies the inequality

llfll, 5 APllfllPl

where A, is as in (2.4). The mapping f C,(G) into itself with norm A,.

H

f

is thus a bounded linear transformation of

(2.7) Remarks. Theorem (2.4) is essentially though not explicitly in Bochner 119391. Helson’s contribution was in formulating Bochner’s conditions in terms of orders on Abelian groups (Helson 119591). The paper Helson (19591 is not concerned with Theorem (2.4), but rather with the Clog’ C case. The paper Helson 119581 deals with the C, case, but only for some special groups. Theorem (2.4) is by no means trivial to prove. A succinct treatment appears in Rudin [1962], Ch. 8, pp. 216-220. The proof is as in M. Riesz 119281 plus some abstract functional

Marcel Riesz’s Theorem oii Conjugate Fourier Series

11

analysis. Theorem (2.6) follows readily from Theorem (2.4), the density of the set of trigonmetric polynomials in C,(G), and the identity (2.5.4). The identity (2.6.1) plainly defines f uniquely, since the Fourier transform of a function identifies the function. We may therefore call .f THE conjugate function of f . Let G be a noncompact locally compact Abelian group whose dual group X admits a Haar-measurable order P. For 1 < p 5 2, and f E L,(G), suppose that there is a function .f also in L,(G) such that ?(z) = - i sgnpi(X) for almost all X E X. Again we will call

f the conjugate function of

f.

(2.8) Connections with M. Ries5’~theorem. Consider the multiplicative group II‘= {exp(it) E CC : --A < t 5 r } . Its character group consists of all functions exp(it) I+ exp(int) for n E Z and so is isomorphic with the additive group Z.The group Z admits exactly two orders. Let P be the order in Z that contains the integer 1. Bochner’s Theorem (2.6) can be interpreted as a generalization of M. Riesz’s theorem (1.6), except of course tha.t the conjugate function f in Theorem (1.6) is explicitly known from Privalov’s Theorem (1.2), while in Theorem (2.6) it is known only as the 2, limit of a certain sequence of unspecified trigonometric polynomials.

The point of view of Theorem (1.6) differs markedly from that of Theorem (2.6). The auxiliary property (1.6.ii) of the conjugate function .f becomes the definiena in Theorem (2.6), while the conjugate function f becomes the definiendum. (2,s) More about the Hilbert transform. Riesz in I19281 does not take up the Fourier transform of the Hilbert transform Hf of f in C,(R) for 1 < p 5 2. He could have, since Titchmarsh 119241 had already proved that the Fourier transform exists for f 6 Cp[R) if 1 < p < 2. For p = 2, this fact is of course Plancherel’s classical theorem. For 1 < p 5 2 and f E E,(R), we know that

i

for almost all t E R. (Here of course R is regarded as its own character group.) The first mention that we have found of (1) is Titchmarsh [1937], Ch. V, p. 120, formula (5.1.8). Now, the additive group R admits exactly two Lebesgue (= Haar) measurable orders (see Hewitt and Koshi 119831). One of these orders is the set { t E R : t 2 O}. Thus Theorem (1.9) provides a perfect analogue of the identity (1.6.ii). The Hilbert transform H f behaves just like the conjugate function f for the two measurable orders of R, for all values of p for which the Fourier transform of H f is defined. Rudin [1962], Ch. 8, p. 226, Theorem 8.7.11, has proved an analogue for (1) for halfspaces in Rk (k € { 2 , 3 , . . .}) but does not construct the Hilbert transform.


12

$3. Themes of this essay.

(3.1) We emphasize that Theorem (2.6) is a pure existence theorem. It offers no way of computing f from f pointwise p-almost everywhere. (The same is true of Plancherel’s theorem, though pointwise methods are well known here.) Thus we have two problems. (3.2) Suppose that we are given some specific discrete Abelian group X containing an order P. As usual, we write G for the (compact, connected) character group of X . Suppose that f belongs to a space of functions on G that contains all of the spaces c,(G) for p > I: for example, C1(G) or Clog+ C(G). Is there a way of constructing a function f on G for which (2.6.i) holds?

(3.3) Suppose that (3.2) has an affirmative answer in some particular case. Do analogues of Kolmogorw’s Theorem (1.4) and Zygmund’s Theorem (1.11) hold? (3.4) Hewitt and Ritter 119831have given reasonably complete answers to (3.2) and (3.3) for all noncyclic subgroups of the additive group $. Each of these groups contains exactly two orders, one of them being the nonnegative rational numbers in the group. They succeeded in constructing f only for f in Clog’ C, and so could not address Kolmogorov’s Theorem (1.4). All else goes through without a hitch, although the constructions and computations are formidable. To our knowledge, this paper is the only published construction of f for any compact Abelian group other than T. (3.5) Now suppose that G is a noncompact locally compact Abelian group with (nondis-

Crete) character group X and that X contains a Haar-measurable order P. Is there an analogue Hf of the Hilbert transform (1.9.i) defined a t least for f E c,(G), 1 < p 5 2, such that

for &almost all x E X ?

(3.6) If Hf exists, can it be explicitly computed? In the remainder of this essay, we explore the questions (3.2)-(3.6).


$4. Orders on

13

Z".

In the present section we classify the orders on Z"(a = 2,3,. . .). Our goal is to obtain analogue of Hewitt and Koshi 119831, Theorem (3.8), for orders in Z a and hence obtain a complete description of orders on Z"as does the theorem of Hewitt and Koshi [1983] for nondense orders in R". (4.1) Definition. (a) Let X be a torsion-free Abelian group. Let A be a subset of X. We say that A is poeifively independent (over Z)if, whenever a l , a 2 , ...,at are in A, nl,n2,. . . , nt are nonnegative integers, and

j=l

we have nl = n2 = . . . = nt = 0. (b) An order P in X is said to be Atchimedean if whenever z and are in P\{O}, there exists a positive integer n such that nz > g/, which is to say nz - g/ E P\{O}. (4.2) Theorem. Let P be an order in Z". (i)The order P is Arcbirnedean if and only if tbere is a vector u = ( a 1 a2,. , . . , a,) in R" sucb tbat the set {alraa,.. . , a"} is linearly independent over Q and

P = {x E Z": u . x 2 0). ( i i ) Let v = (ulrw a r . . . ,u,) be a nonzero vector in R" sucb tbat the set {ul, ua, . . ., ua} is linearly dependent over Q . Then tbere is a non-Arcbirnedesn order Pl sucb tbst {x E La: v .x > 0 } Pl {x E Z": v x 2 0 ) .

5

Proof. Ad (i). Use Theorem (8.1.2.c) of Rudin [1962], p. 194.

Ad (il). Let x l r x 2 ,... , x u be any finite sequence of elements of Za such that v . x( > 0 for C = 1,.. .v. Let a l l (12,. . . , a , be nonnegative integers. Suppose that E t l atxt = 0. We then have Y

U

.

= . . = a, = 0. That is, the Since v x~ > 0 for != 1 , 2 , . . . ,v, (1)implies that a1 = Apply Lemma (2.3) and set {x E Za : v x > 0} is a positively independent subset of Z". Theorem (2.5) of Hewitt and Koshi I19831 to obtain an order Pl on Z"such that 3

(2)

{XE Z":v.x > O}

5 Pl.


14

Note that

z4 = {x € z4: v . x > 0) u {x E L" : v .x= 0) u {x € L" : v .x < o}.

(3)

Since PI is an order, (2) shows that {X

(4)

E Z4: X'V < 0} n P = 0.

The equalities (3) and (2) show that P1

c {x E L" : x - v> O } u {x E z4: x.v = O } = {x E 23" : x *v 2 0).

Since the set {vl,v2,.. . ,v,} is linearly dependent over Q , the set {x E La : x . v = 0) is a nonzero subgroup of Z"which must contain nonzero elements of PI.Hence PI is not a 0 set described by (i), and so Pl is a non-Archimedean order. (4.3) Notation and Remarks. (a) From here on, when dealing with an infinite Abe1ia.n group G, the notions of linear independence of subsets of G, and of a baeie of G, have the same mea.nings as in Hewitt and Ross [1979], pp. 441-442, (A.10). (b) If a basis for a group H contains finitely many elements, say n, then the positive integer n is called the dimension of the group H.

(c) Let G = Z", and consider the elements

where 6tj is Kronecker '8 delta function. We easily check that the set { e t , t' = 1,. . . , a } forms for L".We call this basis the standard baeie for Z".The same is true for G = R". The set { e t , t' = 1,.. . ,Q } is also called the etandard basic for R'. a basis

(d) If A1,A2, ... is a sequence of subsets of R", we say that the eete A, converge to A c R", and we write n-00 lim A, = A if K A n = A = h A , , where

-

limA, =

n( U

A ~ )

n=l p=n

and

u ((7 0 0 0 0

kA, =

n=i p=n

We now present some technical lemmas.

Ap).

Marcel Riesz’s Theorem on Conjugdte Fourier Series

15

(4.4) Lemma. Letb beapositiveinteger. Ifxlrx2,... ,x, arein Z * \ { O } , andifrl, r 2 , .. . ,rv are nonzero real numbers such that the Rb-vectorx = rjxj has positive components,

c,”=,

then there are integers yl,g2,.. . ,gu such that the Rb-vector y = integer components.

Proof. For j = 1 , . . . , v , write x j = ( x l j , q xt > O f o r e = 1 , 2 ,...,b. Wehave xt

t‘ = 1 , 2 . ..,b. Since xf > 0 for zero. Define

E

= rlZtl

j , .

. .,Z b j )

c,”=, has positive

and x = (

yjxj

..,

~ 1 ~ x 2 , .xb),

+ rzxt2 + . .. + r v x t u ,

e = 1 , 2 , .. .,b, not all the z u , e = 1 , . . . ,b, j

= 1 , . . . , v are

is a positive real number. For each integer j = 1 , . . . , v, choose a real number

that 0 5

J E ~ I< E and such that qJ = rJ - ~j ”

E~

such

is a nonzero rational number.

Consider the vector c = C,”=,qixj = (c,, c 2 , . . . ,cb). For

j=l

so that

e = 1,... ,b we have

U

j= 1

For j = 1,.. . ,v and t‘ = 1 , 2 , . . . ,b, either and the choice of E ; we get

xlj

j=l

= 0, so that E j z U = 0, or z t j

# 0. Using (2),

= $Z(

> 0. The relations (3) show that ct is a positive real number. Also, it is clear that, for each e = 1 , 2 , . . . ,b, ct is a rational number. Thus the vector c = (cI,c a r . .. , c b ) has positive rational components. Multiplying the vector c by a suitable positive integer, we obtain the 0 desired vector y.


16

(4.5) Lemma. Let A be an m x n matrix with integer entries. The set

is a linear subspace of Rn spanned by vectors with integer coefficients.

Proof. Apply Theorem 18 and Theorem 1 7 of Birkhoff and MacLane 11965, p. 220 to obtain an m x n matrix B, an m x m matrix P and an n x n matrix Q such that

B = PAQ,

ere I, is the r x r identity matrix and O f , j is the 1 x j matrix of zeros. Theorem 13 and Corollary 1 on p. 216, and the results on p. 219 of Birkhoff and MacLane 119651show that the matrices P and Q are nonsingular matrices that are products of elementary matrices. It is clear that the process of reducing the matrix A to the matrix B can be carried out by using only elementary matrices with rational entries. Hence the matrices P and Q may be taken with rational entries. Let N = {x E Rn : B . x = 0 }. The sets N and K are obviously linear subspaces of Rn.A vector x in Rn is in K if and only if A x = AQQ-'x = 0. Since P is nonsingular, x is in K if and only if PAQQ-'x = 0. Hence K = Q ( N ) . Since Q is nonsingular, the dimensions of N and K are the same, and the image under Q of any basis for N is a basis for K. For x = ( z l , z 2 , .. .,z,,. . ., Zn) in R', (2) shows that the vectors e t , e = r + 1,...,n span the subspace N. Hence the set {Q(ef): t = r 1,.. , ,n} is a basis for K. Since Q has rational entries, we multiply each vector in this basis by a suitable positive integer to obtain the desired basis for K. 0

-

+

(4.6) Lemma. Let P be an order in Z a . Let x l , x 2 , ... , x , be vectors in P\{O}, not necessariJy distinct. Suppose that for some nonnegative real numbers a l , a 2 , . . . ,a, we have n f=1

It then follows that a1 = a2 = . . . = a, = 0. Proof. The proof is by contradiction. If a1 = a ) = .. . = an = 0, the proof is complete. If some at is nonzero, we may consider only the nonzero at's, and so lose no generality in assuming that no at vanishes. Fort = 1,...,n we write x f = ( z l t ,zat,. . . ,z,t).Let A be t h e a x m matrix (zjf);=l Our hypothesis states that the vector LY = ( a l ,0 2 , . . . ,a,) is a nonzero vector in the nullspace of the matrix A . Since the matrix A has integer entries, Lemma (4.5) shows that


17

there are vectors ql,92,.. .,q, (1 5 8 5 m - 1) with integer components, and real numPtqt. Since the vector a has positive real combers PI, Pa, .. . ,be such that a = ponents, Lemma (4.4) implies that there are integers gl, ~ 2 , .. . ,us such that the vector t= gtqf = ( t l , t z , . . . ,t n ) has positive integer components. Thus the vector t E R" is a nonzero vector in the null-space of A. Equivalently, we have

c;=,

ct,

n

j=1

= 0, where ti are positive integers for j = I , . . . , n. Now apply Theorem (2.4) of Hewitt and Koshi I19831 to see that (I) is an impossibility. Therefore the assumption that some at is nonzero is false. 0 We now consider P and Za as subsets of Ra. (4.7) Lemma. Let P be an order in Za. Let

P' =

( e a j x j

: xj E

P, ai E R f }

where n and m are arbitrary positive integers. We then have

P' n p 2 = {o}.

(ii)

Proof. If for some a',

a2,.

..,an,P I , b 2 , . . ., j3m

yl,y2,., .,ym in -(P\{O}) we have

C'!3=1

a3 x3

+ Em 3=1 ~ . ( - y j )= 3

Lemma (4.6) shows that

a1

in R',

.

x ' , x ~., . , x n in P\{O}

c;==,= c,"=, pjyi, then

and

ajxj

0. Since ~ 1 ~ x..2.,x,, , -yz ,... , -Ym are in P\{O}, n = a2 = ... = a, = = . . . = Prn = 0.

(4.8) Lemma. Let P be an order in Za,and let P' be as in (4.74. Then (i)

P' has nonvoid interior;

(ii) P'\{O}

is a positively independent subset of R".

Proof. Let {el,e p , ... ,e,} be the standard basis for R".Since P is an order on Z", just one of ec and -et is in P for e = 1 , 2 , . . . , a . Hence there is a sequence ( E ~ , B .~. ,. ,e a ) where ce = fl such that &tee is in P for e = I, 2, . . . , a . The nonvoid set U = {x = (~1~22, . ..,z,) E Ra, < E ~ Z L< I = 1, 2, . . . ,a} is open and contained in P'. This

t,

N. Asrnar and E. Hewitt

18

proves ( i ) . We prove ( i i )by contradiction. Let X I , xa,. . . , X L be in P1\{O}, let n l , n2,.. . , nl be nonnegative integers, and assume that

If nl = n2 = . . . = n~ = 0, the proof is complete. If some n j is nonzero, we may consider only the nonzero nj's, and so lose no generality in supposing that no n, vanishes. For j = 1 , 2 , . ..,e we write x j = P k j X k j , for some positive real numbers P k j and nonzero elements x k j of P. Then (1) implies that

zyLl

The vectors x k j ( j = 1 , 2 , . . . , e, k = 1 , 2 , ,. . , m i ) are in P\{O} and each is multiplied by the positive real number n , @ k j . By Lemma (4.6), every n3& is zero. Since P k j is positive, 0 each n3 vanishes. This is a contradiction. We now prove a principal fact about orders in (4.9) Theorem. Let

a".

P be any order in La.There exists a nonzero linear mapping Ll

from 25" into IR such that

Furthermore, the following are equivalent: (ii)P is Archimedean;

( i i i ) L ; ' ( { O } )= ( 0 ) ; (i.) P\{o} = L;'(]O, mI);

(u) P = L;'([o,m[). Proof. Let P' be as in (4.7.i). Lemma (4.8.ii) shows that the set P1\{O} is a positively independent subset of Eta. Apply Lemma (2.3) and Theorem (2.5) of Hewitt and Koshi I19831 to obtain an order P* on R" such that (1)

P $ Pl c P*.

Lemma (4.8.i) and (1) show that P* has nonvoid interior. A simple argument, which we omit, shows that P* is a nondense order in the additive group IR'. From (1) we obtain -P 5 Pa c -P*.Since P' is an order on R", we have (-P*) n P* = {0}, and so (-P) n P* = ( 0 ) . We now cite Theorem (3.8) of Hewitt and Koshi [1983]: there is a linear mapping L of Ra onto R such that (2)

5

L-710, a[)P*

s. L-'([o,

031).


5

19

5

Thus we have L-'(] - c o , O [ ) -P* L-'(] - 00,0]). Let L1 be the restriction of L to Za. Plainly L1 is a linear mapping of Za into R. It is clear that L ( e l ) is different from 0 for some e in {1,2,. . . ,a}, since L is linear and not the zero mapping. Hence L1 is not the zero mapping on Za. We have

5

L;'(]O, m[) = L-'(]O, 0 0 [ ) n Za P* n Z a ,

L,'

and

(1 - 00, o[) = L - ~ ( -] 00, o[) n La 5 ( - P * ) n iza.

Also, it is clear that P* n Za = P, and (-P*) n Z"= -P. Putting this together we find that L;'(]O,m[)

(3)

5P

and LT1(] - c o , O [ )

5 -P.

Using the second inclusions in (2), we find that

P = P* n Z a c L-'([o, CO[) n za= L;'([o, 0 0 [ ) .

(4)

The relations (3) and (4) establish ( i ) . Suppose that P is an Archimedean order on Za. Assume that LT1({O}) contains a nonzero vector u. Then nu is in LT1( (0)) for every positive integer n. Given any vector v in P\Lyl({O}), we have Ll(nu - v) = Ll(nu)

- Ll(v)

= --Ll(V)

< 0, and so nu < v for all positive integers n. This contradicts (4.1.b). Therefore (iii) holds. It is now a simple matter to show that ( i i ) - ( i v ) are equivalent. 0 We offer yet another technicality.

Lemma 4.10. Let F = {x(}kl be a nonvoid finite subset of Za. Suppose that there are a vector ul in Ra and a positive real number 6 such that

for e = 1 , 2 , . . . , 8 . Then there is a vector u whose coordinates are a linearly independent set over Q and for which

fore = i,2,.. . , 8 .

Proof. Write a1 = ( a l l ,a21,. . .,a a l ) , XL = zl(,22(, . . . ,z , ~for e = 1 , 2 , . . . , 8 . Let r be a positive irrational number that is not in the linear span over Q of { a l l ,~ 1 ,..., aal } . Since

N Asmar and E. Hewitt

20

F cannot contain 0, for every t' in {1,2,.. . , 8 } there is at least one integer in { 1 , 2 , .. . , a } such that z i t # 0. Choose a positive integer n such that n

the minimum being taken over all j in {1,2,. . . , a } and t' in {1,2,.. . , s} for which zit # 0. Now define

r a3 = aj1 + -

(2)

n

for j' = 1 , 2 , .. . ,a. Since r is not in the linear span over Q of { a l l ,~ ~ 2 1.,.,. aal}, the definition (2) shows that the set {ul, a 2 , . , . ,a,} is linearly independent over Q . For 1 = 1 , 2 , . .. , a , (2) implies that

f:

a j z j t = f:(ajl+

(3)

j=1

r

,)zjt =

f:

aj12jt

+ f:'nz i t . j=1

j=1

j=1

The inequalities (1) show that

" 6

(4)

- _6 -

2'

The relations ( i ) , (3) and (4) imply that This proves ( i i ) .

cq=l

ajzjt

2

for all (zit, zzt,.. . , z a t ) in F.

The next two theorems are vital in our study of orders on 22,.

Theorem 4.11. Let S be a finite subset of Ea\{0}. Let P be an order on Z a . There is an Archimedean order PI in 22, such that

(4

S n P = s n PI

and (ii)

S n (-P)= S n (-PI).

Proof. Clearly it suffices to prove ( i ) . We may suppose that S = S u (-S) = -S. We will find a vector IY = ( a ~a2, , .. . , a,) in R" such that the set {al,aa,. . .,a,}is linearly independent over Q and such that


21

for every x in S n P, and u * y< 0

(2)

for every y in S n (-P). We then define the order PI as in Theorem (4.2.i): Pl = {x E Z a : u ' x 2 O}. It is clear that ( i ) holds for this P l . We proceed to construct the vector a. Let conv(S n P) and conv(S n (-P)) denote the convex hulls of S n P and S n (-P), respectively, in Ra. We have: (3)

Sn(-P) = -(SnP);

k=l

(5)

k=l

conv(S n P) = -conv(S n (-P)).

Since S n P does not contain 0, Lemma (4.6) shows that conv(S n P) does not contain 0. From (5) we see that conv(S n (-P)) does not contain 0. Since conv(S n P) and conv(S n (-P)) are subsets of P' and P2,respectively, not containing 0 (P' and P2 being defined as in (4.7.i)), it follows from Lemma ( 4 . 7 4 that conv(SnP) and conv(Sn(-P)) are disjoint. Plainly ponv(S n P) and conv(S n (-P)) are compact. We apply Theorem (34.1) of Berberian 11s p. 134 (see also p. 122, (30.1)) to find real numbers a l l , ( ~ 2 1 ,.. . ,sol, such that

for all ( z ~ , z z ,. .. ,)'2

in conv(S n P) and a

(7) t= 1

for all ( g l , g2,. . . , ga) in conv(Sn (-P)). The inequality (6) and the equality (5) show that a

L= 1

for all (yl,g2,... ,ga) in conv(S n (-P)). We apply Lemma (4.10) with F = S n P and 6 = 1 to obtain a vector u = ( a l ,a2,. . .,a') such that the set { a l ,a2, . . . , a a }is linearly independent over Q and u*x>0

for every x in S n P. The relations (5) and (3) show that a * y L(xl). F'rom(i) weseethat (x~-x~,O,O)isinP,andso(x~,f,g)+(xz-x~,O,O) (x2,f, y) is in P. Similarly, if (XI,f , g ) is in -P and L(x2) < L(xI) then (x2,f , g) is in

-P. From this we infer that the inequalities -m (1)

< sup{L(x) : x in Ba, {x} x F x

5 inf{L(x) : x in R", {x}x F x

{g}

{g}

c -P}

c P} < 00

hold for every g in H. We claim that the second inequality is in fact an equality. Assume the contrary: there is an interval 10, b[ with the property that, whenever x in B" is such t h a t L ( x ) i s i n ] a , b [ ,then { x } x F x { g } n P + 0 a n d { x ) x F x { g } n ( - P ) # 0 . Itisclear then that if x in R" is such that L(x) is in ] - b, -a[, then {x} x F x {-g} n P # 0 and {x}x F x { -g} n (-P) # 0. Consider the open nonvoid neighborhood V of 0 in R", defined by V = L-'(]a, b[) +L-'(] - b, -a[) = L-'(]a, b[) - L-'(]a, a[). Let x be any element of V. W r i t e x = x l + x 2 whereL(xl)isin]a,b[ andL(xa)isin]-b,-a[. Let f i , f 2 , f s , a n d f 4 be elements of F such that (XI, f i ,g) and (x2,f2, -g) are in P and (xl,f ~g), and (x2,f 4 , -g) are in -P. It follows that (x,f1 fa,O) is in P and (xlfs+ f r , O ) is in -P. From (i) it follows that L(x) = 0. Plainly this is impossible since it implies that L is identically zero. Thus the second inequality in (I) is an equality.

+

For every y in H, let a(g) be the number defined by either the sup or the inf in (1).It is clear from the definition of a ( y ) that (ii) and (iii) hold. We now show that a is a continuous homomorphism. Let y and g' be in H and let x E R" be such that L(x) > a ( g ) + a ( g ' ) . Let x' in Ra be such that L(x) - .(Y')

(2)

> L(x') > .(ar),

and let 5 = x - x'. From (2) we see that L(B) > a(g'). Consequently we have {x'} x F x {g} c P and {B} x F x {g'} c P. These inclusions imply that {x'+ 5 ) x F x {y + y'} c P o r { x } x F x { y + y ' } c P . Wehave a(y

+ d ) = inf{l(x) : {x}x F x {g + y'}

c P}

5 inf{L(x) :L(x) > a(#)+ a(g')} = a(!/)+ ab').


25

+

Similarly, if x in B" is such that L(x) < a ( g ) a(g'),write x = x' + 5 , where L(x') < a ( g ) and L ( 5 ) < a(g'). From ( i i ) it follows that {x'} x F x {y} c -P, and ( 5 ) x F x {y} c -P, so that {x}x F x {g + g'} c -P. We have

That is, a is a homomorphism of H into R". Finally, to establish the continuity of a it suffices to show that a is bounded on a neighborhood of 0 in H. Since P is nondense, there are nonvoid open subsets U,V, and W of B", F and H respectively, such that U x F x W is contained in P and such that U is bounded in a".Let u be a real number such that L ( x ) < u for all x in U. F'rom the definition of a it follows that a ( w ) 5 u for all w in W . Choose any (00 in W , The homomorphism a is bounded above on the neighborhood W - wo of 0, and is bounded below on the neighborhood -W w0 of 0. Thus a is bounded on the neighborhood (W - w o ) n (-W w0) of 0. 0

+

+

We can now classify an important family of orders. (5.4) Theorem. Notation is as in (5.3). The mapping r defined on &" x F x H by

is a continuous homomorphism of Rax

F x IT onto R such that

(ii)

and (iii) (iu)

r-'(] - co,01)

5 -P 5 r - l ( ] - 01). 00,

If H is u-compact, r-l({O}) has Haar measure zero and P is Haar-measurable.

Proof. Parts (i)- ( i i i ) are immediate consequences of Theorem (5.3). To prove ( i u ) we use the fact that every locally null subset of a u-compact group has Haar measure zero. Assume that r - I ( { O } ) is not locally null, and that K c r-'({O}) is compact with positive Haar measure. Then K - K c r-l({O}) contains an open neighborhood of 0. This is impossible since r is not identically zero. 0 The orders describes in Theorem (5.4) are analogues of the Archimedean orders on a", which are identi5ed in Theorem (4.9). We now embark on the study of Haar-measurable orders.


26

(5.5) Lemma. Suppose that P is a Haar-measurable order on Ra x F x H where H is a discrete torsion-free Abelian group. The set P n (R"x ((0,O))) is an order on Ba x ((0,O)) that is measurable with respect to Haar measure on Ra.

Proof. By Theorem (3.8) of Hewitt and Koshi 119831 it is enough to show that P n (Rax ((0,O))) is nondense in Ra x {(O,O)}. Assume the contrary: P n (R" x (0,O))) is dense in Rax {(O,O)}. Since F is compact, Theorem (3.2) of Hewitt and Koshi 119831 shows that the set P n ((0)x F x (0)) is dense in (0) x F x (0). Hence the set P n (R" x F x (O)), which contains P n (R"x ((0,O))) + P n ((0) x F x (0}), is dense in (R" x F x {0}), and so is non Haar-measurable with respect to Haar measure on B" x F x (0). Plainly this 0 contradicts the fact that P is a Haar-measurable subset of R" x F x H . (5.6) Remark. Lemma (5.5) need not hold for nondiscrete H. Consider the group R2= R x R . Let Po be any dense non-Lebesgue measurable order in R (Hewitt and Koshi 119831, Theorem (3.3) and Remarks (3,4,a,b)). The set P = {x = ( X I ,2 2 ) in R2; z1 > 0} U (x = (0,zz) in R2,za in Po} is a Lebesgue-measurable order on R2.

The next lemma, while simple, will be very useful, (5.7) Lemma. Notation is as in (5.5). For every y in H, exactly one of the following holds: ( i ) R" x

Fx

(y)

c P;

( i i ) (Rax F x (y))

(iii)

P n (R" x F x

n P = 0; (y)) is nondense in

R" x F x

{y}.

Proof. Since H is discrete, the set P n (B"x F x (0)) is measurable with respect to Haar measure on Ra x F x (0). It follows from Theorem (3.1) of Hewitt and Koshi 119831 that there are open nonvoid subsets U and V of R"x F such that U X(0) C P and V x (0) C -P. Suppose that neither ( i ) nor ( i i ) holds, and that (xl, f l ,y) is in P and (x2,f 2 , y) is in -P. Then we have (xl,fl,y) + (U x (0)) c P and ( x 2 , f 2 , y ) + (V x (0)) c -P. That is, the set P is nondense in R" x F x {g}. 0 We next construct a useful family of orders. (5.8) Theorem. Let H be a countable discrete torsion-free Abelian group. Let L he a nonzero continuous real-valued homomorphism on Ra and let a be a real-valued homomorphism on H (a may be identically 0). Consider the mapping T defined on R" x F x H

bY

(4

+, f; u ) = L(x) -

and suppose that F # (0) or that a > 1. Then T is a continuous real-valued homomorphism, and there is a nondense order P on R" x F x H such that (ii)

7-1()0,co[)

5 P s; r - ' ( [ o , w [ ) .


21

The set r-'({O}) has Haar measure zero, and the order P is a strongly nondense Haarmeasurable subset of R" x F x H.

Proof. The first assertion is obvious. Next consider the set

The set ( I ) is nonvoid and positively independent. By Remark (2.6) of Hewitt and Koshi 119833, there is an order P on Ra x F x H that contains (1). We have 7-1(]0,00[) c P,r-'(] - q O [ ) c -P,and so P c r-'([O,Co[). I t is easy to show that the inclusions in ( i i ) are strict. This proves ( i i ) . That r - l ( { O } ) is Ha,ar-measurable follows from (5.4.i~). To prove the last assertion, we note that for every y in H there are x1 and x2 in R" such that r(x1, f, y) > 0 and r(x2,f, y) < 0 for all f in F. Thus the set R" x F x {g} contains elements of P and elements of -P. According to Lemma (5.7), the set P n (Ra x F x {y}) is nondense in Ra x F x {y}. That is, P is strongly nondense. 0 (5.9) Remarks. (a) One easily checks that the set H' of all g in H such that Ra x F x {y} contains elements of P and elements of -P is a subgroup of H. It is also easy to see that if ky is in H' for some positive integer k, then g in in H'.

(b) The case H = Zb,where b is a positive integer, is of particular interest. In this case, (a) and Theorem (A.6) of Hewitt and Ross 119791 pp. 450-451, show that there is a basis e\,ea,.. ,eb of Zbsuch that el, ea,. . .,eL is a basis for HI,1 5 h 5 b, whenever H' is not (0).Hence if H' # {0}, every element y in Zb can be written uniquely

.

where

y1

is a linear combination of ei, eh, . . . ,eL and hence is in H'.

In the remainder of this section, we take H = Zb, where b is a positive integer. We continue with our study of Haar-measurable orders. (5.10) Lemma. Let P be any order on Z b , and let S be any finite subset of Zb\{O}. Then there is a real-valued homomorphism (Y on Z b such that a(y) > 0 for ally in S n P 0 and a ( y ) < 0 for ally in s n (-P).

Proof. See the proof of Theorem (4.11).

n

(5.11) Lemma. Let P be a Haar-measurable order on &" x F x Zb. Notation is as in (5.9). Let J be a ffnite symmetric subset of Zb\H'. There is a real-valued homomorphism a1 on Zbsuch that al(ya)> 0 for ally in J for which R" x F x {y} c P and al(y2) < 0 for ally in J for which R" x F x {y} c -P.

Proof. If J is void there is nothing to prove. If H' = {0}, we may appeal to Lemma (5.10). So suppose that J is nonvoid and that H' # (0).Denote by Pl the order Pn ({O,O)} x Zb)

28


on {(O,O)}xZb. Clearly we have ({(o,o)}xJ)nPl = {(O,O)}x {y E J : R"x F x {y} c P}, and ( ( ( 0 , O ) )x J) n (-PI) = {(O,O)} x {y E J : Ra x F x {y} c -P}. Write each y in J &II in (5.Q1b), and let

Weclaimthat {(O,O)}XJZ C PI.Assumethecontrary. Thenforsome(O,O,y) in ({(O,O)}x J) n PI,the corresponding (O,O,y2) is not in PI. Hence ( 0 , O , y ~ must ) be in -Pl, and so (O,O,yg) is in -P. Write R" x F x {y} = Ra x F x {yl} (O,O,ya). It follows that Ra x F x {y} contains elements in -P. This is impossible since y is in (((0,O))x J)n Pl. This establishes our claim. The lemma follows now by applying Lemma (5.10) to the set 0 S = J2 U (-&) and the order PI on E b .

+

We now give a set-theoretic technicality.

(6.12) Lemma. Let X be a locally compact torsion-free Abelian group. Suppose that P and P* are Haar-measurable orders on X and that K is a symmetric subset of X. Denote by px Haar measure on X. The following are equivalent:

Proof. It is easy to see we need only show that ( i ) implies (ii).Suppose that ( i ) , and hence (iu) hold, and assume that (ii)fails. Then there is a subset M of X such that M c K n P*,M n K n P = 8, px(M) > 0. I t follows that M n P = 0, and so M c -P and M n (-P*)= 0. Thus M is contained in (K n (-P))\(K n (-P*)). This contradicts (iu), because px(M)> 0. 0 The following theorem is fundamental. Its proof is long but not conceptually difficult.

(6.1s) Theorem. Let P be a HaaFmeasurable order on the group R" x F x Zb, where a and b are positive integers 8nd F is a compact torsion-free Abelian group. Let K be a compact subset of R" x F x Zb. There is a strongly nondense Haar-measurable order P* on R' x F x Zbsuch that


29

and P ( ( K n ( - p * ) ) \ ~n (-PI)= 0,

(ii)

where p denotes a Haar measure on R" x F x Zb.

Proof. Clearly if the conclusion of the theorem holds with some order P' and some set K , then it holds with the same order P* and all compact subsets of K. Thus we may suppose that K is a compact symmetric subset of Ra x F x Zb. We will construct the order P* by applying Theorem (5.8). Thus we need to define appropriate homomorphisms L on R" and a on Zb. By Lemma (5.5) the set P n (a" x (0,O)) is a Lebesgue-measurable order on R". By Theorem (3.8) of Hewitt and Koshi 119831, there is a real-valued homomorphism on Ra such that

L-'(lol

)I.

x

{(OlO))

5 p n (a"x

{(OlO)))

s: L-'([o,m[) x {(O,~)).

Write K = (us=l(Cj x {yj}) U (U:==,(D( x {B!})), where Cj and Dt are compa.ct subsets of Ra x F, the yj's are in H', and the 5~'s are in Zb\H', where H' is aa in (5.9,b). We will use the decomposition x = x1 +x2 for the element x in Zb,as we did in (5.9,b). By Lemma (5.7), the order P n (R" x F x H I ) is strongly nondense in Ra x F x H'. By Theorem (5.3) there is a real-valued homomorphism a1 on HI such that L-'(] - m,w(r)l) x F x {Y) 5 (-PIn (Rax F x H'), and L-'(]a~(d,m[)x F x {Y} 5 P n (R" x F x H') for each y E H'. Also, from (5.4), the homomorphism rl defined on R" x F x H' by q(x,fly)= L(x)- al(y) is continuous and has the following properties:

5

5

r ~ l ( ] O , m ( )P n (Ra x F x H') r;l([O,

(1)

001);

and p'(rF1({O})) = 0, where p' denotes Haar measure on R" x F x H' I t is clear that this set also has p measure zero. By Lemma (5.11) there is a real-valued homomorphism 1,.. . ,d, we have

>0

(2)

if lRa x F x

a2((5~)2) {5(}

is contained in PI and

of Zb such that for I =

N Asmur and E. Hewitt

30

We distinguish two cases. Case A. v = 0. By (2) and (3), the set K is contained in R" x F x H'. Extend the homomorphism a1 t o a real-valued homomorphism a: on all of Zb. Let r;* be the homomorphism defined on R" x F x Zbby

Notice that r; agrees with rl on the subgroup R" x F x HI. Thus we have

(R" x F x H ' ) n (r:)-'(]O,co[) (4)

H')n r;l(]O,co[), (R" x F x H I ) n (r;)-'([O,co[) = (R" x F x HI)n r;l([O1co[), = (R" x F x

and

(R"x F x XI)n (r;)-'({O})

= (R" x

F x H ' ) n r;'({O}),

Also, the sets (r;)-l({O}) and rc'({O}) have Ha.ar measure zero. (See the proof of Theorem (5.4).) We now apply Theorem (5.8) to obtain a strongly nondense order P* on R"x F x Z b such that

From (4) and (5) we infer that

P* n (R" x F x H I ) n (r;*)-'(]O,co[)=(R" x F x H I ) n (r:)-'(]O,m[) (6)

=(Ra x F x HI)n r;'(]O, coo.

From (I), we see that the set (R" x F x HI)n rF1(]O1co[)differs from (R"x F x H') n P by a set of p measure eero. A simple calculation, which we omit, shows that

p ( ( P n (R" x F x H'))\(P* n (R" x F x H I ) ) )= 0. Since K is contained in R" x F x HI, we get p ( ( P n K)\(P* n K ) ) = 0. Thus we have found the desired order P* in this case. Case

B. v > 0. Define the continuous real-valued homomorphism r on R" x F x Zbby


31

so that r(x,f,Y)= L(x) - 4Y).

According to Theorem (5.8),there is a strongly nondense Haar measurable order P* on R" x F x Z b with the property that

r-'(]o,cm[)

(8)

5 P* 5 r-'([o,co[).

It remains to show that P* satisfies the equalities ( i ) and ( i i ) . Let (x, f, y) be in K n P. We distinguish three cases. C a s e B.I. The element y is in

H' (y2= 0) and L(x) = al(y). Then (x, f,y)belongs to

r-l({O}). This set has Haar measure zero. C a s e B.11. The element y is in

H' and L(x) is different from al(y). Clearly (x,f,y)

belongs to P n (R" x F x H').The relations (1) show that L(x) - W(Y) > 0.

Since the corresponding element y2 to y is 0, it follows that a(y) = a1 (y)and so r(x,f , y) > 0. Thus (x,f,y)belongs to r-'(]O,m[). Fkom (8) it follows that (x,f,y)belong to P*. C a s e B.111. The element y is not in

H'. Then necessarily we have

R" x F x {y}

5 P.

From (2) we get aa(Y2) > 0.

From the definition of v. it follows that

and so +,f,Y) = L(x)- W ( Y J

K:

+ -a2(Y2) cc

2 L(x) - Ql(Y1) + 21. Hence (x,fly)belongs to r - l ( ] O , m [ ) and , so it belongs to P*. From the above cases, we conclude that p ( ( K n P)\(K n P * ) )= 0. By Lemma (5.12) the proof is complete in this case. We have now dealt with all cases.

0

The next and last theorem of this section identifies a remarkable property of all Haarmeasurable orders.

32


(5.14) Theorem. Let P be a Haar-measurable order on a locally compact torsion-free Abelian group X. Let K be any compact subset of X. There are a continuous real-valued homomorphism t) on X and a subset N of X of Haar measure zero such that:

for all z E ( K n P)\(N u (0));

for all z E ( K n -P)\(N u (0)).

Proof. If K is void there is nothing to prove. Also it is clear that if the conclusion of the theorem holds for some set K and a homomorphism t), then it holds with K replaced by any compact subset of K and the same homomorphism t). So we may suppose that K is a symmetric subset of X containing a neighborhood of the identity. Let B be the subgroup of X generated by K. Then B is compactly generated [for a discussion of compactly generated groups, see Hewitt and Ross 119791, p. 35, Definition (5.12)]. A structure theorem for locally compact Abelian groups (Hewitt and Ross [1979],p. 95, Theorem (9.14)) asserts that B is topologically isomorphic with R" x F x Zb,where a and b are nonnegative integers and F is a compact (obviously torsion-free) Abelian group. Since X admits the measurable order P, either it is discrete in which case B p! 12Zb, or a is necessarily a positive integer. In the first case, apply Lemma (5.10). Suppose that we are in the second case. Apply Theorem (5.13) to obtain a strongly nondense Baar-measurable order P* on B for which (5.13.i) holds. Apply Theorem (5.4) to see that P* differs from r 1 ( ] 0 ,co[)only by a subset of r1({ 0} ), which has Haar measure 0 in B . Combining these observations, we find a homomorphism r on B satisfying (i)and ( i i ) . Extend r in any way to a real-valued (plainly continuous) homomorphism $Jon X. Since B is open, $ - I ( (0)) has Haar measure 0 on X, and evidently ( i ) and ( i i ) still hold. 0


33

$8. The H i l b e r t t r a n s f o r m on locally compact A b e l i a n groups. We now begin our task of answering question (3.6).

(8.1) Introduction. Let G be a locally compact Abelian group with character group X. The group X need not be ordered. We suppose instead that there is a nonzero continuous homomorphism r from X into R. Let p denote the adjoint homomorphism of r. The mapping p is also continuous and satisfies the identity

x o p(r) = exp(ir(X)t)

(4

for all r in R and all X in X (Hewitt and Ross 119791, p. 392, (24.37)). With r we associate the function sgn, defined on X by:

1 if

r(X) > 0;

(ii)

For f in l l ( G ) , Lemma (20.6) of Hewitt and Ross 119791,p. 286 shows that the function

+

(2, t) f (z p(t)) is p x A-measurable (here p is Haar measure on G and A is Lebesgue measure on R). We adhere to this G and the present notation through the present section.

Consider the one-parameter group of transformations U' acting on G by translation by p(t). That is, U t ( z )= z + p(t),for all z in G. We will apply the results of Calder6n [1968] in this set-up, taking M = G and T, to be the truncated Hilbert transform on R. I t is easy to check that the results of Calder6n 119681 still hold when M is replaced by any locally compact Abelian group G, not necessarily o-compact (see Asmar [1986], section 2). Let us recall some classical definitions and facts.

(8.2) Deflnition. Let f be in & I (R) and let n be a positive integer. The truncated Ifilbert traneform Hn f off is defined by

The Hilbert traneform H f off is defined by

The mazimal Hilbert tranejorm MHf off is defined by

(8.3) Theorem. I f f is in &,(R)(1 < p < m), the Hilbert transform Hf exists almost everywhere and satisfies the inequality

34


where Mp is as in (1.6.i). ( i i ) Iff is in L,(B), the Hilbert transform Hf exists almost everywhere. Iff is in Lp(R)(1 < p < oo), then the inequality

holds for every positive integer n, where Mp is as in ( i ) . Iff is in L,(R)(l < p < oo), there exists a constant Cp depending only on p such that

Iff is in Ll(R), there exist constants A and B such that for every positive real number g, the inequalities

and

hold.

For ( i ) and ( i i i ) see Zygmund 11959, Vol.II], Ch. XVI, $3, p. 256 Theorem (3.8). For ( i i ) , ( i w ) , ( w ) and (ui) see Garsia 119701, Sections 4.3, 4.4, pp. 112-128,4.3.1, 4.3.9, 4.4.2, 4.4.3.

6

6

We recall that the best constant Mp is tan if 1 < p 5 2 and cot if 2 5 p < oo (see Pichorides [1972]). For p in ]I,oo[,this number will be called M. Rieez’8 conetant and will be denoted by Mp. (6.4) Deflnition. Let p be a number in [l,cm[, and let f be in Lp(G). For every positive

integer n, the function

is defined for palmost all z in GI2. The function HL f is called the nth truncated

HiZbert

fran8form off on G. The mazimal Hilbert tran8form off on G is defined for galmost all 2

bY

(ii)

Combining the results of Calder6n 119681 and the properties of the classical Hilbert transform on R, we obtain the basic properties of the Hilbert transform on G. We proceed to list them.


35

(8.5) Theorem. Notation is as in (6.4). For every function f in f,(G)( I I})5

BP ~

~

~

f

~

~

p

,

G

for all y > 0, where B, = max(A,Cp), A is BS in ( 6 . 3 , ~and ) C, is as in (6.3,iw).

< oo), there is a function Hrf such that Hi f converges to Hrf, palmost everywhere on G and in the L,(G) norm for 1 0, where B, is as in ( i i i ) . The details of the proof of Theorem (6.5) are formidable. To keep this essay within reasonable bounds, we omit the proof. They are available in Asmar [1986]. A diligent reader can also construct them from Calder6n [1968]. (6.8) Deflnition. For f in L,(G), 1 5 p < 00, the function H ' f given by (6.5,iv) is called the Hilbert franuform of f (with respect to the homomorphism 7 ) .

Theorem 8.7. Let p be a number in )1,2],and Jet f be in L,(G).The Fourier transform of H'f satisfies the equality

(4

H7f(X)

= -I' sgn,(X).f(X)

for aJmost all X in X (with respect to Haar measure on X). Proof. It is enough to show that (i) holds for all f in l l ( G )n L,(G).Because of (6.5,iv), (i)will be established for f in Zl(G) f~L,(G)if we succeed in showing, for example, that for almost all X in X, (1)

lim (L-00

HT~(x) = -i sgn,(X)j(X).

36


For every X in X we have

The equality (I) follows now from the classical identity

which can be found, for example, in Zygmund [1959,Vol. I], Ch. II., $7, p. 56, (7.4). (6.8) Remark. It is clear from the uniqueness of Fourier transforms, Theorem (6.7),and the definition in (2.7) of the conjugate function, that the function H'f is the conjugate function f of f with respect t o a Haar-measura,ble order P whenever the equality

(4

sgn,X = sgnpX

holds for almost all X in X. In the following theorems, we list cases where all or some Haar-measurable orders admit r's such that ( i ) holds. (6.9) Theorem. Let B be a torsion-free infinite locally compact Abelian group that is the union of its compact open subgroups, and let a be a positive integer. Let P be any Haar-measurable order in the group R" x B and let p be a number in [l,co[. There is a continuous homomorphism r from R" x B onto IR such that (6.8,i) hold. For f in I,(G), where G is the character group of X, the conjugate function of f is obtained as the pointwise limit of the functions (6.4,i) and has the properties (6.5,i-wi).

f

Proof. Apply Theorem (5.2)and Theorem (6.5).

(6.10) Theorem. Let H be a torsion-free countable discrete Abelian group and let F be compact torsion-free Abelian group. Let P be any strongly nondense order in the group R" x H x F and let p be a number in [I,co[.There is a continuous homomorphism r from Ra x H x F onto R such that (6.8,i) holds. For f in I,(G), where G is the character a


group of R" x

H x F, the coqjugate function

31

f off is obtained as the pointwise limit of

the function (6.4,i) and has the properties (6.5,i-wi).

Proof. To obtain the homomorphism r apply Theorem (5.4).

0

(8.11) Applications. (a) Take G = T,X = Z,and r the identity homomorphism of Z into R. The adjoint homomorphism p of r is the natural homomorphism of R onto R/Z. Theorem (6.5) yields the classical results about the conjugate function on T.The summability method (6.5,iw) for j i s far from new; it can be found, for example, in Zygmund [1959, Vol. I], Ch. 11, $7, pp. 56-57, formula (7.6). (b) Take G = Ra, where a is an integer greater than one, so that X = R".Let P be a Haar measurable order on El". Apply Hewitt and Koshi [1983], Theorem (3.8) to obtain a linear function L mapping Ra onto R such that

Let 1p be the (continuous) adjoint homomorphism of L, mapping R into R". Form the functions

for n = 1,2, ..., and f in l p ( R " )(1 5 p < a).By Theorem (6.5), the functions H,f converge to a function j with the properties given by this theorem. (c) The conjugate function on C,. As we mentioned earlier, the conjugate function on C, is studied in Hewitt and Ritter [1983]. We borrow from Hewitt and Ritter (19831 the notation, and use without proof, several facts about the structure of the a-adic solenoid C, and its character group Q,. The homomorphism r from X into R is in this case the is the character of C, given by "identity" isomorphism, X!lAi H I where Ai

for all ( t , ~ in) C,. (See Hewitt and Ritter [1983], (1.2.4). The (continuous) adjoint homomorphism p of R into C, is given by p(t) =

(t

[t

+ 1/21, [ t + l/2]u)

I

so that

L Ai

= exp(2ri-t). This is shown in Hewitt and Ritter 119831, following (3.2.4). The group Q, admits exactly one order under which 1 is in P. This is also the order for which (6.8,i) holds with r the


38

identity isomorphism. For every f in Ll(En) and almost all (tlx) in C, we have from (6.4,i)

;

f((8

-t

+ [t + 1/21 -

[B

-t

+ 1/2],[8 - t + l/2]u + x - It + 1/2]U))+dt.

Unlltlln

As shown in (6.5) the function Hnf given in (I) converges p-almost everywhere on E n to the conjugate function f of J. The function has all of the properties listed in (6.5). We improve on Hewitt and Ritter 119831, whose construction of .f succeeded only for f E L! log+ L!.

i

(d) The conjugate function on T" ( a an integer > I) wlth respect t o Archimedean orders on Z".An order P on 8" is Archimedean if and only if there are real numbers a l la a , . . . ,a, that are independent over Q for which P = { (21, ~ 2 .. ,~z a.) E 8, : a,zj 1 0) (Theorem (4.2)). The mapping r defined on 8" by

&

a

( ~ 1 r ~ Z , . * * r z aC ) a j=1

(1)

++

jzj

is a continuous isomorphism of 8" onto a dense subgroup of R. The adjoint homomorphism (o of r is the mapping of R into a dense subgroup of TI'" such that

x o p(t) = exp(ir(x)t)

(2)

for every X in 8" and all t in R. Using (I) and (2) we get

(3) Write p(t) =

(p(Q(t),(o@)(t),..

.@ ( , ) ( t ) ) .

From (3) we get

for all t in R. Therefore we must have pi(t) = a j t

[mod 2n]

for j = 1 , 2 , .. . l a rand all t in R. We will simply write @ ( t ) = ajt. For f in Ll(?ra) we have H n j ( x )= 1 (4)

1

f(X - (o(t))+dt

1lnlltlln

=

J((z1

- alt, 21 -

at,.

..

X"

- a,t))l/tdt

1/nlltlln

As shown in (6.5) the functions (4) converge p-almost everywhere to the conjugate function f o f J.

Marcel Riesz’s Theorem on Conjugate Fourier Series

39

$7. The c o n j u g a t e f u n c t i o n on locally compact A b e l i a n groups.

In this section, we achieve the goal of this essay, ub., a construction analogous to Privalov’s theorem (1.2). Our construction yields the conjugate function j p o i n t u h e p-almost everywhere. We begin with a generalization of the well-known fact that trigonometric

, 5 p < co, where G is a compact Abelian group. polynomials are dense in L P ( G )1

(7.1) Theorem. Let G be a locally compact Abelian group with Haar memure p and character group X . Let p be a number in the interval Il,oo[. The linear subspsce of e,(G) n l p ( G ) consisting of the functions with compactly supported Fourier transforms is dense in lP(G). Proof. The case p = 1 is treated in Corollary (33.13) of Hewitt and Ross 119701, p. 301. Throughout the rest of the proof we suppose that p is an arbitrary but fixed number in ]l,co[. Given E > 0 and a nonzero function f in L P ( G ) ,we want to find a function g in &(GI such that 2 has compact support on X and (1)

Ils - f l l P < E .

Apply Theorem (20.4) of Hewitt and Ross [1979], p. 285 to obtain a neighborhood U of 0 with compact closure in G such that

for all y in U. Define the function h in l l ( G ) to be &I,.

We claim that

E

(3)

For every function F in &(G), (:+ and (2), and obtain

(4)

Ilf * h - f l l P < -. 2

5 = l),we use Fhbini’s Theorem, H61der’s inequality,


40

From (4) and Theorem (17.1) of Hewitt and Stromberg 119651, p. 223 it follows that (3) holds. Use Corollary (33.13) of Hewitt and Ross [1970], p. 301 to obtain a function k in l l ( G )such that is in C o o ( X )and

Let g be the function f * k. By Corollary (20.14.ii)of Hewitt and ROSS[1979], p. 293, the function g is in l , ( G ) n l,(G). From the equality 3 = fk, it follows that 5 has compact support. It remains to show that (1) holds. From (3)' (5), and (20.14.ii) of Hewitt and Ross [1979], p. 293, we have M

Ilf - f * kllp IIlf - f * N I P + Ilf * h - f * All, E

B

2

2

+

€1)

= p ( { z : IHnf(z)- H n g ( z ) Hmg(z)- Hrnf(z)J> 8))

IC(({z : IHV - g)(z)l > :)I + P ( { Z : IHrn(f- s)l > f ) ) I2B,P (;)"If

- 911;

< E. The last inequality but one follows from (7.6.ii).Thus the sequence of functions (Hnf)F=l is Cauchy in measure. Hence there is a function H f such that Hnf converges in measure to Hf.That is, ( i ) holds. The inequality ( i i ) holds automatically for Hf. We next show that

H f is the function i of Theorem (7.2).

(7.8) Theorem. Notation is borrowed from (7.7). For f in l,(G) *PO,1 < p < co, the functions Hnf converge in the Lp(G)-norm to Hf. We have

(4

IlHfllP I MPllfllP.

If G is compact, the equality h

(ii)

H ~ ( x=) - i sgnp(x)j(x)


44

holds everywhere on X for 1 < p < 03. If G is noncompact, (ii) holds px-almost everywhere on X for 1 < p 5 2. This is t o say, Hf is the conjugate function of f with respect to the order P. Proof. Suppose that g E L1(G) n L,(G) n %oo(I'). The inequality ( i ) for g follows from (7.7.1) and (7.6.i). The identity (ii)for g follows from (6.7.i). Because of (7.7.i) it is enough to show that the sequence ( H n f ) T Z l is Cauchy in L,(G) to conclude that it converges in the L,(G)-norm to H f . Given f in L,(G) such that

* po, (1 0, choose g in L l ( G ) n Lp(G) n &(I')

Let no be a positive integer such that (7.7.1) holds. Then, for all rn, n 2 no we have

< 8. The last equality but one follows from (7.6.i). Since (i)and (ii)hold for all f in a dense subset of L,(G) * pol they hold for all f in L,(G) * P O . 0 (7.9) Pointwise convergence for the conjugate function. Suppose that we have a sequence (Fn)Fz1 of functions in Ll(G) with the following properties:

(i)finE iio0(r) for all n; (ii)f

* F,,

converges to f pointwise for all f in l , ( G ) *PO,1< p < 03;

(iii) f

* F,

converges in the L,(G)-norm to f , for all f in L,(G) * P O , 1 < p < 03.

Let Kn denote the support of for all m 2 n and that

pn.We lose no generality

in supposing that K, c K ,

00

r=UKn. n=l

Let &,, H n , and H have the same meanings as in (7.4)-(7.7). Then we have

(4

HnFn = HF,

almost everywhere on G for all n. The identity ( i u ) follows from (7.2.1) and the fact that finvanishes off of K,, when we take the Fourier transforms of the two sides of ( i v ) .

(7.10) Theorem. Notation is 85 in (7.9). (i)If (7.Q.ii)holds then the functions (Hnf) * F, converge pointwise palmost everywhere to Hf.


4s

( i i ) If (7.9.iii) holds tben tbe functions ( H n f )* F, converge in the Lp(G)-normto Hf.

Proof. For almost all X in X, we have

( ( ~ ~* ~f n) )

~ (= x HTf(x)$n(x) )

= -i ~ g n @ ~ ( x ) ~ n ( x ) f ? x ) = (Hn &IA( X ) m

= (HF,)A(X)f?X) = - i sgnp(X)T(X)$n(X) = ((Hf) * Fn)'(X). Since Fourier transforms are unique, we get

(Hnf)* F n = (Hf) * Fn p-almost everywhere on G . The conclusions ( i ) and ( i i ) now follow from (7.9.ii) and (7.9.iii). (7.11) Remarks. (a) The existence of sequences as described by (7.9), has been estahlished by Edwards and Hewitt [1965](see also Hewitt and Ross [1970], section 44, pp. 631-679). Edwards and Hewitt were looking for an analogue on locally compact groups of the FejBr-Lebesgue theorem for functions in They found sequences (Fn)r!l for a restricted class of groups. For general groups, an iterated sequence ( F m , n ) ~ =isl ~ ? l needed. All of (7.10) holds for these iterated sequences. The reader may refer to Hewitt and Ross, loc. cit., for details.

el(").

(b) Our construction of Hf is of course restricted to functions in LP(G)*p0(15 p < m). If X is not a-compact, we will need a great gallimaufry of PO'S to accommodate all of vanishing off of a &(G). Naturally every in & ( G ) belongs to some L,(G) * p o l with a-compact subgroup of X.

We complete this section with Kolmogorov's and Zygmund's theorems in our present context . (7.12) Theorem. Let

I be in E1(G), and let po

be such that f = f * po. For 0 < p < 1,

we have

Proof. We showed in (7.7.ii) that H if of weak type ( 1 , l ) . A standard argument now yields ( i ) . See, for example, Hewitt and Stromberg [1965],p. 422, Corollary (21.72). (7.13) Theorem. Suppose that f is in

(i)

I[flll,C

L31

log'L1.

Tbe function Hf is in L l ( G ) ,and

5 B -k c / , I.fh?+ f(&,

46

h? Asmar and E. Hewitt

for certain constants B and C.

Proof. The constant Mp = tan($) is o(&). Then a theorem of Yano can be applied verbatim to give ( i ) . See Zygmund 11959, Vol. 111, Ch. XII, $4, Theorem (4.4.1), pp. 119-120.

0


41

$8. The conjugate function on Ta.

Specific examples are the lifeblood of mathematics. A general theory is a mere wraith until it is fleshed out with particular cases. Therefore we will carry out our construction in detail for T". (8.1) Introduction. Throughout this section, a denotes an integer greater than 1. The symbol P denotes an arbitrary but fixed order on Za. For a function f in .Lp(Ta) (1 _< p < oo),the conjugate function o f f with respect to the order P is denoted by J. We will describe a method of recapturing in a concrete way the function from the function f .

Our method uses the FejCr kernel on Ta.I t enables us to obtain ,f as a pointwise limit of certain trigonometric polynomials obtained from f. Also, this method yields the desired bound on the norm of the conjugation operator. Namely, we obtain the inequality Ilf"llP,lr~ 5 ~ P l l f l I P * l r n

(1)

for all functions f in L p ( T a ) ( < I p < co). (8.2) Let f be a function in L p ( T a ) . Apply Theorem (4.12) to obtain a sequence of Archimedean orders P,,, n = 1 , 2 , .. ., satisfying (4.12.i) and (4.12.3). Since each P, is Archimedean, the construction of Example (6.10.i~)applies and yields the conjugate func-

p.

For each order P,, Theorem tion o f f with respect to P,. We denote this function by (4.9) provides an order-preserving isomorphism r, of Z"into R for which the equality sgnpnY = sgnTnX

obtains for every X in Za. The adjoint homomorphism of r, will be denoted by p,

For a positive integer n, the nth Fejdr kernel on l' is given by

otherwise. For the above definitions and other properties of the FejCr kernel on Stromberg 119651, p. 292.

TIsee Hewitt and

I t is obvious from ( i ) that

(ii) For a positive integer n, let n denote the a-tuple (n, n, . .. , n). The nth Fejir kernel on !ITa is defined by (iii)


48

where K , is given by (1) and t = ( t l , t z , ...,t,) is in ( i i i ) yields

=

T4. Fort=

(!1,!~,...,!,)

in

Z",

nii,(!,). i=1

It fOllOW9 from ( i i ) and ( i v ) that

(4 if

e,

ii,(t) = 0 > n for some j in (1, 2,. . . , Q}. The FejCr kernel on

T4has the following properties:

( w i ) the sequence of functions (f *Kn)FZlconverges to f , palmost everywhere on Tolfor

every function f in tl(T4); (wii) the sequence of functions (f * Kn)F=l converges to f in the ~p('II'4)-norm for every function f in lp(T4) (1 p < 00).

For a proof of ( w i ) , see Zygmund 11959, Vol. 111, Ch. XVII, $3, Theorem (3.1), p. 309. For ( w i i ) , see Zygmund [1959,Vol. 111, Ch. XVII, $1, Theorem (1.23), p. 304. The FejCr kernel on

for all t in

T4is a positive summability kernel: for we have

W4,

for every fixed positive real number 6, where It1 =

dt: + ti + , . . + t f .

Properties (wiii)-(z) are easily obtained from the properties of the Fejkr kernel on listed on p. 292 of Hewitt and Stromberg [1965]. We omit the proofs.

Theorem 8.3. Let p be a real number greater than 1 8nd let f be in 1,(T4).Let Pn, Kn be 8.9 in (8.2). ( i ) The functions function f.

T

p , P,

p , n = 1,2,..., are in L P ( a a ) 8nd converge in the tp(T4)-norm to a

49


( i i ) The function

r" is the conjugate function o f f

with respect to the order P.

* Kn, 1 , 2 , .. ., converge in the L,('Il'')-norrn to the function ,f. (iw) The polynomials F * Kn, n = 1 , 2 , .. . converge p-almost everywhere on ?ra to the

( i i i ) The polynomials

function

f.

(u) The function

satisfies the inequality

Proof. Let g be a trigonometric polynomial on

a". We write g as

where S is a finite subset of Z4 and a, are nonzero complex numbers. In the notation of (4.12.2), choose a positive integer n(g) such that C, contains S for all n 2 n(g). For n 2 n(g), (4.12.3) yields

P, n S = P, n (C, n S ) = (P,n C,) n S = ( P n C,) n S =PnS,

and so (2)

(-P,) n s = ( - P ) n S

for all n 2 n(g). The identities (1) and (2) imply that

P, n S = P,(g)n S and

for all n 2 n(g). The definitions of

inlcn(g)

and

5, (I), (2), and

(3) show that

for all n 2 n(g). We return to our f in f & ( T aGiven ) . a positive real number trigonometric polynomial g such that

(5)

E,

choose a

50

N. Asniar and E. Hewitt I

For every positive integer n, the conjugate function o f f - g with respect to P,,is f" - in. Since the order P, is Archimedean, we may apply (6.10.4)and obtain from (5)the inequality

for every positive integer n. Combining (4) and (6) we see that 8

(7)

111"; - PIIP < 2

for all n 2 n(g). Now, for any positive integers m and n greater than or equal to n(g), (7) shows that E

E

2

2

l

into itself. Its norms on the subspaces C p ( n , r )go to phenomenon Bochner found particularly charming.

00

as p

1 1 and as p T

00.

This

Plainly the definition of the conjugate polynomial f depends on our choice of an order

P in X. It would seem pedantic, however, to use a term like “P-conjugate polynomial.” I t is far from obvious that ( i ) exists as aLebesque integral on R p-almost everywhere. Details of an almost identical construction may be found in Hewitt and Ritter 119831, pp. 823-824. la


54

Bibliography.

Asmar, Nakhlb. The conjugate function on locally compact Abelian groups. Doctoral Dissertation, University of Washington, Seattle, Washington (1986). Berberian, Sterling. Lectures in Functional Analysis and Operator Theory, Graduate Text in Mathematics, 15. Berlin, Heidelberg, New York: Springer, 1973. Berkson, Earl, and T.A. Gillespie. The Generalized M. Riesz Theorem and Transference. Pacific J. Math. 120 (a), Dec. 1985,279-288. Birkhoff, Garrett, and Saunders Mac Lane. A Survey of Modern Algebra, Third edition. New York: The Macmillan Company, 1965.

Bochner, S.. Additive set functions on groups, Ann. of Math. (2) 40 (1939),769-799. Bochner, S.. Generalieed conjugate and analytic functions without expansion. Proc. Nat. Acad. Sci. U.S.A. 46 (1959),855-857. Caldedn, Albert0 P.. Ergodic theory and translation-invariant operators. Proc. Nat. Acad. Sci. U.S.A., 59 (2) (1968),349-353.

Edwards, R.E., and Edwin Hewitt. Pointwise limits for sequences of convolution operators. Acts Math. 11s (1965),181-218. Fatou, Pierre. Skies trigonomktriques et skries de Taylor. Acts Math. SO (1906),335400. F d b r , Leopold. fiber die Bestimmung des Sprunges der Funktion aus ihrer Fourierreihe. J. fur die reine u. angewandte Math. 142 (1913),165-188.Also in Gesammelte Arbeiten, Vol. I, 718-742. Budapest: Akadkmiai Kiad6, 1970.

Ghrdlng, Lars. Marcel Riesz in memoriam. Acts Math. 124 (1970),I-XI. Garsia, Adriano. Topics in almost everywhere convergence. Chicago: Markham Publishing Company, 1970.

Hardy, G. H. Notes on some points in the integral calculus. LVIII. On Hilbert transforms. Messenger of Math. 54 (1925),20-27. Also in Collected Papers of G. H. Hardy, Vol. 111, 121-128. Oxford, England: The Clarendon Press, 1979. Hardy, G. H. and J. E. Littlewood. The allied series of a Fourier series. Proc. London Math. SOC. 2) 22 (1923),ztiii-ztv. Also in Collected Papers of G. H. Hardy, Vol. 111, 167-168. Ox ord, England: The Clarendon Press, 1979.

I

Hardy, G. H., and J. E. Llttlewood. The allied series of aFourier series. Proc. London Math. SOC.(2) 24 (1925),211-246. Also in Collected Papers of G. H. Hardy, Vol. 111, 171-246. Oxford, England: The Clarendon Press, 1979. Helson, Henry. Conjugate series and a theorem of Paley. Pacific J. Math. 8 (1958), 437-446.

Helson, Henry. Conjugate series in several variables, Pacific J. Math. 9 (1959),513-523.

Hewitt, Edwin, and Shoso Koshi. Orderings in locally compact Abelian groups and the theorem of F. and M. Riesz, Math. Proc. Camb. Phil. SOC.93 (1983),441-457. H e w l t t , Edwin, and G u n t e r €Utter. Fourier series on certain solenoids. Math. Ann. 267 (1981),61-83.

Murcel Riesz’s Theorem on Conjugate Fourier Series

55

Hewitt, Edwin, and Gunter Ritter. Conjugate Fourier series on certain solenoids. lkans. Amer. Math. SOC.276 (1983), 817-840. Hewitt, Edwin, and Kenneth Ross. Abstract harmonic analysis I. Grundlehren der mathematischen Wissenschaften, 115, Second edition. Berlin, Heidelberg, New York: Springer-Verlag 1979. Hewitt, Edwin, and Kenneth Ross. Abstract harmonic analysis 11. Grundlehren der mathematischen Wissenschaften 152. Berlin, Heidelberg, New York: Springer-Verlag, 1970. Hewitt, Edwin, and Karl Stromberg. Real and Abstract Analysis. Graduate Texts in Mathematics. Berlin, Heidelberg, New York: Springer-Verlag, 1965. Hilb, E. and Marcel Riess. Neuere Untersuchungen uber trigonometrische Reihen. Enzyklopadie der Math. Wiss., Band I1 C 10, 1189-1228. Leipzig: B. G. Teubner, 1924. Hobson, E. W. The theory of functions of a real variable and the theory of Fourier’s series. 2nd edition. Cambridge, England: Cambridge University Press, 1926. Kolmogorov, A.N.. Sur les fonctions harmoniques conjugukes et les dries de Fourier. Fund. Math. 7 (1925), 24-29. Lichtenstein, L. Review of I. I. Privalov’s “Das Cauchysche Inte ral.” Jahrbuch iiber die Fortschritte der Math. 47 (Jahrgang 1919-1920, publ. 1924/19267, 296-298. Luklcs, fians. Uber die Bestimmung des Sprunges einer Funktion aus ihrer Fourierreihe. J. fur die reine u. angewandte Math. 150 (1920), 107-112. Pichorides, S.K.. On the best values of the constants in the theorems of M. Riesz, Zygmumd and Kolmogorov Stud. Mat. 44 (1972), 165-179. Plessner, A. Zur Theorie der konjugierten trigonometrisch Reihen. Mitteilungen Math. Seminarium Universitat Giessen 10 (1923), 1-36. Pringsheim, A. Ueber das Verhalten von Potenzreihen auf dem Convergenzkreise. Munch. Sitzungsber. SO (1900), 37-100. Privalov, I. I. Sur la convergence des skries trigonom6triques conjugukes. C. R. Acad. Sci. Paris 162 (1916), 123-126. Privalov, I. I. Sur la convergence des skries trigonom6triques conjugukes. C. R. Acad. Sci. Paris 165 (1917), 96-99. Privalov, I. I. Sur les fonctions conjugukes. Bull. SOC.Math . fiance 44 (1916), 100-103. Privalov, I. I. [ n p H B a n o B , H.M.].Cauchy’s integral [ M H T e r p a n Cauchy]. Comm. of the physico-mathematical Faculty of the University of Saratov. Saratov, USSR 1919. Privalov, I. I. Sur les shies trigonomktriques conjugukes.

6K f0

TeOpYIYI COIIpHXCeHHbIX T P H l ’ O H O M e T P H Y e C K A X PHAOB].

Mat. sbornik 31 (1923),

24-228.

Privalov. I. I. On the convergence of conjugate trigonometric series CXOAUMOCTU C O n p R x e H H b I X T p H r O H O M e T p H Y e C K H X PRAOB].

Mat. Sbornik 32

1925), 357-363.

Privalov. I. I. Boundary values of analytic functions [ r p a H Y I Y H b I e CBOYICTBa aHaJ’IHTHYeCKHX f#iyHKI&HH]. Moskva: Izdatel’stvo Universiteta, 1941. 2nd edition, greatly revised and augmented. Editor A. I. MarkuSevit. MoskvaLeningrad: Gostehizdat, 1950.

56

N. Asmar and E. Hewirt

Rieaa, Marcel. Les fonctions conjugukes et les skries de Fourier. C. R. Acad. Sci. Paris 178 (1924), 1464-1467. Riess, Marcel. Sur les fonctions conjugukes, Math. Z. 27 (1928), 218-244.

Rudin, Walter. Fourier Analysis on Groups. Interscience tracts in pure and applied mathematics. New York: Interscience Publishers, 1962. Smlrnov, V. I. Sur les valeurs limites des fonctions analytiques. C. R. Acad. Sci. Paris 188 (1929), 131-133. Tauber, A. fiber den Zusammenhang des reellen und imaginaren Theiles einer Potenzreihe. Monstshefte fur Math. u. Phys. 2 (1891), 79-118.

Tauber, A. Ein Satz aus der Theorie der unendlichen Reihen. Monatschefte fur Math. u. P b p . 8 (1897), 273-277.

Titchmarsh, E. C. A contribution to the theory of Fourier transforms. Proc. London Math. SOC.(2) 23 (1924), 279-289. Titchmarsh, E. C. Reciprocal formulas involving series and integrals. Math. Z. 26 (1926), 321-347.

Titchmarsh, E. C. On conjugate functions. Proc. London Math. SOC.(2) 29 (1929), 49-80.

T i t c h m a r s h , E. C. introduction to the theory ofFourier integrals. Oxford, England: The Clarendon Press, 1937. Young, W. H. Konvergenzbedingungen fur die verwandte Reihe einer Fourierschen Reihe. Munch. Sitzungsber. 41 (1911), 361-371. Young, W. H. On the convergence of a Fourier series and of its allied series. Proc. London Math. SOC.(2) 10 (1912), 254-272. Young, W. H., and G. C. Young. On the theorem of Riesz-Fisher. Quarterly J. Math. 44 (1913), 49-88. Zygmund, A. Sur la sommation des skries trigonomCtriques conjugu6es. Bull. Acad. Polonaise 1924, A, 251-258. Zygmund, A. Sur les fonctions conjugukes. Fund. Math. 13 (1929), 284-303. Corrigenda, ibid., 18 (1932), 312. Zygmund, A. Trigonometric series. 2nd edition, Vols. I & 11. Cambridge, England: Cambridge University Press, 1959, repr. with corrections and additions 1969.

California State University, Long Beach Long Beach, California 90840 and University of Washington Seattle, Washington 98195

Proceedings of the Analysis Conference, Singapore 1986 S.T.L. Choy, J.P. Jesudason, P.Y. Lee (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1988

ON NONLINEAR INTEGRALS Chew Tuan Seng Department o f Ma thema t ics N a t i o n a l U n i v e r s i t y o f Singapore Singapore N o n l i n e a r i n t e g r a l s o f Denjoy t y p e and P e r r o n t y p e a r e g i v e n i n t h i s n o t e . I t i s shown t h a t t h e s e two n o n l i n e a r i n t e g r a l s , as i n t h e l i n e a r case, a r e e q u i v a l e n t t o n o n l i n e a r i n t e g r a l s o f HenstockK u r z w e i l t y p e . I n t h i s note, i t i s a l s o shown t h a t a n o n l i n e a r measure i s nothing but a l i n e a r i n t e g r a l o f a kernel f u n c t i o n . I t i s known t h a t o r t h o g o n a l l y a d d i t i v e f u n c t i o n a l s , which a r e n o t n e c e s s a r i l y

l i n e a r , on c e r t a i n f u n c t i o n spaces can be c h a r a c t e r i z e d i n terms o f l i n e a r i n t e g r a l s w i t h k e r n e l f u n c t i o n s [2; 4; 12; 131.

T h i s t y p e o f f u n c t i o n a l s can

a l s o be c h a r a c t e r i z e d i n terms o f n o n l i n e a r i n t e g r a l s [l;3; 5; 8; 111.

In

[ l ; 5; 111, t h o s e n o n l i n e a r i n t e g r a l s a r e d e f i n e d i n a b s t r a c t spaces. When r e a l i z e d on t h e r e a l l i n e , t h e y correspond t o t h e Lebesgue t h e o r y whereas i n [3; 81, n o n l i n e a r i n t e g r a l s a r e o f Henstock-Kurzweil t y p e , which correspond t o t h e t h e o r y o f Henstock.

I t i s more g e n e r a l .

I n t h i s note, we s h a l l d e f i n e

n o n l i n e a r i n t e g r a l s o f Denjoy t y p e and P e r r o n t y p e i n s e c t i o n one.

Further-

more, we s h a l l p o i n t o u t t h a t , as i n t h e l i n e a r case, t h e s e t h r e e n o n l i n e a r i n t e g r a l s are equivalent.

I n s e c t i o n two, we s h a l l show t h a t a n o n l i n e a r mea-

sure i s nothing but a l i n e a r i n t e g r a l o f a kernel f u n c t i o n .

1. EQUIVALENCE OF THREE TYPES OF NONLINEAR INTEGRALS Let

+

be d e f i n e d f o r s b e i n g r e a l and I a s u b i n t e r v a l o f a compact

= $(s,I)

Moreover, we assume t h a t

i n t e r v a l [a,b].

(N1) (N2)

+

s a t i s f i e s the following conditions :

+(O,I) = 0; +(.,I) i s continuous;

(N3)

g ( s , I1 U 12) = g(s,I1)

(N4)

Given

M

> 0, f o r e v e r y

whenever Isi - tiI
0 t h e r e e x i s t s n > 0 such t h a t

and ItiI 2 M f o r i =1,2

,...,n,

and t11,12

In} i s a p a r t i a l d i v i s i o n o f [a,b].

(N5)

Given

M

> 0, f o r e v e r y

E

n

> 0, t h e r e e x i s t s r~> 0 such t h a t

,...,

58

T.S. Chew

...,n,

whenever Iiyf o r i = 1,2, length less than Obviously,

are pairwise d i s j o i n t intervals with the t o t a l

My

and I s i I

rl

f o r i = 1,2,...,n.

(N4) i m p l i e s ( N 2 ) .

F o r s i m p l i c i t y , we s h a l l f u r t h e r assume t h a t

= +(s,(u,v))

+(s,[u,vl)

= +(S,[U,V))

= @(SY(U,VI)*

We may say t h a t $ ( s , I ) r e p r e s e n t s t h e measure o f t h e s i n g l e - s t e p f u n c t i o n h a v i n g v a l u e s s o n I and z e r o e l s e w h e r e .

Since $ ( s , I ) i s n o t n e c e s s a r i l y

l i n e a r i n s , t h e measure i s a n o n l i n e a r measure. DEFINITION 1 [3;8].

K u r z w e i l i n t e g r a b l e w i t h r e s p e c t t o $ o n [a,b] f o r every

i s s a i d t o be Henstock-

A f u n c t i o n f d e f i n e d o n [a,b]

i f t h e r e i s a number A such t h a t

> 0 t h e r e e x i s t s a f u n c t i o n 6(5) > 0 d e f i n e d o n [a,b]

E

such t h a t f o r

e v e r y d i v i s i o n D g i v e n by a = x and s a t i s f y i n g

ci

-

( l - x ) ( b n

F on ti!, dnd

We say

HC(E), and ( i i ) f o r

and every c l o s e d contiguous i n t e r v a l [a,,

denotes t t i e o s c i l l a t i o n of

which [an,

if (i) F

o f Ca,bl

e

The

Dx.

i s c a l l e d an appro-

i f u,v B Dx, u < x < v and x e Ca,bl. I f A [ a , b l , a A - p a r t i t i o n i s a p a r t i t i o n { a = ao, al, a2, ... , a, = b;

x i m a t e l y f u l l cover ( A F C ) o t Ca,bJ i s an AFC o f xl,

x2,

... , x,,}

w i t h ai-l,

ai

e D

and ai-l

xi

5 xi

5 ai,

i

1, 2 ,

...,

n,

or

S. Darmawijaya and P.Y. Lee

64

x} w i t h u , v e Ux and u < x < v.

a l t e r n a t i v e l y , {[u,v]; A f u n c t i o n f : [a,b]

H*aP [a,b]

f 6

+ R i s s a i d t o be K $ p - i n t e g r a b l e on [a,b]

i f t h e r e e x i s t s A such t h a t t o r every

such t h a t f o r every A - p a r t i t i o n { a = a",

[a,b]

..., xn}

or

{CU,~];

al,

>

U t h e r e i s an AFC, A , o f

...,

a2,

an = b;

XI,

x2,

x } we have \A

-

n 1=1

-

f(xi)(ai

0 t h e r e e x i s t s a p o s i t i v e i n t e g e r nu such t h a t f o r every

E

d i v i s i o n o t [a,bl

a < we have

dl



L E W 2.5

partial

:

bi)

9

-

... < bp < b

F,bi

s

bi

)}I
u.

It i s we1 -known t h a t t h e c h a r a c t e r i s t i c m a n i f o l d s o f RI

are t h e u n i t a r y yroups

Un o f o r d r n, and t h e e l e m e n t i n t h e a n a l y t i c automorphism g r o u p can be r e p r e -

s e n t e d by

W = (AZ + B)(CZ + U ) - l where W , Z E RI and 2nx2n m a t r i x

or)

F = ( : satisfies the followiny three conditions :

d e t F = 1. On Un,

(1.1) chanyes i n t o V = (AU + B)(CU + O)-'

(1.4)

w h i c h t r a n s f o r m s a u n i t a r y m a t r i x U i n t o a n o t h e r u n i t a r y m a t r i x V. Let

fi

and

\i

d e n o t e t h e r e s p e c t i v e volume e l e m e n t s o f U and V,

\j = ( d e t ( C U + D ) L e t a p o i n t Z o f RI become 0 u n d e r (1.1) become V.

then

I-2n fi. and, a c c o r d i n g l y ,

(1.5) a p o i n t U o f Un

Then Hua Luo Geng s t a r t i n y from t h e t h e o r y o f harmonic f u n c t i o n s i n

s e v e r a l complex v a r i a b l e s , d e f i n e d t h e P o i s s o n K e r n e l as f o l l o w s : P(Z,U)

=

det(1 ldet(Z

-

ZT')n

-

U)lZn

Harmonic Analysis on Classical Groups

71

and p r o v e d t h e f o l l o w i n y L e t $ ( U ) be a c o n t i n u o u s f u n c t i o n on Un,

THEOREM 1.1.

The P o i s s o n k e r n e l P(r1,U)

on u n i t a r y g r o u p s i n (1.7)

then

has t h e f o l l o w i n y

expansion P(r1.U)

=

1 pf(r)N(f)xf(U), f

fz,

where N ( f ) = (fl,

..., f n )

i s the order of the sinyle-valued irreducible

u n i t a r y r e p r e s e n t a t i o n A f ( U ) o f Un w h i c h t a k e s f = ( f l , labels ( f l > f2 > characters,

... > f,

..., f n )

f2,

as i t s

a l l are i n t e y e r s ) , xf(U) are t h e correspondiny

and P f ( d

in

P(rI,lJ)xf(U)fi.

=

(1.8)

I f u(U) i s an i n t e g r a b l e f u n c t i o n on Un and i t s F o u r i e r s e r i e s i s u(U) where

-F

c f = w

N(f)tr(CfAf(U)),

I

u(U)Af(U')fi.

'n Then t h e Abel sum o f (1.9)

is (1.10)

pf(r)N(f)tr(CfAf(U)). The c o n c r e t e f o r m u l a f o r p f ( r ) i s i n c l u d e d i n t h e f o l l o w i n g theorem.

THEOREM 1.2.

el > e2 >

I f e l = fl+n-l,

[Z]

... >

0 > E,+~

es

>

..., e k = f k + n - k , ..., en = ... > en ( n > s > 0 ) , we have

fn, when

fl+...+fs-fs+l-...-fn pf(r) = r X

where Ns(a,b)

N( f ,g)N( y , f ) N ( f ) N ( Y)

I:

s>gs+l>...'y,>o

= N(a1,

...,as,bs+l, ...,b n ) ,

g = (yl

,...,yn)

(1.11)

can a l s o be w r i t t e n as

a r e a p e r m u t a t i o n o f 0,1,2

s

X

n-l>vs+l>...>v

I:

r2(gs+1+"'+gn)

Y1+n-l and gZ+n-Z

,...,n - 1

,...,gn

and O>y1>gz>

(e.-vk)(v.-ek) J j = 1 k=l (v.-v ) ( e . - e ) >O J k J k n

n

(1.11)

I

n

n

in

...>ys,s-n.

Z(V~+~+...+V~) r

S.Gong et al.

12

The p r o o f o f Theorem 1.2 i s c o m p l i c a t e d and needs h i g h l y s k i l l f u l c a l c u l a tion.

Here a s k e t c h i s y i v e n and r e a d e r s a r e r e f e r r e d t o [ 2 ]

for details.

We i n t r o d u c e t h e f o l l o w i n g n o t a t i o n s

where q,t

a r e n o n - n e g a t i v e i n t e g e r s , and p i s an i n t e y e r .

I f q = U and p < 0 , t h e n we have

( s i n c e eipe(l-re-ie)-t powers.

"'st

)

P < 0,

for

= 0,

(1.13)

i s a F o u r i e r s e r i e s whose t e r m s have o n l y n e g a t i v e

Similarly,

( qu;

)

for

= 0,

p

> U.

(1.14)

I n virtue of e ipe

( l-rei e)q( l-re-i

lt

( l-re-i')t

( l-rei

,,i ( p + 1 )e

-

,ipe

( l-reie ) q ( l - r e - i ' ) t

'

we have

I n t h e same way, we have

(

q,t

,e-,

(1.16)

I*

Again, f r o m

i t i s deduced t h a t

(1.15),

(1.16)

and (1.17) a r e t h r e e b a s i c r u l e s o f c a l c u l a t i o n .

a p p l i c a t i o n o f (1.15)

( q P, t

Repeated

leads t o =

=

-

( q - Pl , t

1

r(

P

( q-2,t

.....

$1 P+1 )

P+l

+

( q,t

r( q-1,t

I f p < 0 , t h e n i t i s easy t o see from (1.13)

that (1.18)


13

r e p e a t e d l y , we can o b t a i n

By u s i n y (1.18)

I f p < 0, t h e n t h e r e e x i s t s

( Similarly,

q;t

(

of

(

P.4

1.

oyq )

q-'

r-p

1

k=U

(

i t can be deduced f r o m (1.16)

I n v i r t u e o f (1.1Y) of

1=

and (1.2U),

(

P$ 0

).

)(

0 q-k,t

1'

(1.1Y)

t h a t i f p > 0, we h a v e

t h e c a l c u l a t i o n o f (1.12)

F u r t h e r m o r e , by ( 1 . 1 7 ) , and

k-p-1 k

i s reduced t o t h a t

t h e c a l c u l a t i o n can be reduced t o t h a t

On t h e o t h e r hand, i t i s easy t o see t h a t

(

U p,u

=

(

u,q

) = 1 .

(1.21)

The f o r m u l a e m e n t i o n e d above b e i n y a p p l i e d t o (1.8),

a c o m p l i c a t e d and h i y h l y

s k i l l f u l c a l c u l a t i o n can y i e l d s t h a t

i(k x e

e +...+ knen -iB1 1 1 D(e

..., e

,

-ien )dol

... den

(1.22)

1 ................ kn-( n-1)

1

1

..-(

n ,n By u s i n y ( 1 . 1 5 ) ,

(1.16)

and (1.17)

d e t e r m i n a n t can be c a l c u l a t e d o u t . I n v i r t u e o f Theorem 1.2,

repeatedly, t h e value of t h e preceding

Thus t h e c o n c l u s i o n o f t h e t h e o r e m f o l l o w s .

t h e F o u r i e r s e r i e s (1.10)

a b s o l u t e l y converyent, t h e r e f o r e

n,n

of

'j

u(V)P(rU,V)

\j i s

"n (1.23)

S. Gong et al.

74

From Theorein 1.1, i t f o l l o w s t h a t U(U)

=

l i m J u(V)P(rU,V) r + l lln

= liin

3

1 pf(r)N(f)tr(CfAf(U)).

r+l f Thus t h e F o u r i e r s e r i e s o f u(U) i s Abel-summable t o i t s e l f . E v i d e n t l y , as f a r as a p p l i c a t i o n i s concerned a c o n c r e t e t h e o r e m on c o n v e r yence i s s u p e r i o r t o an a b s t r a c t e x i s t e n c e theorem on a p p r o x i m a t i o n .

Thus

Theorem 1.1 sharpens t h e famous Peter-Weyl Theorem. Assuminy t h a t u ( U ) has s u f f i c i e n t smoothness, we can deduce t h e d i f f e r e n c e between SN =

1

N > f > f >...>f 1 2

n

>-N

pf( r)N(f)tr(CfAf

(u))

I n a d d i t i o n , t h i s p r o v e s , i n t h e meantime, t h a t t h e f u n c t i o n system

and u ( U ) .

{ a i J ( U ) } c o n s i s t i n y o f a l l elements o f t h e m a t r i c e s o f t h e s i n g l e - v a l u e d i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n s A f ( U ) = ( a f. ( U ) ) f o r a u n i t a r y y r o u p i s 1J

complete.

As a c o r o l l a r y , we can i m m e d i a t e l y deduce t h e a p p r o x i m a t i o n t h e o r e m

f o r any compact y r o u p and any compact homoyeneous space. L e t us c o n s i d e r t h e r e a l c l a s s i c a l domain Rn c o n s i s t i n y o f a l l r e a l m a t r i c e s o f o r d e r n such t h a t

I

-

XX' > 0,

t h e c h a r a c t e r i s t i c m a n i f o l d o f w h i c h i s o r t h o y o n a l y r o u p O(n) =

{r, rr'

= I}.

The a n a l y t i c autoinorphism

w o f Rn maps o n t o i t s e l f R,

= (AX

+ B)(cx+D)-~

O ( n ) and SO(n) o n t o Rn,

(1.24) O ( n ) and SO(n) r e s p e c t i v e l y .

Lu U i Keny w i t h t h e h e l p o f t h e t h e o r y o f h a r m o n i c f u n c t i o n s i n s e v e r a l r e a l v a r i a b l e s , d e f i n e d P o i s s o n k e r n e l s on Rn as f o l l o w s : (1.25)

As i n Theorem 1.1, we can p r o v e t h e f o l l o w i n y .

THEOREM 1.3.

[l] I f

u(r)

i s a c o n t i n u o u s f u n c t i o n on r o t a t i o n y r o u p SO(n),

then

I n t h e e a r l y 1 9 6 0 ' s , Zhony J i a (ling, u s i n g t h e method o f y e n e r a t i n y f u n c t i o n , p r o v e d t h e e x p a n s i o n o f P o i s s o n k e r n e l s on r o t a t i o n groups.

THEOREM 1.4.

[7] The P o i s s o n k e r n e l P ( r r , r ) o f r o t a t i o n g r o u p s has t h e

Harmonic Analvsis on Classical Groups

75

(1.26)

(1.27) or, e q u i v a l e n t t o

(n-2) ( r )

...

I

I n (1.27),

..., q,,-k-l

when n = 2 k + l , we t a k e (q1,q2,

1)

n - k - 1 > 41 >

2)

qi

... > q n - k - l

(n-2) ( r ) 5,-1

) from

6

which s a t i s f i e s

> 0,

+ q j # i+j-1, f o r a l l i # j ,

and

n-k-1

1

if

qi =

u,

qi,=

3, 4

1

n-k-1

-1,

if

(mod 4 ) ;

when n = 2k, i t i s t a k e n f r o m E w h i c h s a t i s f i e s

...

1)

n-k > q1 >

2)

qi

# i f o r a l l i,

3)

qi

+ qJ. # i+j f o r any i

and

E(qlS

Moreover, N(m1,

...,mk,

qn-k-1

. * . 9

91,

"3

#

j

qn-k-l ) = (-l)(ql+'"+qn-k-l)/'

...,q n - k - l )

i s t h e o r d e r of t h e i r r e d u c i b l e u n i t a r y

r e p r e s e n t a t i o n o f a u n i t a r y group o r o r d e r n-1 which takes (ml, qn-k-l)

ml

SentatiOn

ml

>

> m2 >

as i t s l a b e l s , inl

> m2 >

... > rnk = 0,

O f

... > mk

> 0.

...,mk,ql ,...,

I f n = 2 k + l o r n = 2k and

t h e n um(r) i s t h e c h a r a c t e r o f t h e i r r e d u c i b l e r e p r e -

so(n) which takes

... > m k >

(1.2Y)

(r) m

0, t h e n u

Ill

= (ml,...,mk)

as i t s l a b e l s .

1f

= 2k and

i s t h e sum o f t w o c h a r a c t e r s o f t h e i r r e d u c i b l e

r e p r e s e n t a t i o n s o f s o ( n ) w h i c h t a k e s (Inl,

...,*Ink)

as i t s l a b e l s .

S. Gong el al.

16

............ Sn-l(r) =

+

,2n-2k-3

for

n = 2k+l,

-

,2n-2k-4

for

n = 2k.

(1.30)

The p r o o f o f Theoren 1.4 i s c o m p l i c a t e d and needs h i g h l y s k i l l f u l c a l c u l a t i o n .

F o r d e t a i l s , see [7]. I-et us c o n s i d e r a domain c o n s i s t i n g o f q u a t e r n i o n m a t r i c e s X o f o r d e r n such

-

that I

Xx'

> U, t h e c h a r a c t e r i s t i c m a n i f o l d o f w h i c h i s t h e u n i t a r y syrnplec-

t i c y r o u p IJSP(2n).

As m e n t i o n e d above, c o n s i d e r i n y t h e a n a l y t i c automorphism

y r o u p on t h e domain I

- XX'

> 0, we can o b t a i n t h e c o r r e s p o n d i n g P o i s s o n

kernels (1.31) where 0 < r < 1 and U

B

USP(Zn), by u s i n y t h e t h e o r y o f harmonic f - u n c t i o n s on

t h e q u a t e r n i o n domain. E m p l o y i n g t h e y e n e r a t i n y t u n c t i o n inethods used by Zhong J i a Q i n g i n t h e p r o o f o f Theoren 1.4 l e a d s t o t h e f o l l o w i n y .

THEOREM 1.5.

(He Zu Qi and Chen Guang Xiao, see ( 1 1 ) .

I n t h e expansion o f

P o i s s o n k e r n e l s on u n i t a r y s y i n p l e c t i c g r o u p s P(rI,U)

=

$ pf(r)N(f)xf(U).

t h e c o e f f i c i e n t s have t h e e x p r e s s i o n

1

where

5 (r) 1

.

r

fl+2n

..... t n ( r )

.

S p )

f ,+n+l

r

,~

n n-1 n+l + r ~ + ~ =( rr ,) ~ , + ~ ( r =) r


77

A s i n t h e case o f u n i t a r y y r o u p s , we a r e a b l e t o s t u d y t h e Abel summation o f F o u r i e r s e r i e s on r o t a t i o n o r u n i t a r y s y m p l e c t i c y r o u p s , and on t h e h a s i s o f Theorems 1.4 and 1.5 we o b t a i n t h e f o l l o w i n y c o r r e s p o n d i n y r e s u l t : The F o u r i e r s e r i e s o f any c o n t i n u o u s f u n c t i o n on t h e l a t t e r t w o c l a s s i c a l y r o u p s i s a l w a y s Abel-summable t o i t s e l f . 2. The Cessaro Sumnation The s e r i e s o f methods e s t a b l i s h e d i n t h e r e s e a r c h f o r t h e h a r m o n i c a n a l y s i s on u n i t a r y y r o u p s a r e w i d e l y a p p l i e d t o t h e r e s e a r c h f o r t h e h a r m o n i c a n a l y s i s on c l a s s i c a l y r o u p s and on compact L i e y r o u p s .

F o r example, t h e methods " f r o m

sums t o k e r n e l s " and " f r o m k e r n e l s t o sums" t o b e i n t r o d u c e d i n t h i s s e c t i o n j u s t come f r o m t h e i d e a s used i n t h e r e s e a r c h f o r t h e h a r m o n i c a n a l y s i s on u n i t a r y yroups.

By a p p l y i n g t h e s e t w o methods, t h o s e r e s u l t s o b t a i n e d by u n i t a r y

y r o u p s i n t h i s s e c t i o n and t h e subsequent ones can be e s t a b l i s h e d on c l a s s i c a l y r o u p s and on compact L i e y r o u p s .

I n f a c t , as f a r as we know, t h e s e t w o

methods a r e a l m o s t a p p l i c a b l e t o v a r i o u s t y p e s o f summation, c e n t r a l o p e r a t o r s and c e n t r a l m u l t i p l i e r s e s t a b l i s h e d on c l a s s i c a l y r o u p s and on compact L i e y r o u p s a t home and abroad. The summation c o e f f i c i e n t s o f t h o s e summations and c e n t r a l m u l t i p l i e r s e s t a b l i s h e d by t h e method " f r o m k e r n e l s t o sums",

such as A b e l - and Cesaro-

summation i n t h i s a r t i c l e and t h e c l a s s o f c e n t r a l m u l t i p l i e r s e s t a b l i s h e d t h r o u y h t h e F o u r i e r t r a n s f o r m a t i o n f o r L i e a l y e b r a s by R . S . S t r i c h a r t z [14], a r e u s u a l l y very complicated.

1.5,

1.4,

2.7

2.3,

and 2.9

H e r e o n l y t h o s e c o e f f i c i e n t s i n Theorems 1.2,

a r e c o n c r e t e l y y i v e n and t h e i r d e t e r m i n a t i o n depends

on t h e c o m p l i c a t e d c a l c u l a t i o n and s k i l l f u l methods m e n t i o n e d above. For studyiny t h e properties o f Fourier series, C e s a r o summations i n t h i s s e c t i o n , t a r y yroups.

L e t u(U)

such as t h e c o n v e r g e n c e o f

t h e f o l l o w i n y method i s e s t a b l i s h e d on u n i -

e L ( U n ) , and qJ,(V)

= c-1

J

u(uwvw-l)fi.

u" The method i s t o s t u d y F o u r i e r s e r i e s o f u ( U ) t h r o u y h t h e c l a s s o f f u n c t i o n s {$,,(V),

U 6 Un}.

As qJU(V) i s a c l a s s f u n c t i o n ,

iel

F o u r i e r s e r i e s of a c l a s s of f u n c t i o n s (qJu(e

iel where JIU(e

,

..., e

ien

we o n l y need s t u d y m u l t i p l e

,

..., e

iBn

) , U E Un} on t o r u s ,

) a r e t h e v a l u e s o f $u(V) a t t h e maximum t o r u s , i . e .

diayonal u n i t a r y matrices.

at

L a t e r on, t h i s rnethod was a l s o used i n t h e r e s e a r c h

f o r t h e h a r m o n i c a n a l y s i s on c l a s s i c a l y r o u p s and on compact L i e y r o u p s . r e l a t e d examples can be f o u n d i n [ l O ] e x c l u d i n y t h o s e c o n d u c t e d a t home.

The

I n s p i r e d by t h e Abel-summation on u n i t a r y g r o u p s we have d e f i n e d C e s a r o means of F o u r i e r s e r i e s (1.9)

on u n i t a r y y r o u p s i n

[el.

N o t o n l y can t h e k e r -

n e l s be e x p l i c i t l y r e p r e s e n t e d by m a t r i c e s , b u t b o t h t h e summation c o e f f i c i e n t s

S. Gong et al.

78

and t h e r e l a t e d i n t e y r a l c o n s t a n t s can be c a l c u l a t e d o u t e x p l i c i t l y .

For

u n d e r s t a n d i n g o f t h e y e n e r a l Cesaro means, F e j e r means, w h i c h i s one o f t h e most t y p i c a l and most i m p o r t a n t example o f Cesaso means, was s t u d i e d c a r e fully.

T h i s example i n d i c a t e d t h a t t h e o t h e r c o e f f i c i e n t s and c o n s t a n t s

r e l a t e d t o t h e y e n e r a l Cesaro means can be o b t a i n e d i n t h e same way. L e t u ( U ) be an i n t e y r a b l e f u n c t i o n on Un and t h e Cesaro (C,a) F o u r i e r s e r i e s (1.9)

means o f i t s

be

where

H;(N)

=

xy 1 r~/ xf(V)K;(V)i

(2.2)

un and Ka(V) i n (2.2) N

i s Cesaro (C,a)

kernel which i s equal t o

N

1 $1;

detn[

VK(I

-

V'2kf1)j

k=O

(2.3)

detn B;=c

where

1

1

N

:A:;

/

k=O

un

(2A;)n

2

Aa =

and

Vk(I

b+a+)

N

detn(I

... ( a + l )

N!

- V'2kt1) -

v')

0,

,

moreover, a > -1 i s needed. F o r Cesaro means on u n i t a r y y r o u p s we have

THEOREM 2.1 C3l.

Cesaro (C,a)

means (2.1) o f F o u r i e r s e r i e s ( l . Y )

o f any

i n t e y r a b l e f u n c t i o n u(U) on u n i t a r y y r o u p s can be e x p r e s s e d as

1 C /

u(V'U)K;(V)t.

(2.5)

'n

PROOF.

From (2.2)

T h e r e f o r e (2.5)

and ( 2 . 3 ) ,

we have

can be w r i t t e n as

w h i c h i s j u s t t h e (2.1). I t can be i m m e d i a t e l y seen t h a t t h e r e l a t i o n between (C,a) k e r n e l s o f Cesaro

(C,a)

means d e f i n e d as above and P o i s s o n k e r n e l s o f Abel means d e f i n e d by Hua

Hurmoriic Anul-vsis or1 Classical Groups i s t h e same as i n t h e case o f F o u r i e r s e r i e s ,

i.e,

(C,a)

19

k e r n e l s become P o i s s o n

kernels i t a tends t o i n f i n i t y .

c31 L e t u ( U ) be c o n t i n u o u s on (In. Then, when

THEOREM 2.2. s e r i e s (1.9)

o f u ( U ) i s (C,a)

(O
U

BN'( ( 2 n ) !)n( - 1 ) n ( n - 1 ) / 2 ( 2 n - 1 ) ! (2n-2)!

kn>O

2n

(2n-l+kl)!

2n

...n! (N+1 )n2 k1>0

(2n-l+kn-1)!

n kn>U

..................... 2n

2n

(n+l+kl-l)!

sl=u 1

k l ! s p n - s p

(n+l+kn-l)!

k !s ! ( 2 n - s n ) ! sn=O n n

"*"

kn>U

kl>U

(2.15)

.

where kl

.... kn .an+nN-(N+l)sn;

'llfnN-(N+l)sl,

.... 1' ,

Ek = f k + n - k ,

.

f,.

S i m p l i f y i n y (2.15)

.

B

n

(0,O

.....0 ) .

)n(n-l)!

2

(N+I)"

2n ... 1 (-1) s =o s =o

1

Sl+.

n kn>U

where k . = n - j + n N - ( N + l ) s . J J

... 2 ! 1 !

... n !

(zn-l)!

2n

1 kl>U

....

fl+n-1,

Thus

= (-l)n('-')/'(n!

x

.

E s p e c i a l l y , we have H f ( N ) = 1 if

c a l c u l a t i o n e a s i l y l e a d t o (2.1U). f

al

f u r t h e r and a p p l y i n y i n g e n i o u s

..+sn

.

'gn

n+kl

1

(N+l)(n-s.)-j, J

... n:C j

n+kn Cn N((N+l)sl n

.l , Z , . . . , n sj .

.....( N + l ) s n ) (2.16)

. ....( n - 1 )

I t i s known f r o m t h e d e f i n i t i o n o f k j t h a t U,l, f o r k j > 0 i f N > n-1. (2.16) The u s u a l method b e i n y used [lJ,

i s necessary becomes

A s e r i e s of s k i l l f u l c a l c u l a t i o n r e l a t e d t o (2.17) h a v i n y been made [3], (2.11)

i s obtained.

Generally, series i s

p=-"

summation

l e t u(0) be an i n t e g r a b l e f u n c t i o n on 0
6 > 0 and

= 1,2,

...,n ) ,

where H,

TI,

> 6, a r e c o n s t a n t s dependent o n

"m m only. Then t h e T-means (2.21)

o f F o u r i e r s e r i e s o f u(U) converges t o u ( U ) i f u(U)

i s c o n t i n u o u s on UnF o r t h e summation o f F o u r i e r s e r i e s on u n i t a r y y r o u p s s e t up by t h e method " f r o m sum t o k e r n e l " , we may b e y i n w i t h y i v i n y a c o r r e s p o n d i n y sum o f F o u r i e r s e r i e s on u n i t a r y y r o u p s by (2.18)

[l]


T,(u)

f

where e

,

( U 1)

(2.24)

..., e

(2.25)

ien a r e t h e c h a r a c t e r i s t i c r o o t s o f V, D(x1,x2

Obviously,

1t r ( C f A

o f T-summation o f t y p e I 1 a r e

The k e r n e l s T;(V)

iel

Uman N ( f

2

T-summable t o s o f t y p e I 1 i f r m ( U ) + s when m t e n d s t o a

and c a l l (2.19)

limit.

1 lJmelUmk

=

83

, . . . I

Xn) =

n

l t i < jt n

(Xi

-

and

x.). J

i f we t a k e

... vmLn

as t h e summation k e r n e l s o f m u l t i p l e F o u r i e r s e r i e s and r e w r i t e i n (2.24)

,...,‘

, t h e n we d e f i n e a summation on u n i t a r y g r o u p s , t h e

as

1 n k e r n e l s o f w h i c h can be o b t a i n e d by c h a n y i n y km(el) km(e1,e2,..

.,en).

...k m ( e n )

i n (2.25)

into

and t h e n t h e Abel summation o f t y p e I 1 i n (2.1Y)

Take urp = 1-1’1,

i s g i v e n ( s e e C21). Choose uNk = A!(N) C e s a r o (C,a)

= A!

= r ( a + N - l k ( + l ) r ( N + l ) / ( r ( a + N + l ) r ( N - J k ( +and l)) then

summation o f t y p e 11 T,(U)

=

N>el>.

I:

..>en>-N

A’

‘1

... A:

N(f)tr(CfAf(U))

(2.27)

n

i s y i v e n [see C21). The k e r n e l (2.25)

c o r r e s p o n d i n g t o summations (2.26)

and (2.27)

t a k e s one-

d i m e n s i o n a l P o i s s o n k e r n e l and o n e - d i m e n s i o n a l Cesaro k e r n e l r e s p e c t i v e l y as k, ( 0 ) r e s p e c t iv e l y

.

F o r Abel and Cesaro summation o f t y p e 11, t h e f o l l o w i n y t h e o r e m i s v a l i d .

THEOREM 2.5.

[2]

L e t u(U) be a f u n c t i o n h a v i n g c o n t i n u o u s p a r t i a l d e r i v a t i v e s

up t o o r d e r n ( n - 1 ) / 2 , converges t o u ( U).

t h e n t h e A b e l - o r Cesaro-summation o f t y p e I 1 u n i f o r m l y

S. Gong et al.

84

Many Shi Kun and Dony Dao Zheny d e f i n e d Cesaro k e r n e l s on r o t a t i o n groups

SO(n) N-j

N

K:(r)

+ 1

= (B:)-ldet((A:I

(rJ+r'J)

j =1

where

r

8

1

A:-1)/A:)n(n-1)'2,

(2.28)

r=U

on S O ( n ) i s

i s a number such t h a t t h e i n t e y r a l o f K;(r)

S O ( n ) and B:

equal t o 1.

If' u ( r ) i s i n t e y r a b l e on S O ( n ) , i t s F o u r i e r s e r i e s i s

u(r) where

h(r) a r e

m = (ml m

,...,mk)

> m2 > 1

- m1

N(m)tr(CmAm(r)),

(2.29)

t h e i r r e d u c i b l e r e p r e s e n t a t i o n s o f SU(n) w h i c h t a k e

...

> > mk > U a r e i n t e y e r s i f n = 2 k + l and > l m k ( > U a r e a l s o i n t e y e r s i f n = 2k, N(m) = N(ml

as i t s l a b e l s , ml

... > mk-l

,...,

m k ) i s t h e o r d e r o f A m ( r ) , and

where c i s t h e volume o f SO(n) and L e t Xm(r) = tr(A,(r))

?

i s t h e volume e l e m e n t .

and (2.31)

I t i s e a s i l y seen t h a t

i f n = 2 k , m1 > coefficients

... > mk

B i 1,. ..,m

Cesaro (C,a)

> 0. T h e r e f o r e , we o n l y need t o c a l c u l a t e t h e f o r ml > > mk > U.

...

k

means o f (2.29

are (2.32)

THEOREM 2.6.

(Wany Shi Kun and Dong Dao Zheny, see [ l ] )

t i n u o u s f u n c t i o n on SU(n), s e r i e s (2.29)

of

u(r)

1)

JCi(r) -

2)

ll:(r)

3)

IC;(r) -

r

6

i s (C,a)-summable

t o i t s e l f and, when

u ( r ) J < A ~ N - P ; i f a(n-1)tz-n u(r)(

L e t u ( r ) be a con-

S o ( n ) , t h e n , when a > ( n - Z ) / ( n - l ) ,

< A ~ N - P ~ ON,Y

,

the Fourier L i p p,

> p;

i f =(n-1)+2-n

n-2-a( n-1) u(r)) < A ~ N

u(r) e

= p;

i f a ( n - l ) + Z - n < p.

When a = 1, t h e Cesaro summation i s j u s t t h e F e j e r summation and i t s k e r n e l s are


KN ( r )

=-

1

I

N

lihen n = Zk, (2.33)

. N-j+l

1 r-' N+l

det(1 + 2

j =1

BN

85

I (n-1)/2

)

(2.33)

becomes

and

THEOREM 2.7.

(liany Shi Kun and nony Dao Zheny, see [ I ] )

On S 0 ( 2 k ) , t h e F e j e r

summation c o e f f i c i e n t s r e a d

n

((2k-1)!)k

.m k

(s2-j2)

4k-2

O<j< s < k - 1

-

B ml..

N( i n ) ( 2 k - 1 ) !

...

sl+ ...+ s

4k-2

1

Sk'0

1 el>l-k

3k-2+el

k '4k-2 s k

(-1)

...

1=u

...( 4 k - 3 ) !BN(N+1) k ( 2 k - 1 )

'4k-2

".

'2k-1

sk

3k-2+ek '2k-1

*"

ek>l-k

... N( ( n - l - s l

) ( N+l)-ml,

where e. J ( 2 k - l ) N - ( N + l ) s . - m . - k + j , 3 3 3 i n t e y r a l constants are ((2k-l)!)k B

=

(Zk-l)!(Zk+l)!

..., ( n - l - s k )

j = 1,2

,...,k

and m l >

O < jn< s < k - l ( s 2 - j 2 )

... ( 4 k - 3 ) ! ( N + l ) k ( 2 k - 1 )

where e j = ( P k - l ) N - ( N + l ) S j - k + j ,

j = 1,2

( N+1 )-mk),

,...,k.

On S 0 ( 2 k + l ) t h e F e j e r summation c o e f f i c i e n t s a r e

4k-2

1=o

1 el>l-k

... > mk > U and

...

its

S. Gong et al.

86

((2k)!)k

B,

;...

(k

- );

n

...

=

N(m) ( 2 k ) ! ( 2 k + 2 ) ! 4k

... Sk'04k1

1

x

sl=o el>l-k

.+S

where e . = Z k N - ( N + l ) s . - k . , J J J constants are ((2k)!)k B

N

C4k s1

-1)

...C 4ksk '2k3k-l+el ...C2k3 k - l + e k

ek>l-k

1

-

((n-1-s. ) ( N+l)-L. J J

lI j =1

i)2)

+

(4k-2) !BN(N+l) k ( 2 k

.

sl+. (-1)

k x

( ( sj (-2)+- j !-

U<j<s ( 2 n - 2 ) / ( 2 n + l ) , o f u ( U ) i s (C,a)-summmable

IT;(U)

t o i t s e l f , and when u ( U )

- U(IJ)( < I~;(u) - u ( u ) ( < l r ~ ( ~- ) u ( ~ ) 2 p;

A ~ N - P l o g N, i f ( 2 n + l ) a - ~ n + 2 = p; A ~ N ~ ~ - ~ - i(f ~( ~~n ++l ) a~- 2) n +~ 2 l-n

(-1)

S1+"'+Sn

C4n+2 s1

kn>l-n

3n+kn 'kn+n-l

3n+kl ...c4n+2 Ck+n-l ... 'n

N( (n+( 2 n + l ) N - ( N+l)sl-fl,.

where k . = (2n+l)N-(N+l)sj-(fj+n-j+l), J

.., Z n - l + ( 2 n + l

j = l,Z,.

..,n,

and

)N-(N+l )sn-fn),

S. Gong et al.

88

4n+2

4n+2

3n+kn

3n+kn

kl>l-n x

n

Sl+...+S

“ ’ ckn+n-l(-l)

‘kl+n-l kn>l-n

,..., 2 n - 1 + ( 2 n + l ) N - ( N + l ) s n ) .

N(n+(2n+l)N-(N+l)sl

L i Shi X i o n y and Zheny Xue An d e f i n e d and d i s c u s s e d Ceasro k e r n e l s and Cesaro summation o f F o u r i e r s e r i e s c o n n e c t e d w i t h compact L i e y r o u p s . w i t h , Cesaro (C,a)

a l g e b r a i s one o f t h e compact L i e a l g e b r a s (A,,)U, (F4lu,

F6

(E6)u,

TO b e g i n

k e r n e l s a r e d e f i n e d on any compact L i e groups whose L i e

( E 7 l u y ( E 8 ) u and un = ( A n - l ) u

@

(Bn)U, (Cn)u,

H1,

(G2)u,

6B H1,

Y2 =

e6 = (Efj)u f3

= ( E s ) u @I H2,

e7 = (E7),, 6B H1 and Hn w h i c h i s t h e L i e a l g e b r a o f t o r u s

Tn w i t h d i m e n s i o n n.

These L i e a l y e b r a s a r e u s u a l l y c a l l e d t h e b a s i c compact

Hl,

L i e alyebras.

F o r a g e n e r a l compact L i e g r o u p G, t h e L i e a l g e b r a o f I; can be

decomposed e i t h e r as a d i r e c t sum w h i c h c o n s i s t s o f t h e b a s i c compact L i e a l g e b r a s l i s t e d above e x c e p t (An)U, (G2)u,

(E6)u,

e6,

( E 7 ) u , o r as a d i r e c t sum

w h i c h c o n s i s t s of t h e b a s i c compact L i e a l y e b r a s l i s t e d above e x c e p t Hn and a t l e a s t one o f (An),,,

( G Z ) ~ , (E6)u, e6, (E7),,i s i n c l u d e d i n i t .

t h e r e y u l a r d e c o m p o s i t i o n o f a compact L i e a l g e b r a .

This i s called

Here t h e Cesaro k e r n e l o f

G i s j u s t a p r o d u c t o r some r e s t r i c t i o n o f t h e p r o d u c t o f Cesaro k e r n e l s o f s e v e r a l b a s i c compact L i e groups m e n t i o n e d above, w h i c h c o r r e s p o n d t o t h e r e g u l a r d e c o m p o s i t i o n o f t h e L i e a l g e b r a o f G.

THEOREM 2.10. ( L i Shi X i o n y and Zheny Xue An) group. whose L i e a l g e b r a i s one of (An)u, (Es),,,

(E7),,,

.-,

,

( E B ) ~ , un, 92, e6, e6

We h a v e ( 1 ) L e t G be a compact L i e

(Bn)us ( C n ) u , (Dn)u, (Gillu,

(F4Iu,

e7, and a g a i n l e t t h e c r i t i c a l v a l u e s a.

c o r r e s p o n d i n y t o t h e above-mentioned b a s i c compact L i e a1 gebras b e

respectively.

Then t h e Cesaro means

o f F o u r i e r s e r i e s of any c o n t i n u o u s f u n c t i o n f ( x ) on G u n i f o r m l y c o n v e r g e s t o f ( x ) i f a > ao, where x denotes convolution; satisfy

6

G, K’(x)

n

s t a n d f o r Cesaro ( C , a ) k e r n e l s on G,

and if f ( x ) b e l o n y s t o L i p p and . a

= a/b,

*

t h e n T;(x)


d)

Iri(X)

-

f ( X ) ) < AIN-P,

b)

IT:(x)

-

f ( x ) ( < A2N-P

C)

IT;(x)

-

f(x)

I

89

i f ab-a > p;

l o g N, i f ab-a = p;

< A3Na-ab,

i f ab-a < p;

where a and b a r e g i v e n i n ( 2 . 3 8 ) . 2)

L e t G t a k e L as i t s L i e a l g e b r a and t h e r e y u l a r d e c o m p o s i t i o n o f L be L = L1 f3 L 2 f3

... f3 L k .

By a o ( L j ) we d e n o t e t h e c r i t i c a l v a l u e s c o r r e s p o n d i n g t o L j , j = 1,2,...,k,

and

set

Then t h e Cesaro summation o f F o u r i e r s e r i e s o f any c o n t i n u o u s f u n c t i o n f ( x ) on

G u n i f o r m l y c o n v e r g e s t o f ( x ) i f a > ag.

Moreover, i f a0 = a/b,

then a), b),

and c ) c o r r e s p o n d i n g t o 1 ) a r e a l s o v a l i d . The T-summation and T-summation o f t y p e I 1 o f F o u r i e r s e r i e s on u n i t a r y g r o u p s e s t a b l i s h e d by t h e methods " f r o m k e r n e l t o sum and f r o m sum t o k e r n e l " have s i m i l a r e x t e n s i o n s on compact L i e g r o u p s .

The r e l a t e d d e t a i l i s o m i t t e d .

3. The Cubical Partial Sums o f Fourier Series I n t h i s s e c t i o n we c o n s i d e r b r i e f l y t h e d e f i n i t i o n o f t h e c u b i c a l p a r t i a l sums o f F o u r i e r s e r i e s on u n i t a r y g r o u p s and i t s e x t e n s i o n s on c l a s s c a l g r o u p s and on compact L i e y r o u p s . I n t h e p r o o f o f Theorem 3.1,

i n which t h e concrete expression f o r D i r i c h l e t

k e r n e l s o f t h e c u b i c a l p a r t i a l sums o f F o u r i e r s e r i e s on u n i t a r y y r o u p s was e s t a b l i s h e d , a b a s i c method o f s t u d y i n y c l a s s f u n c t i o n s was s e t up.

The

essence o f t h e method i s t o t r a n s f o r m a r e s e a r c h p r o b l e m on c l a s s f u n c t i o n s t o a p r o b l e m on F o u r i e r s e r i e s o f t h e f u n c t i o n s ( s u c h as g(el,

...,en)

i n (3.6))

on

t o r u s w h i c h a r e made by t h e p r o d u c t o f v a l u e s a t t h e maximum t o r u s o f c l a s s f u n c t i o n s and t h e Weyl f u n c t i o n .

T h i s method i s a l s o w i d e l y a p p l i e d t o

r e s e a r c h f o r c l a s s f u n c t i o n s on c l a s s i c a l g r o u p s and compact L i e groups.

Some

r e s e a r c h e r s a b r o a d such as R. J . S t a n t o n and P. A. Tomas a d o p t e d t h i s method i n t h e i r studies

on t h e a l m o s t e v e r y w h e r e c o n v e r g e n c e o f F o u r i e r s e r i e s o f c l a s s

f u n c t i o n s on compact L i e y r o u p s . The c u b i c a l p a r t i a l sum o f F o u r i e r s e r i e s on u n i t a r y g r o u p s have t w o f o r m s

o f e x t e n s i o n s on compact L i e g r o u p s .

One i s made by R. J. S t a n t o n and P. A.

Tomas They, s t a r t i n g f r o m t h e convex p o l y h e d r o n ( i n c l u d i n g t h e o r i g i n as i t s i n t e r i o r p o i n t ) on C a r t a n sub-a1 g e b r a s o f L i e a1 g e b r a s o f compact L i e g r o u p s w h i c h i s i n v a r i a n t u n d e r Weyl g r o u p s , d e f i n e d t h e p o l y h e d r a l p a r t i a l sums, f o r

S. Gong e l al.

90

w h i c h one o f t h e fundamental p r o p e r t i e s f o r t h e c u b i c a l p a r t i a l sums d e f i n e d on u n i t a r y g r o u p s was used.

A n o t h e r i s made by L i S h i X i o n g and Zheng Xue An.

They, s t a r t i n g f r o m t h e r e g u l a r c o o r d i n a t e s f o r t h e h i g h e s t w e i g h t s i n a cube o r a p o l y h e d r o n , d e f i n e d t h e c u b i c a l and p o l y h e d r a l sums o f F o u r i e r s e r i e s on compact L i e groups, f o r w h i c h a n o t h e r b a s i c p r o p e r t y f o r t h e c u b i c a l p a r t i a l

sums d e f i n e d on u n i t a r y groups was used. F o r e x p r e s s i n g D i r i c h l e t k e r n e l s e x p l i c i t l y , a d i f f e r e n t i a l o p e r a t o r was e s t a b l i s h e d on u n i t a r y groups, by means o f w h i c h D i r i c h l e t k e r n e l s on u n i t a r y groups c o u l d be s i m p l y e x p r e s s e d by D i r i c h l e t k e r n e l s o f m u l t i p l e F o u r i e r series.

Moreover, when we e s t a b l i s h T-summation k e r n e l s o f t y p e I 1 i n s e c t i o n

I 1 and when we deduce t h e i n t e g r a l r e p r e s e n t a t i o n s o f t h e s p h e r i c a l means summation, t h i s o p e r a t o r a l s o p l a y an i m p o r t a n t r o l e .

Wany Shi Kun, Dony Uao

Chen Guang X i a o e s t a b l i s h e d c o r r e s p o n d i n g d i f f e r e n t i a l Zheng, He Zhu Qi, o p e r a t o r s on r o t a t i o n y r o u p s and u n i t a r y s y m p l e c t i c groups r e s p e c t i v e l y .

Li

S h i Xiong and Zheng Xue An e s t a b l i s h e d c o r r e s p o n d i n g d i f f e r e n t i a l o p e r a t o r s on g e n e r a l compact L i e groups and gave t h e i r r e p r e s e n t a t i o n s under v a r i o u s systems o f coordinates e x p l i c i t l y . Some r e s e a r c h e r s abroad such as J . L. C l e r c [ l l J e s t a b l i s h e d c o r r e s p o n d i n g d i f f e r e n t i a l o p e r a t o r s on ( s e m i - s i m p l e ) compact L i e g r o u p s w h i c h were e x p r e s s e d as d i r e c t i o n a l d e r i v a t i v e s . When d i s c u s s i n g t h e p r o b l e m a b o u t t h e c e n t r a l m u l t i p l i e r on compact L i e groups, K. Coifman, G. Weiss [ l o ] and N. J. Weiss [15] e s t a b l i s h e d t h e d i f f e r e n c e o p e r a t o r s s i m i l a r t o t h e d i f f e r e n t i a l o p e r a t o r s on u n i t a r y yroups. The c u b i c a l p a r t i a l sums o f F o u r i e r s e r i e s (1.9)

o f an i n t e g r a b l e f u n c t i o n

u(U) on t h e u n i t a r y group Un a r e d e f i n e d by

where L~ = fl+n-l,

k2 = f2tn-2,

..., in = fn.

Let

then

SN(U,U) = U * U N ( U ) = c-1

J

U(UV')UN(V)i,

'n

p ( V ) i s c a l l e d the D i r i c h l e t kernel. N

THEOREM 3.1.

121

d N ( a ) = I p = - N eipe,

Let

ia1

,

.,.,

t h e n we have

ie

n be t h e c h a r a c t e r i s t i c r o o t s o f V

e

Un,


-

( - i )n(n-1)/2

'5,...,e l e n ) ( n - l ) ! ...l ! D ( e

PROOF.

(8 ))

d e t (din-J

(3.3)

.

( x ) = ( d / d x ) '-JdN( x )

where

91

The f u n c t i o n

c .>tn>-N

N>tl>..

DN ( V ) ,

as Abel-means o f

Pf(r)N(f)xf

i s a class function,

V)

9

hence we o n l y need t o c o n s i d e r t h e

following series ( D ( e iel

,...,e i'n)l-l N>tl>

From t h e d e f i n i t i o n (1.8)

i S

iel

,...,e i e n ) ,

z ...

pf(r)N(f)Hf(e

of pf(r),

i t i s easy t o see t h a t t h e s e r i e s i n

>tn>-N

(3.4)

t h e c u b i c a l p a r t i a l sums o f t h e m u l t i p l e F o u r i e r s e r i e s o f t h e f u n c t i o n

y(el

,...,8,)

Thus (3.5)

iel = l(1-re

iel

2

i8, )...(l-re

) ) - 2 n ( l - r 2 ) n D(e

,...,e

ie, ).

(3.6)

can be e x p r e s s e d as

(3.7)

I n v i r t u e o f t h e skew-symmetry o f g($l,...,$n)

,...,$,)

+ ($j

,...,$jn ), 1 2n

(n!)-1(2n)-n

I ... J

0

0

(3.5)

2n g(Q1

under t h e permutation

can a l s o be e x p r e s s e d as

,...,$n)P($l ,...,$n;

81,...yen)d$l...d$n

S. Gong et al.

92

c l a s s f u n c t i o n and i t s v a l u e f o r d i a g o n a l m a t r i c e s i s

By Theorem 1.1 t h e v a l u e o f (3.8)

$1

,...,$,

a t p o i n t $1 =

... = $n

i s t h a t o f t h e continuous f u n c t i o n o f

= 0 when r + 1.

By a r e s u l t i n [l], this is

Thus t h e c o n c l u s i o n f o l l o w s f r o m t a k i n g l i m i t i n ( 3 . 4 ) . F o r t h e u n i f o r m c o n v e r g e n c e o f t h e c u b i c a l p a r t i a l sums o f F o u r i e r s e r i e s on u n i t a r y groups, t h e f o l l o w i n g r e s u l t s a r e v a l i d .

[4]

THEOREM 3.2.

I f u(U)

8

Cn(n-1)/2tp(Un)

(0 < p < l ) , t h e n t h e c u b i c a l

p a r t i a l Sums SN(U,U) o f i t s F o u r i e r s e r i e s c o n v e r g e t o u(U) and (SN(u,U) Let

u(r)

-

u(U)

1

2

< A max{ (N-'loyn

N)l/(ntl),

N-pl~gn-lN}.

be an i n t e g r a b l e f u n c t i o n on S 0 ( n ) , and t h e n t h e c u b i c a l p a r t i a l

sums o f i t s F o u r i e r s e r i e s (2.29) a r e d e f i n e d b y (3.10) i f n = 2k and

r

€

S0(2k+l);

The f o l l o w i n g lemma

s needed :

...

qk b e i n t e g e r s such t h a t q1 > q2 > > qk > 0, p j ( q s ) L e t ql... be a f u n c t i o n dependent o n l y on qs, j = 1,2 k, and l e t N be a p o s i t i v e

LEMMA 1. C8l

,...,

i n t e g e r , a and b be any r e a l numbers, t h e n

The D i r i c h l e t k e r n e l o f t h e c u b i c a l p a r t i a l sums d e f i n e d by (3.10) (3.11)

are vN(r)

therefore

"al>.

c..>en>O

N(m)om(r),

and (3.12)

Harmonic Analysis on Classical Groups where t h e meaniny o f a,(r)

THEOREM 3.3.

[81

93

Then by Lemma 1, we have

i s y i v e n i n Theorem 1.3.

I f n = 2k. then (3.13)

and i f n = 2 k + l , t h e n (3.14) where d N ( e ) = s i n ( N

...4 ! 2 ! ,

aZk = Z1-!2k-2)!

(ej)), c ( e )

det(C

1

n / 2

- [n/2],

U < p

and i n p a r t i c u l a r i f f ( y )

6

Ck*P(G)

91

,

where k = [ n / 2 ]

< 1, t h e n t h e F o u r i e r s e r i e s f o r f ( g ) c o n v e r g e s

a b s o l u t e l y and u n i f o r m l y , a c c o r d i n g t o t h e d e f i n i t i o n o f ( 3 . 2 1 ) .

If f(g)

2)

L b y s ( G ) , where k i s a n o n - n e g a t i v e i n t e g e r , U < r ( ( 3 / p ) - 1 / 2 ) m + q ( r - ' + p - ' - l ) . I f f ( g ) f L b y s ( G ) , where k i s a n o n - n e g a t i v e i n t e g e r , 0 < r < 2,

3)

1< p
(3/p-3/2)m+q(l/(2r)+l/p-l). The p r i n c i p a l r e s u l t s abroad p a r a l l e l t o t h o s e on t h e c u b i c a l p a r t i a l sums of F o u r i e r s e r i e s i n t h i s s e c t i o n and t o t h o s e on t h e summations o f F o u r i e r s e r i e s i n s e c t i o n 2 a r e as f o l l o w s . In

[lo],

K.

Coifman and G. Weiss s t u d i e d t h e r e l a t i o n between t h e c e n t r a l

m u l t i p l i e r s f o r F o u r i e r s e r i e s on compact L i e g r o u p s and t h e m u l t i p l i e r s f o r multiple Fourier series. Let

They p r o v e d t h e f o l l o w i n g r e s u l t .

H be t h e C a r t a n s u b - a l g e b r a o f t h e L i e a l g e b r a o f a compact L i e g r o u p

exp be t h e e x p o n e n t i a l mapping, and

E

be t h e u n i t e l e m e n t o f G.

Ino ( d x m h ) C h ( r )

G,

If (3.22)

hEG

d e f i n e s t h e bounded m u l t i p l i e r on L ( H / e x p - l E ) , P

then

1- mhdxXx(Y) x fG d e f i n e s t h e bounded c e n t r a l m u l t i p l i e r on L P ( G ) ,

(3.23) where p > 1.

I n addition, the

p r e c e d i n g c o n d i t i o n s a r e a l s o n e c e s s a r y f o r p = 1. I n (3.22),

'I E

H, C,(T)

=

1 oew

eiB(a,ar),

(3.24)

S. Gong et al.

98

where W d e n o t e s t h e Weyl g r o u p , and B(

,

) represents t h e i n v a r i a n t inner

p r o d u c t on t h e L i e a l y e b r a f o r G. The d i f f e r e n c e o p e r a t o r

0

i n (3.22)

brinys

rn

where a l , a2

,..., a,

are a l l p o s i t i v e roots.

R. J . S t a n t o n and P. A. Tomas ( s e e 1121 and L131) d i s c u s s e d t h e p o l y h e d r a l p a r t i a l sums o f F o u r i e r s e r i e s on compact L i e y r o u p s d e f i n e d as f o l l o w s . Suppose t h a t 9 i s a c l o s e d connex p o l y h e d r o n w h i c h t a k e s t h e o r i g i n as i t s i n t e r i o r p o i n t and i s i n v a r i a n t under t h e t r a n s f o r m a t i o n o f t h e Weyl y r o u p i n t h e C a r t a n s u b a l y e b r a H o f t h e L i e a l y e b r a o f a compact L i e y r o u p G, and l e t Rt = { t x l x E R}.

They d e f i n e d t h e p o l y h e d r a l p a r t i a l sums o f f

6

L p ( G ) , p > 1,

by

and p r o v e d t h e f o l l o w i n y :

1) L e t G be a s i m p l y c o n n e c t e d s i m p l e compact L i e yroup, T be a maximal t o r u s of G, d i m G = n, dim T = q and Ly(G) be a l l c l a s s f u n c t i o n s i n L p ( G ) . p > 2 n / ( n + q ) and f

e

If

L ~ ( G ) ,t h e n S N f ( x ) a l m o s t e v e r y w h e r e c o n v e r g e s t o f .

2 ) When G, T, n, q a r e t h e same as I n l ) , and p < 2 n / ( n + q ) o r p > 2 n / ( n - q ) , t h e r e e x i s t s f E L y ( G ) such t h a t S N f ( x ) does n o t c o n v e r y e i n t h e sense o f L p norm.

3)

When G, T, n, q a r e a l s o t h e same as l ) , t h e r e e x i s t s a number p ( R ) ,

2 n / ( n t q ) < p(R)

(2n-2q+2)/(n-q+2)

such t h a t S N f ( x ) c o n v e r y e s i n t h e sense of

L p norm, t o f ( x ) f o r p ( R ) < p < p ( R ) ' and f

4)

6

Ly(ti).

When G i s a s i m p l e c o n n e c t e d s e m i - s i m p l e compact L i e yroup, t h e n t h e

r e s u l t s c o r r e s p o n d i n g t o 1)

-

3 ) can be composed by c o m b i n i n g t h e r e s u l t s o f

i t s s i m p l e subgroups. ) t h a t S N f ( x ) does n o t c o n v e r g e i n 5 ) When p # 2, t h e r e e x i s t s f E L ~ G such t h e sense o f L p norm.

6)

When p < 2, t h e r e e x i s t s f 6 Lp(G) such t h a t S N f ( x ) a l m o s t e v e r y w h e r e

does n o t c o n v e r g e t o f ( x ) .

R . A. Mayer ( s e e [18])

discussed F o u r i e r series f o r G = SU(2).

He p r o v e d

the followiny.

1) L e t f 6 C 1 ( t i ) , t h e n t h e F o u r i e r s e r i e s f o r f c o n v e r g e s u n i f o r m l y , and t h e r e e x i s t s g 6 C1(G) such t h a t i t s F o u r i e r s e r i e s does n o t c o n v e r g e absolutely. 2)

Let f

e L2(G) and f b e l o n g t o c l a s s C1 a l m o s t everywhere, t h e n t h e

F o u r i e r s e r i e s o f f a l m o s t e v e r y w h e r e c o n v e r g e s t o f.

Here f t h a t b e l o n g s t o


99

c l a s s C1 a t one p o i n t means t h a t f i n a neighborhood o f t h e p o i n t i s equal t o a function i n c ~ ( G ) . 3)

L e t f E L1(G) and f be equal t o z e r o i n a neighborhood o f a p o i n t b e G.

Moreover, F o u r i e r s e r i e s Then

I;= p, nf(x)

f o r f s a t i s f i e s P n f ( b ) + 0, when n +

-.

p n f ( b ) converges t o zero.

L a t e r , Mayar s t u d i e d v a r i o u s problems about F o u r i e r s e r i e s on S U ( 2 ) systematically. I n [lY], (3.21))

PI.

E. T a y l o r discussed t h e a b s o l u t e converyence ( i n t h e sense o f

o f F o u r i e r s e r i e s on compact L i e groups and proved : l e t G be a compact

L i e yroup, dim G = n, and l e t s > n/4 be an i n t e y e r .

I f f e HZs and i n p a r t i -

c u l a r i f f E CZs(G), t h e n t h e F o u r i e r s e r i e s o f f converges a b s o l u t e l y and u n i f orml y

.

0. L. Ragozin (see 3 ) o f [20])

discussed t h e problem o f t h e a b s o l u t e con-

vergence o f F o u r i e r s e r i e s on compact L i e yroups i n t h e f o l l o w i n g sense and t h e problem o f t h e r e l a t i o n between t h e convergence and t h e d i f f e r e n t i a b i l i t y o f f :

where t h e meaning o f t h e r e l a t e d n o t a t i o n s i s t h e same as i n (3.19)

and (3.20),

and t r ( l C a I P ) i s d e f i n e d as f o l l o w s : L e t xl,

x2,

...,

be t h e c h a r a c t e r i s t i c r o o t s o f

non-negative and

Cayi.

Then t h e y a r e

d.

B. D r e s e l e r (see C161 and C171) s t u d i e d Lebesgue c o n s t a n t s f o r s p h e r i c a l p a r t i a l Sums o f F o u r i e r s e r i e s on compact L i e groups and proved t h a t t h e Lebesyue c o n s t a n t s a r e O(N(n-1)/2).

Moreover, he gave t h e e s t i m a t e s from above

and from below, n b e i n g t h e dimension o f t h e yroup.

4. Sumnation by S p h e r i c a l Means The d e f i n i t i o n o f summation by s p h e r i c a l means i n harmonic a n a l y s i s on u n i t a r y yroups and t h e r e l a t e d methods (see [6])

a r e w i d e l y used i n t h e

r e s e a r c h f o r harmonic a n a l y s i s on c l a s s i c a l groups and on compact L i e yroups. The s p h e r i c a l means summation o f F o u r i e r s e r i e s on u n i t a r y groups, essentially,

i s such a summation t h a t t h o s e terms o f F o u r i e r s e r i e s c o r r e s p o n d i n g t o

t h o s e f u n c t i o n s h a v i n g t h e same c h a r a c t e r i s t i c values o f L a p l a c e o p e r a t o r i n t h e r e p r e s e n t a t i v e r i n g o f a u n i t a r y group a r e m u l t i p l i e d by t h e same c o e f f i cient.

T h i s can e a s i l y be done by t a k i n g

adding a f a c t o r f u n c t i o n e x p ( - i ( n - l ) ( e l +

(4.4).

ak = fk + ( n - Z k + l ) / Z i n (4.1) and

...+en)/2)

t o the i n t e g r a l expression

S.Gong el al.

100

I n t h e r e s e a r c h work on t h e s p h e r i c a l means summation i n u n i t a r y groups, a method based on t h e F o u r i e r t r a n s f o r m a t i o n on C a r t a n s u b a l g e b r a s was e s t a blished.

As a C a r t a n s u b - a l g e b r a ,

u n d e r t h e i n v a r i a n t i n n e r p r o d u c t , con-

s t i t u t e s an E u c l i d e a n space, a v a r i e t y o f t o o l s o f t h e F o u r i e r t r a n s f o r m a t i o n i n t h e E u c l i d e a n space c a n be a p p l i e d .

Some r e s e a r c h e r s abroad such as

H.

S.

S t r i c h a r t z ( s e e C141) a d o p t e d a r e s e a r c h method, whose b a s i s i s t h e F o u r i e r t r a n s f o r m a t i o n on t h e L i e a l g e b r a .

C o m p a r a t i v e l y , t h e f o r m e r n o t o n l y can g i v e

an e x p l i c i t e x p r e s s i o n and r a t h e r a c c u r a t e r e s u l t s b u t a l s o can g i v e more F o r example, a w i d e c l a s s o f bounded o p e r a t o r s

r e s u l t s t o a l o t o f problems. on L ( G ) i n Theorem 4.12

( 1 ) w h i c h i s e s t a b l i s h e d by t h e methods on u n i t a r y

y r o u p s c a n n o t be o b t a i n e d by t h e methods on L i e a l g e b r a s i n [14].

But t h e

l a t t e r c e r t a i n l y has some advantages o v e r t h e f o r m e r i n some r e s p e c t s . F o r F o u r i e r s e r i e s ( l . Y ) y we c o n s i d e r t h e

L e t u ( U ) be i n t e g r a b l e on Un. f o l l o w i n g sum

I:

I:

m

fl>.,.>f

e;+.. ek = f k +n-k, k = l,2,...yn. L e t 4(t) be a f u n c t i o n on 0
" . > f

N(f)tr(CfAf(U)). n

e 2l + . ..+ez=m O b v i o u s l y , when u(U

i s i n t e g r a b l e and R i s a c o n s t a n t , (4.2)

c o n v e r g e n t f o r a l m o s t a l l U E Un,

i s uniformly

provided

1 (@(J;/R)N(f)(

0 is Si(u;U) By (4.3),

6(fi/R)-l

1 6(~/R)N(f)tr(CfAf(U)).

8

Un,

t h u s (4.2)

a )(Hb a ,..., -

acn

(4.8)

1

t h e s e r i e s on t h e r i g h t s i d e o f (4.8)

almost every U As D(

-

and (4.4)

(n-2)/2

1 1 1 l-n)

( 15 )

i s a b s o l u t e l y convergent f o r

a r e equal f o r almost, every u

6

Un.

can be c a l c u l a t e d by r e c u r r e n c e

formula i t takes i t s o r i g i n a l s i g n o r t h e o p p o s i t e s i g n under t h e permutation (cl

,...,5,)

+

(cj 1

,...,5 .

'n

) a c c o r d i n g t o t h e p e r m u t a t i o n b e i n g even o r odd.

102

S. Gong et al.

Thus i t i s e q u a l t o (-1)n(n-1)/2D(cl,...,cn)H and (4.7)

6 (n

a r e equal.

F o r 6 ( t ) i n t h e s p h e r i c a l means ( 4 . 2 ) ,

2 ( ~ E ( ) I ~ ,I i ~. e .- ~(4.4)

-2)/2

t h e most i n t e r e s t i n y examples a r e t h e

f o l 1owi ny :

1)

6 ( t ) = e-t,

2)

g ( t ) = e-t2,

3)

6(t) =

{

t h e Poisson-Abel summation, t h e Gauss-Sommerfeld summation,

:1-t2)6

for

o
1. K ti 3 ) I f u ( U ) i s i n t e g r a b l e on Un, t h e n SR(u;U) c o n v e r g e s t o u ( U ) f o r R +

-

-

a l m o s t everywhere.

THEOREM 4.4.

then

converges t o u(u) u n i f o r m l y , f o r R +

s;(u;u)

1)

I f 6 > (n2-1)/2,

(see [S]).

-

if

U(U)

i s continuous on

And i f u ( U ) 6 L i p a, (1 < a < 1, t h e n

U.,

2)

+

( n2 - 1 ) / 2 > 6 ;

(SR(u;U)

-

u ( U ) I < AgK - 6 + ( n 2 - 1 ) / 2 ,

b)

lSi(u;U)

-

u ( U ) ) < A6R-alog R , i f a + ( n 2 - 1 ) / 2 = 6;

c)

IS;(~;U)

-

u ( u ) ( < A ~ R - ~ i, f a

I f u(U)

6 Lp(U,),

for R +

Lp(Un),

3)

6

a)

-,

p > 1, t h e n S;(U;u)

(n2-1)/2 < 6 . c o n v e r g e s t o u ( U ) i n t h e norm o f

and I l S ~ ( u ; U ) i lP < A811u(U)II P’

I f u(U) i s i n t e g r a b l e on Un,

everywhere f o r R +

+

if a

then Si(u;U)

c o n v e r g e s t o u(U

-.

I n Theorems 4.2, 4.3

and 4.4,

t h e numbers Ao,

A1

a1 inos t

ndependent o f

R.

THEOREM 4.5.

(see [6])

V a l , and (4.3),

i f U < (51 < 1/R,

where p > 0, Then,

(4.5),

L e t a ( t ) be a b s o l u t e l y c o n t nuous on any f i n i t e i n t e r and (4.6)

Eloreover, we have

and

i f 1/R < (51
1, f o r j = 1,2,3, o p e r a t i o n s on Lp(G) and

t h e n SQ,,(f;g)

a r e bounded l i n e a r

l l S ~ , R ( f ; y ) i i p < A(G,$,j ,R) ilfll P ' and S'? ( f ; g ) i s r e y a r d e d as an i n t e g r a b l e f u n c t i o n f o r w h i c h t h e F o u r i e r J ,R s e r i e s i s j u s t (4.17-j).

&

1, and i f P ( h ) P ( ) { $ ( h ) } 6 L(H), then f o r I f f ( g ) E Lp(G), p S'? ( f ; g ) i s a bounded l i n e a r o p e r a t o r on Lp(G) and J ,R

2)

j = 1,2,3,4,

and S'? ( f ; g ) i s r e g a r d e d as an i n t e g r a b l e f u n c t i o n f o r w h i c h t h e F o u r i e r J ,R s e r i e s i s j u s t (4.17-j).

&

) { $ ( h ) } ( < A ( l + (hI)-"', and j = 1,2,3,4, 3) I f IP(h)-lP( b e s i d e s 2 ) o f t h i s theorem, t h e f o l l o w i n g r e s u l t s a r e v a l i d . a)

SQ,,(f;y)

almost everywhere converges t o f ( g )

b)

SQ,R(f;g)

u n i f o r m l y converges t o f ( g ) f o r R +

C)

SUP

R>U

M f ( y ) = sup r>O

(sj,R(f;g)I

/

< A(G,$,j)(Mf(Y)

If(t)(dt(B(g;r)(-',

+

/

h(t )

G

6

-

E

L(G) f o r R +

> 0, t h e n

m.

i f f ( g ) i s continuous.

-'

I I f ( g t - ' ) 1 d t ) , where

and B ( g r ) d e n o t e s a l l t

E

G from which

B(g;r)

t h e Riemann d i s t a n c e t o g i s l e s s t h a n r;

[ { s uR p I S ? ,R ( f ; g ) )

d) 4)

If

1-

> y } l < A(G,$,j)y-lllfllLl.

I$((A+e)/R)ldA
0.

And, i n t h e meantime,

i s equal t o S'? ( f ; g ) f o r a l m o s t e v e r y g 6 G and j = 1,2,3 o r j = 4, J ,R i f $ ( h ) s a t i s f i e s t h e f i r s t c o n d i t i o n o r t h e c o n d i t i o n i n 3) r e s p e c t i v e l y , (4.17-j)

b e s i d e s t h e above m e n t i o n e d c o n d i t i o n s .

6 ) From t h e P o i s s o n summation f o r m u l a t h e summation k e r n e l s K ? ( 9 ) c a n be J YR deduced w h i c h s a t i s f i e s


S$

THEOREM 4.13. U

1

f ( g t - l ) KQ,R(t)dt. G Take + ( t ) = ( 1 - t 2 k ) 6 , ( L i S h i x i o n g and Zheng Xuean).

J ,R

(f;g) =

109

t < 1 and 0 f o r t > 1, k b e i n g a p o s i t i v e i n t e g e r .

Thus (4.17-1)

(4.18) for defines

t h e R i e s z summation of o r d e r 6 and d e g r e e 2k o f F o u r i e r s e r i e s on compact L i e When k = 1, i t i s t h e u s u a l 1 R i e s z S Z k s 6 ( f ; g ) denotes S2k*6(f;g). 1,R summation d e n o t e d by S i ( f ; g ) . Then S i k S 6 ( f ; g ) s a t i s f i e s t h e f o l l o w i n y :

groups.

i s valid f o r Sik9&(f;g) i f 6 > (n-l)/2.

1)

The c o n c l u s i o n of Theorem 4.12

2)

I f f ( g ) i s c o n t i n u o u s on G and 6 > ( n - 1 ) / 2 t h e n

-

1Sik"(f;g)

3)

The s a t u r a t i o n o r d e r o f

THEOREM 4.14.

f(g)

Siky6

1

< A(G,k,G)w(f;l/R).

i s R-2k.

( L i S h i x i o n g and Zheng Xuean).

I f 6 > (n-1)/2,

then

s < 2k-1.

< A(G,k,G)

1

i f f ( g ) c C2k(G).

Ilfl12kR-2k

dxtr(C,A,(g))

-

f ( g ) l l m } , C x an a r b i t r a r y

x+BlO

b)

S i k s 6 ( f ; y ) c o n v e r g e s t o f ( y ) a l m o s t everywhere;

c)

l i m iisiky6(f;g) I?+-

-

)iitilp;

= 0.

f(g)ii,

I

f ( y ) d y , where g,y a G. B(g;r) G t h e r e e x i s t s r o > 0 such t h a t

4) g

< A(G,P,k

a)

Let f*(g;r)

=

f*(g;r+2s)

-

+ f*(g;r)

2f*(g;r+s)

I f f o r almost every

= o(s/log

s)

i s v a l i d u n i f o r m l y f o r s < r < ro, t h e n t h e F o l l o w i n g r e s u l t o f t h e Salem t y p e 2 k ,6 i s v a l i d : sR O(f;g) converges t o f ( g ) ( f o r R + W ) almost everywhere i f I t ) l o g + l f ( i s i n t e g r a l f o r G b e i n g a t o r u s of d i m e n s i o n i n t e g r a l f o r G b e i n g o t h e r compact L i e group, Dini-,

n >

2 or i f f i s

S i m i l a r l y , we can g i v e t h e

t h e J o r d a n - and t h e L e b e s g u e - t e s t f o r S ~ " ' " O ( f ; g )

on compact L i e g r o u p s

by use o f t h e f u n c t i o n f * ( y ; r ) .

E. PI. S t e i n ( s e e [ 9 ] ) d i s c u s s e d t h e f o l l o w i n y s p h e r i c a l means o f F o u r i e r

where x E G, f E L ( G ) , and he p r o v e d

2)

where t > 0, f ( x ) € Lp(G), p > 1. P' p t i s a s e l f - c o n j u y a t e o p e r a t o r on L ~ ( G ) .

3)

f > 0 i m p l i e s t h a t ptf

4)

l i m -P= t f - f

1)

llPtfUp c Ufll

t+O

t

-(-A) l / Z f

0.

,

where

5) u(t;x) equation

I Ptf(x)

E C"(Gx(0,-)),

6)

u(t;x)

X2,

5

+

A)U

I 0.

converges t o f ( x ) f o r t + 0 i n t h e norm o f L ( G ) , where

..., Xn

i s a b a s i s o f t h e L i e a l g e b r a o f G,

n A =

1

i,J=l

s a t i s f i e s t h e Laplace

2

(

XI,

and a l s o u ( t ; x )

( a i j ) = (-B(Xi,

n

a..X.x., 'J 1 J

AA ( x ) = -p A ( x ) , A f ( x ) = A X

1

a. . X . x . f . 1J 1 J

i ,j=1

L e t f be a r e a l v a l u e d f u n c t i o n w h i c h b e l o n g s t o C"(G)

and d e f i n e

Xj))".

Harrnonic Analysis on Classical Groups

( v f (2 ( x ) If f

6

Cm(tix(O,-)),

111

n a..(xif)(x.f). i ,j=1 1J J

1

=

then

S t e i n d e f i n e d t h e L i t t l e w o o d - P a l e y f u n c t i o n o f f f Lp(G) as

Then E. M.

m

I0 t l v u ( t ; x ) ( 2 d t ) 1 / 2

(

Y(f)(X) =

9

and p r o v e d t h e f o l l o w i n g :

7 ) Let f 6 Lp(G), 1 < P < Ap such t h a t

-.

Then g ( f )

IIg(f)ll Conversely, i f

I f(x)dx

= 0,

P

6

< A Ilfll P

Lp(G) and t h e r e e x i s t s a c o n s t a n t

P '

then there e x i s t s a constant B

G Ilfllp

8)

P

such t h a t

< Bpllg(f)llp.

L e t t h e R i e s z t r a n s f o r m a t i o n on G be K . f = X.(-A)-"'f, J J

...,n,

where f E C"(G).

1 < p < -,

Then R . , j = 1,2, J from which f o l l o w s

J. L. C l e r c ( s e e [ l l ] )

j =1,2

,...,n,

a r e bounded o p e r a t o r s on Lp(G) f o r

d i s c u s s e d t h e summation o f F o u r i e r s e r i e s on compact

L i e y r o u p s by R i e s z means o f o r d e r 6.

H i s m a i n r e s u l t s a r e as f o l l o w s :

L e t G be a compact L i e g r o u p o f d i m e n s i o n n and r a n k q, D ( e x p h ) be C l e y l ' s f u n c t i o n o f G and then, 1) 2)

S i f + f f o r 6 > ( n - 1 ) / 2 i n t h e norm o f L p ( G ) , p sup ( S i f ( x ) I < C ( M f ( x )

f

K*lf((x)),

1.

6 > (n-l)/2.

R>O

3)

If 6 > ( n - 1 ) / 2 , f E L(G) and m i s t h e Haar measure, t h e n m{sup I S i f I > a] < A llflll , R

and, from t h i s , S i f c o n v e r y e s t o f a l m o s t e v e r y w h e r e ;

4) that

I f 1 ( n - l ) ( l / p - 1 / 2 ) ,

IlSUP

R

R.

S.

t h e n t h e r e e x i s t s a c o n s t a n t Ap such

6

I S R f ( II

P

< A

P

llfll

P

.

S t r i c h a r t z ( s e e C141) d i s c u s s e d t h e m u l t i p l i e r t r a n s f o r m a t i o n on

compact L i e a l g e b r a s and groups.

S. Gong et al.

112

L e t G be a compact L i e y r o u p and $I be i t s L i e a l g e b r a , H be a C a r t a n subalgebra o f

9,

dp

be ad- n v a r i a n t f i n i t e measure o n

9.

E s p e c i a l l y , when dp

i s absolutely continuous, t h e r e e x i s t s a f u n c t i o n F(x), x E i n t e g r a b l e and a d - i n v a r i a t (i.e.

9,

which i s

F ( h ) ( P ( h ) I 2 i s i n t e g r a b l e on ti, h E H such

t h a t dp = F ( x ) d x . R. S. S t r i c h a r t z p r o v e d :

1)

If

then

$(A)

or

=

$(A) =

(*I J @ ( A + B - ~ ~ ~ B(**I) ~ Y @(A+B)

G

a r e bounded o p e r a t o r s on L ( G ) . 2 ) L e t @ ( x ) be t h e same as i n 1 ) and d e f i n e o r ( x ) = @ ( x / r ) . : an o p e r a t o r 0P(@) on

and ( * ) o r ( * * ) d e f i n e s a n o p e r a t o r o p ( $ ) on G. t h a t 0 P ( @ ) i s bounded on L ( D.

L. R a g o z i n ( s e e [ n o ] ) ,

9)is

Then d e f i n e s

Then t h e n e c e s s a r y c o n d i t i o n

t h a t o p ( g r ) i s u n i f o r m l y bounded when r +

m.

u s i n g i m b e d d i n g method i n t o t h e E u c l i d e a n space,

p r o v e d t h e Jackson Theorem, t h e B e r n s t e i n Theorem and o t h e r r e s u l t s on compact L i e groups and on compact homoyeneous spaces. As t o t h e harmonic a n a l y s i s on u n i t a r y groups and i t s e x t e n s i o n on c l a s s i c a l y r o u p s and compact L i e groups, t h e r e a r e many r e s u l t s such as : a v a r i e t y o f theorems o f Tauber t y p e , a v a r i e t y o f p r o b l e m s on how t o s t u d y t h e h a r m o n i c a n a l y s i s on c l a s s i c a l domains t h r o u y h t h e harmonic a n a l y s i s on c l a s s i c a l yroups, and many r e s u l t s on t h e a p p r o x i m a t i o n t h e o r y . omitted,

A l l these r e s u l t s are

f o r w h i c h t h e r e a d e r s a r e r e f e r r e d t o [l] - [ 6 ] and o t h e r a r t i c l e s .

REFERENCES

El1

Gony Sheny (Kuny Sun), Harmonic A n a l y s i s on C l a s s i c a l Gr-0ups ( i n Chinese S c i e n c e Press, B e i j i n g China, 1983. , Acta. Math. S i n i c a , 1 0 ( 1 Y 6 0 ) , 239-261 ( i n Chi nese c21 , i b i d 1 2 ( 1 9 6 2 ) , 17-31 ( i n C h i n e s e ) , C31 , i b i d 1 3 ( 1 9 6 3 ) , 152-161 ( i n C h i n e s e ) . C41 , i b i d 1 3 ( 1 9 6 3 ) , 323-331 ( i n C h i n e s e ) . [51 , i b i d 15(1Y65), 305-325 ( i n C h i n e s e ) . C61 1 7 1 2 h o n g J i a q i n g , J o u r n a l o f Chinese U n i v e r s i t y o f S c i e n c e and ’echnol ogy , 9(197Y), 31-43. [ 8 ] Gony Sheny, J o u r n a l o f Chinese U n i v e r s i t y o f S c i e n c e and Technology, 9 ( 1 9 7 9 ) , 25-30. [9] S t e i n , E. M., Annals i n Math. Study, P r i n c e t o n , 1970, No. 63. [ l o ] Coifman, R . & Weiss. G., B u l l . Amer. Math. SOC. 8 0 ( 1 9 7 4 ) , 124-126. [ll] C l e r c , J. L., Ann. I n s t . F o u r i e r . Grenoble, 2 4 ( 1 9 7 4 ) , 1:14Y-172. [12] S t a n t o n , R. J., Trans. Amer. Math. SOC. 218(1976), 61-81.


[13] [14] [l5] [16]

[I71 [18]

[lY] [20] c211

c221 [23]

S t a n t o n , R. J . & Tomas, P. A., Amer. J. Math. 1 0 0 ( 1 9 7 8 ) , 477-493. S t r i c h a r t z , R. S., T r a n s . Amer. l l a t h . SOC. 1 9 3 ( 1 9 7 4 ) , 99-110. Weiss, N. J . , Amer. J. Math. 9 4 ( 1 9 7 2 ) , 1U3-118. D r e s e l e r , R., M a n u s c r i p t a Math. 3 1 ( 1 Y 8 0 ) , 17-23. , F o u r i e r A n a l y s i s and A p p r o x i m a t i o n Theory, Ed. G. A l e x i t s and P. Turan, V o l . I ( 1 9 7 6 ) , 327-342. Mayer, R. A., Duke Math. J . 3 4 ( 1 Y 6 7 ) , 549-554. T a l o r , M. E., Amer. Math. SOC. 1 Y ( 1 9 6 8 ) , 1103-1105. K a y o z i n , D. L., Trans. Amer. Math. SOC. 1 5 0 ( 1 9 7 0 ) , 41-53. , I l a t h . Ann. 1 9 5 ( 1 9 7 2 ) , 87-94. , i b i d , 2 1 9 ( 1 9 7 6 ) , 1-11. Zheny Xue An, Advances i n Math., V o l . 1 3 , 2 ( 1 9 8 4 ) , 103-118.

113

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115

INTERPOLATION OF OPERATORS IN LEBESGUE SPACES WITH MIXED NORM AND ITS APPLICATIONS TO FOURIER ANALYSlS Satoru IGARI Mathematical Institute Japan

T6hoku University

Sendai,980

INTRODUCTION This note was prepared for the lectures of a workshop and a conference at the National University of Singapore. Our objective is to discuss an interpolation method of linear operators on functions in a product measure space and apply i t to some problems arising in Fourier analysis on Euclidean space. For a function f on the d-dimensional Euclidean space Rd and & 2 0 let s8(f) be the Riesz-Bochner mean of order & , which s defined by the Fourier transform

for

le

< 1

and

= 0

(1-1412)~1(5)

=

s"(f)^(e)

otherwise, where = 1 JRdf(x) e-iexdx.

d = 1, sE is a bounded operator of Lp(R) to Lp(R) for & 2 0 and p > 1 by M.Riesz's theorem. Let d 2 2 . Then s o is bounded on LP((Rd) if and only if p = 2 ( FeffermanC81) and sE ( & > 0 1 is unbounded on LP(Rd) unless 2d/(d+1+2&) 4 2 0, then sE is bounded on L (R 1 ( Carleson and SjolinC33). In Chapter I 1 1 we shall give the following estimate of sE(f), & > 0, applying Parseval relation and an L4 (R 2 )-argument due to Carleson-SjolinC31 and CordobaC61; If

THEOREM.

for all

If

f

d > 2

in

and

Co(Rd),

&

> 0, then

where

5

= ( x o ,..., x ~ - ~ and )

=

(Xd-2'xd-1)' Since the operator sE is rotation invariant, we can choose any pair (xi.x ) , 0 i i,j < d, as a coodinate system of Rd in 1 the inequality ( 1 ) .

S. Igari

116

Now we observe that there are d-2 variables in the inner integrals in (1) and 2 variables in the outer integrals. Thus i t may be speculated that the average u of the exponents of f and sE would satisfy l/u = C(d-2)/2 + 2/41/d = (d-l)/2d, that is, u 2d/(d-1), which is just the Herz-Pollard bound. This is motive of our interpolation method. For f in the xM and s , t > 0 define product measure space Mm+n = MxMx

[SMn(s

...

.

.

I f I 'dxo. .dxm- ) t/sdxm. Mm dxm+n- 1 Illt, where p denotes the coordinate system of Mm In Chapter I 1 we shall consider the linear operator T which S bounded on the spaces with mixed norm. Under this condition we get some information on the boundedness of T in LU(Mm+n) Applying our interpolation theowhere l / u = (m/s + n/t)/(m+n). rem we get

the mixed norm Ilfll(t,s:p) =

Ils"(f)

'2d/ (d+l )

for f in L2d/(d+1)(Rd) fd-l(xd-l), where E > 0.

*

uf112d/(d+l) of product form f(x) = fo(xo)

...

In 55 2 and 3 in chapter I 1 1 we show a restriction theorem of Fourier transform and an estimate of K a k e y a , ~maximal operator for functions of product form applying our interpolation theorem. Plan Chapter I Preliminaries Poisson intfgral and the space H 2 1.1 1.2 The space N 1.3 Riesz-Thorin interpolation theorem Chapter I 1 Interpolation of operators in Lebesgue spaces with mixed norm 2.1 Lebesgue spaces with mixed norm 2.2 Auxiliary functionals ; the case m 2 n 2.3 Auxiliary functionals : the case m < n Interpolation theorems 2.4 Chapter 1 1 1 Applications of interpolation theorem to problems arising in Fourier analysis Estimates of Riesz-Bochner means 3.1 Restriction problem of Fourier transform 3.2 to the unit sphere Kakeya's maximal function 3.3 PRELIMINARIES CHAPTER I 1.1. Poisson integral and the space H2 For l z l < 1 the Poisson kernel P(z,B), -71 i I3 < 7 1 , is defined by 1 -r 2 Ptz,B) = Re e ' 9 , 2 , z = r ei t e i B - z - 1-2rcos(t-~)+r

Interpolation of Operators in Lebesgue Spaces

Q(z,0)

and the conjugate Poisson kernel

I I7

by

-

For an integrable function u the Poisson integral the conjugate Poisson integral u are defined by

u(z)

and

and

respectively

Suppose that

u

is real valued. I f + iiicz),

f(z) = u(z) then f(z) 3.n f(0) = 0.

holomorphic in

s

121

< 1

(1.1)

Re f(z) = u(z)

and

DEFINITION. Let p > 0. A function f(z) holomorphic in is said to belong to the space Hp if

A

function f sup Iftz)l

f E C:(Rd)

0, then

and

p

E

P.

Theorem 3 implies that llsE ( f 1I (4.2 :p)

By duality

'

'

'

(4/3,2 :p) Applying Theorem 2(ii) we get SE (f

'

'fu(4,2:p)* 'If'(4/3,2:p)'

'

f

for

ws"(f) '2d/ (d+ 1 1'f112d/(d+l) of the product form fo(xo)fl(xl)...fd-l(xd-l).

By an interpolation theorem for multilinear operators

113

)

THEOREM 4.

Let

E

> 0 and & '

2d/(d+I))

I s (f)llU i

for

f

of product form.

c

Ilfllu

< u

i 2. Then

(

see

S. lgari

126

3.2.

Restriction problem of Fourier transform to the unit sphere Suppose that

where da is an area element on Sd-' and f € CI(Rd). Put po(A) 2d/(d+1+2A). By Fefferman's remark ( see 1911, if po(X) 2d(d+l)/(d+3). In 1 1 1 1 we showed the following THEOREM 5 .

If

d P 2 , 1 i u i 2d/(d+l)

and

f E CZ(Rd),

then

(3.3) be the family of d-1 indices in ( 0,l. d-1 ) and I p , p E P, be disjoint arcs in aD of length 2x/d. Let liP(z) be functions in H 2 such that !Re bp(eie) = 1 a.e. in IP and = 0 otherwise, and 3m Ip(0) = 0. Then 6 ( 0 ) = l/d. Define P a mapping TZ by d- 1 bj(Z) TZf(4) = ? ( f ) n I f j l j=o Applying an analytic operator version of Theorem l(ii) with M R, N = Sd- 1 and uo 2, u l = 1, we get Let

...,

P

THEOREM 6. I f Droduct form

d

2

2, 1 i u i 2d/(d+1)

and

f in

CI(Rd)

is of

3.3. Kakeya's maximal function Let 91 be a family of on-empty bounded open sets in Rd. For a locally integrable function f on Rd the maximal operator MR related to I is defined by MIf(x)

=

sup - lRIfldx. x€R€!R

When I is the family of all open balls in Rd, Mlf is HardyLittlewood maximal function. For given N > 2 and a > 0 let !R

Interpolation of Operators in Lebesgue Spaces

127

be the family of all rectangles in Rd with size ax ...xaxaN, but with arbitrary direction. When d = 2, the operator M9 has arisen in the work of Fefferman C91 and Cordoba 151 to estimate Riesz-Bochner operator. In fact Cordoba C61 proved that when d = 2

#M9fl12 i C(l0g N)l12 Ilfll,. Recently, Christ, Duoandikoexea and Rubio de Francia 141 showed that if d 2 3 and 1 0 . In Igari C121 we have shown the following. For

-

-

..,xdml) denote THEOREM 7.

x = (x,,...,x~-~) and

There exists a constant

C

x = (xo,xl,..

% = (xd-2 SXd-l ) '

such tha I )2dG.

(3.5)

(3.5) implies that %fl+2,m:p)

i

c

1 o g 3 l 2 ~iifH(2,m:p),

where p = (d-2,d-1). Thus by Theorem 2(ii) we get THEOREM that

8.

Let

d 2 3 . Then there exists a constant

C

such

#M9flld i C log 12N Ilfll, for

f

in

Ld (Rd 1

of product form.

REFERENCES

C11 C21

131

C41

C51 161

J.Berg and J.Lofstrom, Interpo ation Spaces, An Introduction, Springer-Verlag, Berlin,Heidelberg/New York, 1976. A.P.Calderon and A.Zygmund, On the theorem of HausdorffYoung and its extensions, Ann.Math.Studies,25(1950),166-188. L.Carleson and P.Sjolin, Oscillatory integrals and multiplier Problem for the disk, Studia Math.,44(1972),287-299. M.Christ,J.Duoandikoetxea and J.L.Rubio de Francia, Maximal operators related to the Radon transform and the CalderonZygmund method of rotations, Duke Math.J.,53(1986),189-209. A.Cordoba, The Kakeya maximal function and the spherical summation multipliers, Amer.J.Math.,99(1977),1-22. A.Cordoba, The multiplier problem for the polygon, Annales

S. Igari

128

C71 C81

C93 ClOl 1113

of Math.,lo5(1977),581-588. P.Duren, Theory of Hp Spaces, Acad.Press,New York/London, 1970. C.Fefferman, The multiplier problem for the ball, Annales of Math.,94(1974),330-336. C.Fefferman, A note on spherical summation multipliers, Israel J.Math.,15(1973),44-52. C.Herz, On the mean inversion of Fourier and Hankel transforms, Proc.Nat.Acad.Sci.USA,40(1954),996-9. S.Igari, Interpolation of operators in Lebesgue spaces with mixed norm and its applications to Fourier analysis,Tohoku Math.J.,38(1986),469-490.

1121 C131

C143

S.Igari, On K a k e y a , ~maximal function, Proc.Japan Acad. R.Rochberg and G.Weiss, Analytic families of Banach spaces and some of their uses, Recent Progress in Fourier Analysis, ed.by I.Peral and J.L.Rubio de Francia, North-Holland 1985, 173-201. P.A.Tomas, A restriction theorem for the Fourier transform, Bull.Amer.Math.Soc.,81~1976),477-478.

Proceedings of the Analysis Conference, Singapore 1986 S.T.L. Choy, J.P. Jesudason, P.Y. Lee (Editors) 0 Elsevier Science Publishers B.V. (North.Holland), 1988

129

SHIFT INVARIANT MAFXOV MEASURESAND THE ENTROPY MAP OF THE SHIFT*

CHOO-WHAN K I M

Department o f Mathematics and S t a t i s t i c s Simon F r a s e r U n i v e r s i t y Burnaby, B . C . CANADA V5A 1S6

.

m

Let R = xo{O,l, . . . ,s-11, s ? 2 , and l e t T b e t h e s h i f t on R Let P(R,T) b e t h e compact convex s e t o f a l l s h i f t i n v a r i a n t normalized B o r e l measures on R , and l e t M(R,T) be t h e set o f a l l Markov measures i n P(R,T). Let Ml(R,T) b e t h e s e t o f a l l Markov measures i n M(R,T) t h a t a r e i n d u c e d by i r r e d u c i b l e s t o c h a s t i c m a t r i c e s , and l e t M2(R,T) b e t h e s e t o f a l l Markov measures i n M(R,T) t h a t a r e induced by r e d u c i b l e r e c u r r e n t s t o c h a s t i c m a t r i c e s . We show t h a t b o t h M(R,T) and Mz(R,T) a r e compact, connected nonconvex s u b s e t s o f P(R,T), and M1(R,T) i s an open, c o n n e c t e d , s t r o n g l y nonconvex dense s u b s e t of M(R,T) such t h a t M(R,T) = M,(Q,T) U Mz(R,T). We a l s o show t h a t t h e e n t r o p y map o f t h e s h i f t i s an a f f i n e upper semicontinuous f u n c t i o n from P(Q,T) o n t o [ O , l o g s ] and is a c o n t i n u o u s f u n c t i o n on M(R,T) which maps Ml(R,T) o n t o LO, l o g s l . 1.

INTRODUOION Throughout t h i s p a p e r ,

that

s 2 2.

s

w i l l denote a f i x e d b u t a r b i t r a r y i n t e g e r such

S = { O , l , . . . ,s-11

Let

be endowed w i t h t h e d i s c r e t e t o p o l o g y ,

l e t R = xmS b e endowed w i t h t h e p r o d u c t t o p o l o g y , and l e t B be t h e 0 5 - a l g e b r a o f B o r e l s e t s i n R . Note t h a t R i s a compact m e t r i z a b l e s p a c e . Each element

R

x :

let

-t

w

c R

i s a sequence

(wn)n20 where

be t h e c o n t i n u o u s s u r j e c t i o n d e f i n e d by

S

transformation

R

T:

+

R

d e f i n e d by

(To),

=

on+l

t i n u o u s s u r j e c t i o n and i s c a l l e d t h e s h i f t on

Let

P(R)

c

P(Q), l e t

subset of Let

rI+

=

Ti.l 6 P ( R )

=

o

n Z 0

.

be such t h a t Tp =

(Tp)(E) = u(T-'E)

~ 1 Note . that

0,

The i s a con-

(R,B)

endowed w i t h

i s a compact convex m e t r i z a b l e s p a c e .

P(R)

Z

.

P(R,T)

f o r each

For each

8

E

.

i s a compact convex

be t h e s e t of a l l

s x s

s t o c h a s t i c matrices

ll be t h e s e t o f a l l p r o b a b i l i t y v e c t o r s p = ( p i ) i is called positive i f

{p

matrix

xn(w)

f o r each

n

P(R).

M(s x s)

and l e t

p C II

Then

P(R,T) = {p c P(R):

Define

R

For e a c h

d e n o t e t h e s e t o f a l l p r o b a b i l i t y measures on

t h e weak* t o p o l o g y . p

wn c S .

< II:

p > 0).

P 6 M(s x s)

pi > 0

for all

i

c

S , denoted by

P = ( p ij . .) i j. C S A vector

p > 0

.

Let

i s c a l l e d a s t a t i o n a r y d i s t r i b u t i o n of a s- 1 pP = p , i . e , Z p . p . . = p . f o r each j < S. For i = o 1 11 1

A vector

if

c II

s.

p

*Research s u p p o r t e d by NSERC Canada.

C.-W. Kim

130

each of

c

P

M(s x s ) , l e t

, i . e . , II(P)

P

TI(P) # @ P

c M(s

and

{p 6 II: pP = p}.

=

is irreducible iff

P

f o r each

n

?

=

For any

p

and

P

p

u is PP Markov measure.

. The measure

S

o r the

(p,P)

denote t h e s e t o f a l l Markov measures, i . e . ,

M(R)

c TI,

M(R) = {upP: p

P

c

M(s

M(R,T) = M(R)

Then we have measures i n

M(R,T)

M2(R,T)

and l e t

X

s)),

n

P(R,T).

s h i f t i n v a r i a n t Markov measures. let

.

{p} f o r some p C I'

i o , . . . , i ni n

c a l l e d t h e Markov measure induced by Let

=

denoted by

P(R),

and each sequence

0

n(P)

such t h a t PP ' i k , 0 5 k 5 n) = pi p . . , . . . , p i i 0 loll n-1 n

theorem, a unique measure i n ppp(xk

P c M(s x s ) ,

Note t h a t , f o r each

p c TI, t h e r e e x i s t s , by t h e Kolmogorov e x i s t e n c e

and any

s)

X

denote t h e s e t o f a l l s t a t i o n a r y d i s t r i b u t i o n s

I[(P)

M(R,T)

Elements o f

Let

=

M(R): pP = p } . {p PP M(R,T) a r e c a l l e d t h e

be t h e s e t o f a l l Markov

M1(R,T)

t h a t a r e induced by i r r e d u c i b l e s t o c h a s t i c m a t r i c e s , and

be t h e s e t o f a l l Markov measures i n

M(R,T)

t h a t a r e induced

by r e d u c i b l e r e c u r r e n t s t o c h a s t i c m a t r i c e s . For each

p

=

P

(pi) C TI, we d e f i n e t h e s t o c h a s t i c m a t r i x

= (p. J .

ij

by

p . . = p . f o r a l l i , j C S , t h e n pP = p . I n t h i s case t h e 11 1 Markov measure i s c a l l e d t h e p B e r n o u l l i measure, denoted by

B e r n o u l l i measure n

up

, we have p ( x

and each sequence

? 0

a l l B e r n o u l l i measures, and l e t

B(R,T)

B(R,T)+

s).

M(s

x

s)

M(s x s )

a l l irreducible matrices i n x

Pi

-

S

.

Let

=

{p

c

B(R,T)

B(R,T):

0

.

,

C~ S

(p,P) P ,pi

.

For each f o r each

3 . ' .

denote t h e s e t o f

p > 01.

Note t h a t

M(R,T).

C

I n S e c t i o n 2 , we show t h a t M(s

i k , 0 5 k 5 n) -

=

P k i o , . .. , i n i n

i

i s a compact convex s e t and t h e s e t of i s an open, convex dense s u b s e t o f

On t h e o t h e r hand, t h e s e t o f a l l r e d u c i b l e m a t r i c e s i n

i s a compact connected s u b s e t of

M(s x s ) .

M(s x s )

The n o t i o n o f s t r o n g l y nonconvex

s e t i s i n t r o d u c e d i n D e f i n i t i o n 2.15.

In S e c t i o n 3 , we show t h a t convex s e t , and s e t of

B(R,T)+

B(R,T)

i s a compact, connected s t r i c t l y non-

i s an open, connected, s t r o n g l y nonconvex dense sub-

B(R,T).

The main r e s u l t s o f t h i s p a p e r a r e s t a t e d i n S e c t i o n 3 . M(R,T)

and

M2(R,T)

a r e compact, connected nonconvex s e t s and

open, connected, s t r o n g l y nonconvex dense s u b s e t o f M(n,T)

=

We show t h a t both

M(R,T).

Ml(R,T) i s an

We a l s o have

M1(R,T) U M2(R.T).

I n S e c t i o n 5 , t h e r e s u l t s o f S e c t i o n s 3 and 4 a r e used t o show t h a t t h e e n t r o p y map o f t h e s h i f t i s a continuous f u n c t i o n on M1(R,T)

and

B(R,T)

onto

M(R,T)

which maps both

[ O , log s].

For background i n f o r m a t i o n on Markov measures and on Markov c h a i n s w e r e f e r t h e r e a d e r t o B i l l i n g s l e y [ l ] , Chung [ 2 ] , Denker e t a1 131, F e l l e r [ 4 ] , and Walters [ 7 1 .

Markov Measures

131

PRELIMINARIES

2.

Given

o f t h e s t o c h a s t i c p r o cess { x 1 n n 1 0 (R,B,p) i s d e f i n e d by

c P(.Q), t h e s t a t e space E

p

d e f i n e d on t h e p r o b a b i l i t y space E = { i c S : v(xn = i) > 0

s

of

f o r some

n Z 01.

Clearly,

i s a nonempty s u b s e t

E

.

Proposition 2.1. (ii)

Let

C P(R).

The f o l lo w i n g a s s e r t i o n s a r e e q u i v a l e n t : P c M(s x s )

i s t h e Markov measure induced by

( i ) 1~

The p r o c e s s

s t a t e s p ace

(pi)i

Note t h a t

i C S - E, so that

(i) = ( i i )

p . = p(xo = i ) Zi

pi = 1

p(xn = i ) > 0

such t h a t

.

so that

f o ll ow s from Theorem 1 o f Chung for all

c

and

E

.

i c E

Let

p i j = 1 f o r each

Cj

i

pi = 0

i

c

E

for all

Then t h e r e e x i s t s an

p . . = p ( ~ =~ j Ixn + ~= i ) 11

I t f o l l o ws t h a t

and t h e

E

E .

Proof. The i m p l i c a t i o n

[2, p . 7 1 .

.

.).

11 1 , J

i n i t i a l distribution

c II.

p

(R,B,p) i s a Markov c h a i n w i t h t h e

d e f i n e d on

Cxn'n t 0

, t h e s t a t i o n a r y t r a n s i t i o n m a t r i x (P.

E

and

for all

, so t h a t

(p. .). .

11 1,1

n t 0

c

j

S.

is a

c E

stochastic matrix. (ii) matrix each

=)

(i):

Suppose

P ' = (PI . ) . .

(i,j) c E

11 1 > 1

x

5

by

s

13

Let

11

f o r each

11

13

i 6 S

.

Define t h e s t o c h a s t i c

( i , j ) c ExE, p!.

for

0

=

11

(i,j) c F x S

f o r each

p = (pi)i

f o r each

F = S - E.

Let

p!. = p . .

F, p . . = 6 . .

Kronecker's d e l t a . pi = p(xo = i )

( i i ) holds.

where

6..

denotes

13

be t h e p r o b a b i l i t y v e c t o r d e f i n e d by Then

p

is t h e (p,P')

Markov measure.

We s t a t e wi t h o u t p r o o f s t h e f o ll o w in g t h r e e p r o p o s i t i o n s .

Proposition 2.2.

Let

p C P(R).

Then t h e f o l l o w i n g a s s e r t i o n s a r e

equivalent : (i) (ii)

u c P(R,T). The p r o c e s s

Proposition 2 . 3 .

{xnjn

d e f i n e d on

~

6 P(R,T).

Let

is stationary.

(.Q,B,p)

Then t h e f o l l o w i n g a s s e r t i o n s are

equivalent: (i) (ii)

p

is the

The p r o c e s s

(p,P)

Markov measure where

Cxnln

d e f i n e d on

c h ai n with t h e s t a t e space

E

Let

u

.

is a s t a t i o n a r y Markov

, t h e s t a t i o n a r y t r a n s i t i o n m a t r i x ( p 11 . .). . 1,J

and t h e s t a t i o n a r y i n i t i a l d i s t r i b u t i o n Proposition 2.4.

pP = p

(R,B,u)

C M(R,T).

( p i) i

where

E

C

E

S.

Then t h e f o l l o w i n g a s s e r t i o n s a r e

equivalent: (i) (ii)

u

C B(n,T). The p r o c e s s

txnln

d e f i n e d on

(R,B,u) i s

a sequence o f

independent, i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s w i t h f i n i t e mean.

Let

R and

C(n) let

denote t h e u s u a l Banach s p a c e o f a l l continuous r e a l f u n c t i o n s on lA denote t h e i n d i c a t o r f u n c t i o n o f a s e t

c o l l e c t i o n of a l l t h i n cy l in d er s e t s : where

n 2 0

and

io,.

. . , i nC

S

Z(io , . . . , i n )

, t o g e t h e r with

=

A

C

(5 =

n.

Let

s

be t h e

ik, 0 5 k 5 n)

t h e empty s e t

@

.

I t is

C-W. K i m

132

i s a c o u n t a b l e b a s e f o r t h e p r o d u c t topology o f

S

e a s i l y seen t h a t

i s a clopen s e t , i . e . ,

S

in

lAC C(R)

for all

.

s

A 6

.

8

i s a s e m i a l g e b r a which g e n e r a t e s t h e o - a l g e b r a

R and

Note a l s o t h a t each s e t

a s e t which i s b o t h c l o s e d and open, s o t h a t Using t h e S t o n e - W e i e r s t r a s s theorem, we show

r e a d i l y t h a t t h e family o f a l l l i n e a r combinations o f i n d i c a t o r f u n c t i o n s o f S

sets i n

i s dense i n

C(n).

By t h e p r e c e d i n g remark, we o b t a i n at once t h e f o l l o w i n g p r o p o s i t i o n s . P r o p o s i t i o n 2.5.

Let

p,pn C P ( R )

where

n = 1,2,.

..

.

Then t h e

following assertions are equivalent :

un

(i) (ii)

.

1-1

+

lirn pn(A) = p(A)

n P r o p o s i t i o n 2.6.

f o r each

Let

pn

s .

A E

be t h e

(pn,Pn)

Markov measures where

P, = ( p i ( n ) I i s , pn = ( ~ ~ ~ ( n ) ) ,~n ,= ~1 , 2 ,... Markov measure where p = ( p i ) i , P = (pij)i,j

.

Let

be the (p,P)

p

.

l i m pi(n) = pi each i E S , and lirn p . . ( n ) = p . . f o r each n n 11 1J un + 1-1 . ( i i ) I f 1-1, * u , t h e n lirn p . ( n ) = p . f o r each i t S , and n 1 l i m p. . ( n ) = p . f o r e a c h i C S w i t h pi > 0 and each j C S . n 11 ij P r e p o s t i o n 2 . 7 . Let p be t h e p B e r n o u l l i measure, and l e t un be t h e

(i)

If

i . j C S, then

pn

B e r n o u l l i measures where

f o r each

i C S

n

For b r e v i t y , we s h a l l denote P = ( p . .) 6 M(s x s ) .

.

i,j

11

Define

P

0

.

of

is called a state.

13

pn * p

iff

(pij).

Let

pij

by

P

Pn t M(s x s )

Note t h a t

I

13

(6. .).

for a l l

n

13

?

An element

.

C p(nl = m n = l 11 i s c a l l e d a b s o r b i n g i f p . . = 1.

0

P

n - t h power of t h e m a t r i x

p!1)

S

Then

(pij)i,j

The m a t r i x

where

=

... .

= I , t h e unit matrix, i . e . ,

n , the

integer

1,2,

=

,

denotes t h e s e t of a l l m a t r i c e s i n M(s x s ) h a v i n g M (s x s ) denotes t h e s e t o f a l l r e d u c i b l e m a t r i c e s i n

s).

W e s h a l l always assume t h a t

topology of

[0,1ln

with

M(s n = s2 .

x

s ) i s endowed w i t h t h e r e l a t i v e p r o d u c t Then we o b t a i n t h e f o l l o w i n g p r o p o s i t i o n ,

Markov Measures

133

t h e e a s y p r o o f o f which we l e a v e t o t h e r e a d e r . Theorem 2 . 9 .

M(s

Theorem 2.10.

Proof. P

c

M1(s

x

s)

Pn

+

P

i s a compact connected s u b s e t o f Pn C Mr(s x s )

where

An = I

+

BS-'

there exists a positive integer

n'

a contradiction.

set of

M(s

A:-'

To prove t h e connectedness o f C A

E

,

E

-.

We s e e r e a d i l y t h a t +

such t h a t x s)

Since

,n

A:-'

Mr(s

and

gS-l ?

s)

x

11

.

where E ' = S - E

of

c

Mr(s

.

S

M(S

s), l e t

x

s

Assume

n' , are positive,

is a compact sub-

be t h e c o l l e c t i o n o f a l l

Mr(s

2

unit matrix, s o t h a t it i s

2

X

i s n o t convex as t h e

P,Q C M 2 ( s

s)

X

by

, ~ , . . . , s - z } , ps-l,s-l

c

i.j

1,

=

{ l , ,..., ~ s-11.

c C ( O , l ) , cP + ( I - c ) Q

M2(s x s )

by

p. .In) = 6 . . f o r 0 11

n = 1,2,

5 i 5

c

M1(s

X

s).

Then

On t h e o t h e r hand, is

M2(s x s )

is n o t c l o s e d .

M2(s

Define Pn

x

s)

1 s-3, 0 5 j 5 s-1, p s - 2 , s - 2 ( n ) = I - = ,

= l / n + l , P,-l,s-2(")

... .

EAI

x

X

M2(s x s)

Then

3.

?

We a l s o n o t e t h a t

Ps-2,s-l(")

{E : 1 5 n 5 Z S - s } .

f o r each ( i , j ) c E

we show, by a m o d i f i c a t i o n o f t h e p r o o f o f Theorem 2 . 1 0 ,

13

=

s , En) is convex and 2s-2 Consequently, M ( s x s ) = U M (s x s , En) i s n=l

f o l l o w i n g example shows. Define 1 p.. = -for all i , j c C O 11 s-1 1 for all qoo = 1, 913 .. = s-1

W e obtain, f o r each

o

P.. =

x s):

M2(2 x 2 ) c o n s i s t s o f t h e

Remark 2 . 1 1 . compact convex.

h

We may assume t h a t A

I t i s p l a i n t h a t each

connected.

where

1

11

contains t h e unit matrix I .

connected.

{A

n n 2 1 is positive,

let

x s , E ~ =) { ( p . .)

M,(S

n

If

x s).

is a positive

s).

x

p r o p e r nonempty s u b s e t s

For each

as

P C Mr(s

Therefore,

.

B = I + P

and

Pn

+

r e d u c i b l e and

M(s x s ) .

P c M(s

and

s ) , t h e n , by Gantmacher 15, p . 511, ( I + P)'-l

Let

are also

i s a compact convex s e t .

M (s

Suppose

x

matrix.

s)

x

=

PS-1,S-1 (n) = 1 / 2

converges t o t h e m a t r i x

Pn

P c M3(s

x

s)

d e f i n e d by p..

6.. f o r 0

=

13

11

5 i 5 s-2,

S i m i l a r l y , we prove t h a t prove

M3(s

pij(n)

=

x

0 5 j 5 s-1,

M3(s

x

s ) is not closed, l e t

6 . . for 0 5 i

5

s)

ps-1,s-2

=

_ -1 ps-1,s-1 - 2

.

i s connected and i s n o t convex.

Pn

M3(s

X

s)

To

be such t h a t

1 1 s-Z,O 5 j 5 ~ - 1 , p ~ - ~ , ~ -= ~n( ' nP s) - l , s - 2 ( n ) = 1 - - .

11

where

n

=

1,2,

i s n o t convex.

...

.

Then

Pn

+

I C Mz(s

X

s).

Note a l s o t h a t

Mr(s

X

s)

C-W.Kiwi

134

The s e t o f a l l p o s i t i v e s t o c h a s t i c m a t r i c e s i s dense i n

P r o p o s i t i o n 2.12. M(s x s ) .

In p a r t i c u l a r ,

Proof.

c S

ji

(i)

such t h a t 1

Pik(n) where

i

+

13

.

c

1.1

.

‘ikjk

P

1

S - {j.}

{PnIn

k

for all

is irreducible stochastic.

Ck

. . , by

.

Choose

i k , j , 6 Ck

r b e a p o s i t i v e i n t e g e r such t h a t

Let

such t h a t 1 > - for

p. . lk’k

Qn = ( 9 . . ( n ) ) . . 11 1 9 1

Define t h e m a t r i c e s (n)

p. .

=

ikjk

1 - -

I-

by

’

r+n

.

\

c M1(s

=q. . ‘mJ1

s)

x

= -

1 r+n

a

and

+

’

.

I’

By ( i ) , we o b t a i n a

of p o s i t i v e s t o c h a s t i c m a t r i c e s which converges t o

{Pnln

(iii)

=

are p o s i t i v e s t o c h a s t i c m a t r i c e s and con-

~

q. .(n) = p . . elsewhere 11 11 where n = 1 , 2 , . . . Then

matrix

5).

( p . .(TI)). . , n = 1 , 2 , . IJ 1.3

Define t h e m a t r i c e s

q . . ( n ) = q . . (n) = . . . = q i . l1J 2 ’2’3 m-13,

sequence

M(s x

~

> 0, 1 5 k 5 m pikjk all k

i s a dense s u b s e t o f

P = ( p . .) ( M2(s x s ) . Then t h e r e e x i s t s a (unique) ‘1 , 2 5 m 5 s , o f S such t h a t each m a t r i x

{Ckll

= (p. .).

M1(s x s )

P = ( p . .) C M1(s X s ) . For each i c S , t h e r e i s 11 > 0 . Let r be a p o s i t i v e i n t e g e r such t h a t

(s-1) ( r + n )

Then

Suppose

partition

k

. .

P

.

i

Pik

=

S

(

verges t o (ii)

p.. IJi

for a ll

piji >

P

Suppose

P

.

P = ( p . . ) . . c M3(s x s ) . Using t h e c a n o n i c a l form o f t h e 1 3 1.1 ( s e e S e n e t a [ 6 , p . 1 5 ] ) , t o g e t h e r with a m o d i f i c a t i o n o f t h e

Suppose P

i n M1(s x s ) Qn’n 2 1 Using ( i ) a g a i n , we complete t h e p r o o f .

argument i n ( i i ) , we o b t a i n a sequence { verges t o

P

.

M1(s x s ) i s a convex s u b s e t o f

I t is plain

M(s

which con-

s ) , s o t h a t , by

X

theorem 2.10 and P r o p o s i t i o n 2 . 1 2 , we o b t a i n t h e f o l l o w i n g p r o p o s i t i o n . Theorem 2 . 1 3 .

M1(s x s )

D e f i n i t i o n 2.14.

F o r any

denote t h e p o i n t all at

m c {O,l,.

{a}, i . e . , EJA)

i o , . . . , i n - lC S

R

~

. .,n-l}.

i s an open, convex, dense s u b s e t o f

such t h a t w

F o r each p o i n t

= lA(w)

c R ,

A c R

for all

M(s

X

s).

n 2 1, l e t [ i o , . . , i 1 n- 1 %n+m = im f o r a l l k 2 0 and where let

E

.

w

denote t h e u n i t mass

For convenience, we i n t r o d u c e t h e f o l l o w i n g c o n c e p t . D e f i n i t i o n 2.15.

A subset

cp + ( 1 - c ) v C P(R) - E

for all

E

i

P(R)

1.1,v

c E

i s c a l l e d s t r o n g l y nonconvex i f with

We s h a l l always assume t h a t e v e r y s u b s e t o f

1.1 # v P(Q)

and a l l

c

C (0,l).

discussed i n the r e s t o f

t h i s p a p e r i s endowed w i t h t h e r e l a t i v e weak* t o p o l o g y o f P ( Q ) .

F o r concepts

and n o t a t i o n n o t e x p l a i n e d i n t h i s p a p e r , we r e f e r t o t h e s t a n d a r d works: s e e f o r example B i l l i n g s l e y [11, Denker e t a1 [31, and W a l t e r s [71.

135

Markov Measures 3.

BERNOULLI IEASUES We b e g i n by p r o v i n g t h e f o l l o w i n g b a s i c lemmas. Lemma 3 . 1 .

and

'The mapping

p

+

p

ll o n t o

from

P

B(R,T]

i s a homeomorphism,

i s a compact connected s e t .

B(R,T)

P r o o f . I t i s p l a i n t h a t Il i s a compact convex s u b s e t o f t h e s p a c e __ [ 0 , l I s , s o t h a t it i s a compact connected s e t . On t h e o t h e r hand, we s e e r e a d i l y t h a t t h e mapping

p

up

-f

and

B(R,T)

ii

u

(i]

If

(ii)

p

=

(pi)i

and

A(p)

q

(qi)i

=

A(p)

n

A(q) = @

n

A(q) #

be p r o b a b i l i t y v e c t o r s

{ i 6 S: p . > 0 1 , A(q)

=

be t h e p - B e r n o u l l i measure and t h e

c

c

311

Let

p # q , and l e t

and

T h e r e f o r e , t h e mapping i s a homeomorphism,

i s a compact connected s e t .

Lemma 3 . 2 . such t h a t

Il and B(R,T] and

i s a b i j e c t i o n between

i s c o n t i n u o u s by P r o p o s i t i o n 2 . 7 .

, t h e n cp

{i c S : q . > 0 ) . Let

=

.

q - B e r n o u l l i measure

- B(R,T)

+ (1-c]u C t l ( R , T )

for

(0,l).

If

A(p]

4 , then

cu + ( 1 - c ) v c P(R,T]

M(R,T) f o r a l l

-

c c (0,l). Proof,

Let

be a r b i t r a r y and

c E (0,l)

s o t h a t , by P r o p o s i t i o n 2 . 2 , { x n l n Let

A1 = A(p)

r

A2 = A(q].

$ ,

(r.).

=

Clearly both

.

B = S - A1

.

r . = cp. + ( 1 - c ) q i

that is, that

and

E = A1 U

Define

p = cp + ( 1 - c ) v .

A1

For e a c h

i

and

c S ,

We s e e r e a d i l y t h a t

c

P'(R,T],

(R,B,p).

a r e nonempty s e t s .

A2

d e f i n e r . = p(xO

r. > 0

is a positive probability vector.

l l t E

Then p

i s a s t a t i o n a r y p r o c e s s on

~

iff

i

c

=

i],

E, so

Define t h e s t o c h a s t i c

by r . = p(xl = j Ixo = i ) , t h a t i s , R = ( r . .]. . 11 1 , J E Ij r 11 . . = ( c p1. p1. + ( l - c ) q 1 . q1 . ) / r i . I t i s e a s i l y s e e n t h a t C. . . = r J. 1 c E r 1. r11 each j E E , t h a t i s , r R = r .

matrix

(i)

Assume

r..

=

n

0

11

Let

.)

n

A1

{i ",..., i n }

A1, =

r1( Xn+

1

=

{i,

,..., i n }

t h e n , f o r each

jlx,

,... , i 1

{i,

Then w e o b t a i n e a s i l y t h a t

A2 X E . p . f o r each ( i , j ) t A1 x E , r . . = q . f o r e a c h ( i , j ) 1 11 I and l e t i o , . .. , i nC S be such t h a t p(xk = i k , 0 5 k 5 n) > 0 ,

equivalently, e i t h e r

If

4 .

A2 =

= i k ,0 5

c A2,

j[xk

P ( X , + ~=

that

p

n Z 0

c

jlx,

p(xn+l

-

c

j

and any

E

p E M(R,T).

.

io,. . , i n , j

=

, =

=

i k , 0 5 k 5 n) = p,

=

C

A2

.

If

E ,

then, f o r each i k , 0 5 k 5 n)

=

{ i O , . .. , i n }

or

c A1 j

k 5 n)

By P r o p o s i t i o n 2 . 3 , we o b t a i n

f o r any

for

c

.

j / x n = i ) = q . = r. 1 In;

.

Since

# cp.

A1,

j j xn = in) = p . = r . . 1 Inl

1

= P ( X , + ~= j ]

i t f o l l o w s from P r o p o s i t i o n 2 . 4

M(R,T) - B(R,T).

To prove ( i i ) , we s h a l l c o n s i d e r t h e f o l l o w i n g two c a s e s . Case 1.

A1

n

A2 # @

p # q, there exist

i

c

and A2

,

B

n

A2 = @

j C A1

, or equivalently,

such t h a t

A2 c A1

i # j , pi # qi

.

Since

, p j # q 1.

.

C.-W. Kim

136

Then we have p(x2 = j I x o = i , x1 = i) = r . . , t h a t i s , p c M(R,T). 11 2 2 (cp / t ) P . + ( ( l - c ) q i / t i ) q = (cPi/ri)P. + ( ( l - c ) q i / r i l q j l i j j J 2 where t . = cp , so t h a t c p i / t i = c p . / r . . I t f o l l o w s pi = qi , + (l-c)qi 1 1 a c o n t r a d i c t i o n . Consequently, p c P(R,T) - M(R,T). Suppose

;

A1

n

# j

.

Case 2 . Clearly

i

equivalently, tradiction.

#

A2

and

@

If

c

p

n

B

.

A2 # @

M(R,T), t h e n

, j c B n A2

i c A1 fl A2

Choose

p(x2 = jlx,

= j , x1 = i ) = r . .

'

11

.

Or

q . = ( 1 - c ) q . q . / ( c p i + (1-c)qi) s o t h a t cpiqj = 0 , a conJ 1 1 Hence p c P(R,T) - M(R,T). Thus t h e p r o o f i s complete.

From Lemma 3.2 we o b t a i n P r o p o s i t i o n 3.3.

Both

B(R,T) and

B(R,T)+

are s t r i c t l y

nonconvex s e t s .

The n e x t theorem f o l l o w s immediately from Lemma 3 . 1 and P r o p o s i t i o n 3 . 3 . B(R,T)

Theorem 3 . 4 .

i s a compact, connected s t r o n g l y nonconvex s e t .

We s h a l l now prove t h e f o l l o w i n g r e s u l t .

Theorem 3 . 5 . subset of Proof, -

i s an open, connected, s t r i c t l y nonconvex dense

B(R,T)+

B(R,T).

TI+

It is easily seen that

3.3, i t remains t o show measure i n

B(R,T)+

.

B(R,T) - B(R,T)+

p r o p e r s u b s e t of

.

S

a positive integer

no

probability vector

p, 1

n

P i (n) = Pi 0 0

We have t h e n

pn

Remark 3.6. homeomorphism (l-c)pq

c II+ , n , pi(n)

+

Let

B(R,T).

t, r = s - t

=

B(R,T).

By P r o p o s i t i o n

uP

E = { i : pi > 0 1

pi > l / n o

such t h a t

for all

p i ( n ) = pi

i s dense i n

Then t h e s e t

card E

Let

TI , s o t h a t ,

i s a convex open s u b s e t o f

i s a connected open s u b s e t o f

by Lemma 3 . 1 , B(R,T)+

io c E

and

for a l l

i

i c S - E

,

c

E .

b e any i s a nonempty

.

There e x i s t s Define t h e

> no , by 1

=

for all

i t E - Ci

p , equivalently

0

p

1

.

P,

+

.

An immediate consequence o f Lemmas 3.1 and 3 . 2 i s t h a t t h e

p

+

3

up

from TI

Fcp + ( l 0 c ) q

onto for a l l

cup

+

4.

SHIFT INVARIANT MARKOV MEASURES

B(R,T)

is not affine, i . e . ,

p , q 6 TI

p # q

with

and a l l c

c

(0,l).

The p r i n c i p a l aims o f t h i s s e c t i o n i s t o prove analogues o f Theorems 2 . 9 , 2.10 and 2 . 1 3 f o r s h i f t i n v a r i a n t Markov measures.

We b e g i n w i t h t h e

following d e f i n i t i o n . Definition 4.1.

Let

M (R,T)

1

b e t h e s e t of a l l Markov measures i n

t h a t a r e induced by i r r e d u c i b l e matrices, i . e . , M1(R,T) = 111 PP S i m i l a r l y we d e f i n e

c

M(R,T):

P 6 M1(s x s ) , pP = p1.

M(R,T)

Markov Measures

Mn(R,T)

=

Mr(R,T)

=

Note t h a t

from

II

where

n = 2,3,

U M3(R,T).

M2(R,T)

Theorem 4 . 2 .

n

M(R,T): P C Mn(s x s ) , p C IL(P)}

M(R,T) = M1(R,T) U Mr(R,T).

_ P r o_ of.

where

c

PP

137

M(R)

i s a compact, connected nonconvex s u b s e t o f

Il x M(s

I t is clear t h a t

X

.

s + s2

=

X

s)

By p a r t ( i ) o f P r o p o s i t i o n 2 . 6 , t h e mapping

M(s x s )

M(R)

onto

P(R).

i s a compact convex s u b s e t o f [O,lIn (p,P)

is a continuous s u r j e c t i o n , so t h a t

-+

P(R) . By p a r t ( i i ) o f Lemma 3 . 2 , M(R)

i s a compact connected s u b s e t o f

PP

M(R)

is

n o t convex. Lemma 4 . 3 .

Proof. that

i s a compact nonconvex s u b s e t o f

M(R,T)

I t f o l l o w s from Theorem 4 . 2 , t o g e t h e r w i t h

M(R,T) i s compact.

Lemma 4 . 4 .

P r o o f . For each

c

P

P .

M1(s x s ) , l e t

M1(s x s )

p = f(P)

=

M(R) 3 P ( R , T ) ,

i s n o t convex.

By p a r t ( i i ) o f Lemma 3 . 2 , M(R,T)

M1(R,T) i s homeomorphic w i t h

d i s t r i b u t i o n of

M(R). M(R,T)

and i s connected.

b e t h e unique s t a t i o n a r y

By an e l e m e n t a r y argument, we show t h a t t h e mapping

II i s c o n t i n u o u s , s o t h a t , by p a r t ( i ) o f P r o p o s i t i o n 2 . 6 , t h e mapping F: M1(s x s ) * M(R,T) d e f i n e d by F(P) = p where p = f ( P ) i s a l s o PP f : M1(s

X

s)

+

To prove

continuous.

i s an i n j e c t i o n , suppose F(P) = F(Q) f o r some

F

P,Q C M1(s x s ) , t h a t i s ,

f o r some VpP = pqQ, p = f ( P ) , q = f(Q) Then we o b t a i n a t once p = q S i n c e b o t h p and

.

P , Q C M1(s x s ) .

p o s i t i v e , we a l s o g e t

Note t h a t

P = Q.

M1(R,T)

=

f o l l o w s from p a r t ( i i ) o f P r o p o s i t i o n 2.6 t h a t t h e mapping M1(s x s) and

morphism between

Ml(R,T).

M1(R,T) = F(M1(s x s ) ) i s connected. By a p a r t i t i o n o f

S

Since

F

Ml(s x s)

are It

i s a homeo-

i s a convex s e t ,

T h i s completes t h e p r o o f .

, we s h a l l always mean a f i n i t e c o l l e c t i o n

5

p a i r w i s e d i s j o i n t nonempty s u b s e t s

q

[ F ( P ) : P 6 M1(s x s ) } .

of

S

such t h a t

S =

2 n S s .

U Ck k=l

{Ckjc=l

of

where

Lemma 4.5. M2(R,T) i s connected.

Proof. {Ck}L=l

ci =

Assume

An

Let of

be t h e f i n i t e c o l l e c t i o n of a l l n - s e t p a r t i t i o n

S

where

2 5 n 5 s

a

Let

2 5 n 5 5-1.

=

.

{%jc=l

An .

be any element o f

Let M1(Ck)

(p. . ) . . . Let ’1 ‘1 ‘k Tk b e t h e s h i f t on R k , and

be t h e s e t o f a l l i r r e d u c i b l e s t o c h a s t i c m a t r i c e s

Rk let

=

xm

i=O

Si

where

Ml(Rk,Tk)

Si

for all

=

i , let

b e t h e s e t o f a l l s h i f t i n v a r i a n t Markov measures d e f i n e d on

+

Rk t h a t are induced by m a t r i c e s i n M1($). Each of M1(!+.Tk) i s o f t h e form p where Pk M1(Ck) and pk i s t h e unique s t a t i o n a r y distribution of measure i n

M (R , T ) l k k

P(R)

PkPk Pk . by

We s h a l l always e x t e n d e a c h measure uk(E)

i s connected.

= uk(E

Let

IIn

n

Rk) f o r each

E c R

.

pk

t o a unique

By Lemma 4 . 4 ,

be t h e convex s e t o f a l l n - d i m e n s i o n a l

C - W. Kim

138

Define t h e connected s e t c = ( c )" k k=l' ~x M ~ ( R ~ , xT ... ~ ) x M ~ ( R ~ , .T ~ )

probability vectors

x ~ = nn ,

Define t h e mapping

Fn: X

n,a

?(R,T)

+

'n,a

by

by

F n ( C > P 1 >+*. > P , , ) =

C spk * k=l I t i s s t r a i g h t f o r w a r d t o show t h a t t h e mapping

i s continuous, s o t h a t

Fn

Fn(Xn,a) i s connected.

As

I t i s evident t h a t

M2(R,T:

Let

denotes t h e p a r t i t i o n o f

S

i n t o one-point sets.

As)

denote t h e s e t of a l l Markov measures induced by t h e u n i t s-1 s-1 m a t r i x . Then M2(R,T:As) = C % E [ ~ ] : 0 5 ck 5 1, C c = 1 ) . C l e a r l y k=O k=O k M2(R,T: As) i s convex We now have

s- 1 1

(ua p, Fn(Xn,a)) M2(Q,T: As) . n= 1 n To prove t h e connectedness o f M2(R,T), i t i s enough t o show t h a t M2(R,T) =

i s n o t s e p a r a t e d from any of c1

= { C k l i = l 6 An

ci,

If

= {i},t h e n

p . . (m) = 1 11

{ i , ,...,i . 1 where I 1 for a l l p . .(m) = 1 - 11 m+ 1

$

p .1

1

(m) = p 1 . 1 . (m) =

1 2

2 3

p,

let

pm be t h e

urn 6

Fn(xn,a), m

Lemma 4 . 6 .

5

n

5 s-1, c1 C An

M2(R,T: . Let

As)

+

I

.

...

< i . , then 1

i C C

k '

1 . . (m) = m+ 1 . i,(m) = p 1.1 1-1 I 1 1 be t h e uniform p r o b a b i l i t y v e c t o r on

...

Let

.

il l / n o

such

S

Let

for a l l

'kJ k k .

Let

t l = 1 and l e t

tk

c

(0,1],

C-W. Kim

140 2 5 k 5 m

, that

w i l l be determined l a t e r .

Pn = ( p . . ( I I ) ) ~ , ~

irreducible s t o c h a s t i c matrix p . . (n) lk'k

=

p. . (n) 'mJ 1

=

p. . lkJk

t

(n)

' 'ikjktl

=

k

,

1 5 k 5 m-1

for

tm tm - , p . . (n) = p . . - lm'm lmJm

p. .(n) = p . . 11 11 Let pn =

elsewhere. b e t h e unique s t a t i o n a r y d i s t r i b u t i o n o f t h e m a t r i x P

We s h a l l show t h a t t h e d i s t r i b u t i o n s there is a probability vector Since

by

11

-

n

n > no, d e f i n e t h e

F o r each

pn

a r e independent o f

p = (pi)i

such t h a t

p = pn

n

.

.

That i s ,

for all

n > no

.

pnPn = pn , we o b t a i n

=

T h e r e f o r e , we g e t

c.

c1 p i ( n )

pi (n)

=

m

pi ( n ) t m

m

1

pi ( n ) t k = pi (n)tkcl k k+1

( p i ( n ) t m - pi

+

for

.

(n)):

.

1

S i m i l a r l y , we a l s o o b t a i n

1 5 k 5 m-1,

so that

pi (n) = pi ( n ) t k f o r 2 5 k 5 m . 1 k From t h e above e q u a t i o n s , t o g e t h e r w i t h t h e e q u a t i o n s pj(n)

=

Z. p.(n)p.. for all j 1 c ck 1 1J

Z p.(n) j=o

Pi(n) of

n

where

such t h a t

a i k P i k ( n ) = a i k P i 1( n ) / t k

=

- {ikl, k

Then we o b t a i n m

pk

=

ci

ck -

=

1,. . . ,m

.

Note t h a t each

aik

i s independent

m

{ikl

aik. Define t h e p r o b a b i l i t y v e c t o r p = ( p i ) i.

m Pil pi

=( =

,..., m

1,

a. > 0 ik

i

.

Ck - { j k } , k = 1 , 2

J

there exist

for all

=

f

((%

+

1 ) / t k ) F 1 , pi

(aik/t )p. l1

= pi / t k

k

k=l for

for

2 5 k 5 m ,

1

i f Ck - { i k } , 1 C k 5 m

,

c s

by

141

Markov Measures Then we o b t a i n Since and

pPn

pppn

p p

=

and

Pn

m p =

such t h a t cl/(A1+l)

-.

-f

Using

-f

, we o b t a i n

m

pP = P , p

I t remains t o choose

ck = Ci

, 1

pi

5 k 5

itk]*

m , we o b t a i n pi

=

1

'\+

1) ( A +1) , 1 5 k 5 m k 1 We may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t

(ii)

6 M1(R,T)

ppn

k

.

=

1 5k 5 q < m

and c

k

.

r = 2 , 3 ,...

where

n

n

and c1

tk

as

.

C ckrk k= 1

as

P

+

.

n > no

for all

%(R,T)

ppp

+

p

=

= 0

for

Then

pr

t h e r e i s a sequence i n

q + l 5 k 5 m.

y(n,T)

M1(R,T)

Put

and

pr

0 < ck

u = m-q.

+

which converges t o

r

as

p

vr.

for

Define

-t m

.

By ( i ) ,

T h i s completes

the proof. We o b t a i n a t once from Lemmas 4 . 3 and 4 . 4 , t o g e t h e r w i t h P r o p o s i t i o n 4 . 8 , t h e f o l l o w i n g analogue o f Theorem 2 . 9 . Theorem 4 . 9 .

i s a compact, connected nonconvex s e t .

M(R,T)

W e a r e now i n a p o s i t i o n t o prove t h e f o l l o w i n g analogue o f Theorem 2.13.

Theorem 4.10. s u b s e t of

i s an open, connected, s t r o n g l y nonconvex dense

M1(R,T)

M(R,T).

By Lemma 4 . 4 , Theorem 4 . 7 and P r o p o s i t i o n 4 . 8 , i t remains t o show t h e

f o 11owing propos i t i on. P r o p o s i t i o n 4.11.

Let

Markov measures and l e t

and

P

stationary distributions

p

v

Q

and

q

be t h e

be any matrices i n

, respectively. (q,Q)

M1(s x s)

Let

Markov measure.

with

b e t h e (p,P)

p

Then t h e

f o l l o w i n g a s s e r t i o n s are e q u i v a l e n t : (i)

P = Q

(ii)

1-1 = v

. .

(iii)

cp + (1-c)v

(iv)

cp + ( 1 - c ) v

c c

M(R,T)

for all

M(R,T)

f o r some

c

c c

(0,l).

c

(0,l).

The p r o o f o f t h i s p r o p o s i t i o n f o l l o w s from t h e f o l l o w i n g two lemmas. Lemma 4.12.

Let

P = ( p . .) 11

and

with s t a t i o n a r y d i s t r i b u t i o n s

v

i',j' (i)

c

S

M(R,T)

f o r some

be any m a t r i c e s i n

11

p = (pi)

be t h e (p,P) Markov measure and cp + ( 1 - c ) v

Q = (9. .)

the

and

q

=

(qi),

M1(s x s)

respectively.

(q,Q) Markov measure.

Let

If

pilj,

# qilj,

f o r some

qi qi ...qi ilpil 0 0 1 n-1

for all

n Z 1 and

c

(O,l),

and

, then

pi p . . . . . p i ilqil 0 loll n- 1 all

io,...

=

c s ,

p

C-W. Kim

142

(ii) (iii) (iv)

PiPii'qi,

qiqiiIPi1

=

pipijpjilqil

9 1. 911 . . q ~. . i l p fi o~r a l l

Let

p = cp + ( 1 - c ) v

Note t h a t b o t h

p

q

and

Suppose t h a t

c

p

c

0 < c

where

r = (r.).

S

.

Clearly p

,

1

e q u i v a l e n t l y , p c M1(R.T), By P r o p o s i t i o n 2 . 3 , we o b t a i n

M(R,T),

S.

R

c

P(R,T).

Define t h e

r . = p(xo = i )

by

R = ( r ..). . 11 1 > 1 s by I t is e a s i l y s e e n t h a t

and d e f i n e t h e s t o c h a s t i c m a t r i x r . . = (cp.p.. + (1-c)q.q. .)/r. . 1 1J 1 1J 1 1J rR=r.

i ' , j l

c

are positive probability vectors.

positive probability vector

f o r some

i,j

.

i' # j '

___ Proof.

=

i C S ,

for a l l

c

=

cp. + (1-c)q.

1'

M1(s x s) and

piljl # qi,j,

and

P ( X ~ =+ j~l I x o = i o , . . . , x n-1 = in - 1 ' xn = i ' ) = r i. ' j l , f o r any n Z 1 and any io, . . . , i c S such t h a t p(xo = i o , ... ,xnTl = i n-1' x n = i ' ) > O , n- 1 o r equivalently,

f o r any

n Z 1 and any

Pi i . . ' P i

i'

n-1

0 1

.

i o , .. , i

C S

n- 1

qi i .'.qi i 0 1 n- 1

+

.

> 0

l

such t h a t p i l j l # q i l j l , we o b t a i n

Since

CPiOPiOil.~ . P i n- 1i ' cp. p . . . . . p i i l 0 ' n-1 f o r any

pi

n

?

1 and any

i'

...pi

qi

+

n-1

0 1

f o r any

0 1

n 5 1 and any

p. . ...pi

i o , .. . ,i

n- 1

...q. . ioil...qi

.

io,. . ,i

c

S i l

S

n- 1

> 0 . i l n- 1 Both (ii) and ( i i i ) f o l l o w from ( i ) .

1011...qi

'0'1

n-1

that

pililpilqil

pilil = qilil Lemma 4 . 1 3 . in

M1(s

x

.

p..

ij

If =

> 0 , or equivalently,

such t h a t

Thus ( i ) h o l d s . On t h e o t h e r hand, we g e t from ( i i ) qil > 0 ,

Thus, ( i v ) h o l d s .

Let

P = ( p . .) 11

and

Q = ( 9 . .)

p

be the

Pjqi p.q. qij

for all

be any two d i s t i n c t m a t r i c e s

11

(p,P)

cp + ( 1 - c ) v t M(R,T)

1 1

(l-c)qil

such t h a t

with s t a t i o n a r y d i s t r i b u t i o n s

s)

+

n- 1

s o t h a t , s i n c e P . >~ 0 and

qililpilqil

Let

respectively. measure.

=

CP i CPiI

i l

n- 1

.

+ q.

i'

-

+ ( 1 - c ) q i. oq i. o l. l . . * q i

i,j

f o r some

c

p

=

(p.)

Markov measure and

S ,

and

v

c 6 (O,l), then

q

=

the

(qi), (q,Q) Markov

Markov Measures __ Proof. p

c

0 < c < 1 , and

p = cp + ( 1 - c ) v ,

Define

A

P # Q , we g e t

Since

If

Suppose

M1(R,T).

= {j

c

Q,

.

1 A1 #

143

# q.

S: p .

p

f o r some

c

equivalently,

M(R,T),

k C Sl

and B

= S - A1. Ik lk I t f o l l o w s from p a r t ( i i ) of Lemma 4 . 1 2 t h a t

A1 = S , t h e n we a r e done.

Suppose

c

(pij'i

A

1

# S , e q u i v a l e n t l y , B # Q, . S i n c e i s a nonzero m a t r i x .

B, j c

is irreducible, the matrix

P

Define

> 0 f o r some k C A l l . C l e a r l y , A2 # Q, and A1 n A2 = 4' . ik be any s t a t e i n A2 and l e t k C A1 be such t h a t p . > 0 . Again Ik u s i n g p a r t ( i i ) o f Lemma 4 . 1 2 , we g e t

A

2 Let

= {j C B: p.

j

= (1.4. 1 I ]. qj.k p k

pipijpjkqk

for all

i

c

S ,

so that

Therefore, we get

If

S = A1

U

A2

, t h e n we are done.

O t h e r w i s e , by r e p e a t i n g t h e above

p r o c e d u r e f i n i t e l y many times, we o b t a i n a p a r t i t i o n such t h a t t h e e q u a t i o n s h o l d on each o f t h e s e t s

S

{'%I1 5 k 5 n X

\, 1 C

k 5 n

s

of

.

This

completes t h e p r o o f . Proof o f P r o p o s i t i o n 4 . 1 1 .

Since the implications ( i ) (iv) = ( i ) .

h o l d t r i v i a l l y , i t remains t o show holds.

Suppose

for a l l

P

i,j C S

.

j=0

qi p.(-q. .) = p . j = 1 J q j 11

'J

for a l l

I f we d e f i n e t h e m a t r i x

i

c

Q' = ( q i j ) i , j

S

. by

4.

9 . . = 2 q.. for all i , j c s , 1J 9 i 11 t h e n Q ' i s an i r r e d u c i b l e s t o c h a s t i c m a t r i x . s-1

c

j=o

p.q!. J I'

=

pi

r ,

(iii)

=)

# Q. Then by Lemma 4.13, we o b t a i n p . . = p . q . 4 . . / p . q . s- 1 11 1111 1 1 S i n c e Z p . . = 1 f o r e a c h i 6 S , we o b t a i n

s- 1

I

(ii)

To t h i s e n d , assume ( i v )

,

s-1

c

j=o

q.q!. J 31

=

q.

We a l s o have

(iv)

C-W.Kim

144

c

By t h e uniqueness o f s t a t i o n a r y d i s t r i b u t i o n s o f Q' , we P.4 have p = q s o t h a t p . . = A 9 . . = 4 . . f o r a l l i , j C S , a c o n t r a d i c t i o n . 1J P i q j 11 1J Remark 4.14. We give an a p p l i c a t i o n of P r o p o s i t i o n 4.11. A sequence for a l l

i

S.

o f f u n c ti o n s {Pn'n 2 0 conditions i f

pn:

S

X:

+

i s s a i d t o s a t i s f y t h e co n si st en cy

[O,l]

s- 1

s- 1 (ii)

.

C ~ ~ + ~ (. ,ii n ~ , i, )=, p n ( i o , . i =O

i o , .. . , i nC S

Any p a i r

II

(p,P) 6

.. , i n )

for all

n i? 0

and a l l

.

x M(s

g i v e s r i s e t o such a sequence {pn 1n ? O

s)

X

. There a r e sequences {pnIn p (io,.. .,i ) = p. p. ...pi ' 0 'oil n - 1 in s a t i s f y i n g t h e c o n s i s te n c y c o n d i ti o n s t h a t do n o t a r i s e from any p a i r formula

(p,P) C II X M(s X s). and Let (p. . ) . . 13

6

1 3 1

hij)i,j

s

c s

pn(io,.

n

for all

Let

.. ,i )

c 6 (0,l) = cp.

and a l l

2 0

p. .

be a r b i t r a r y .

10 1011

.. .

i o , .. . ,inC S

(pi)i

0

.

and

(qiIi

I

Define

+ ( l - c ) q i qi

'in-lin

s a t i s f y t h e co n s is t e n c y c o n d i t i o n s .

~

be any d i s t i n c t i r r e d u c i b l e

s t o c h a s t i c m at r i c e s w it h s t a t i o n a r y d i s t r i b u t i o n s respectively.

by t h e

0 1

...

'in-lin

C lear l y t h e sequence

{pnIn

~

By P r o p o s i t i o n 4 .1 1 , t h e sequence

i s n o t g e n e r a t e d by any p a i r (p,P) c II x M(s x s ) . By t h e {pnIn Kolmogorov e x i s t e n c e theorem, t h e sequence {pnIn d e f i n e s a unique ~

measure

?

c P(Q,T) - M(R,T)

for all

n

?

5.

and a l l

0

THE ENTOPY MA!'

such t h a t T ( X =~ i o , . , . , x

i o , ,. . , i

S

= i n ) = p n ( i o ,..., i n )

.

OF THE SHIFT

The o b j e c t of t h i s s e c t i o n i s t o show t h e en t r o p y map o f t h e s h i f t i s a s u r j e c t i o n from

B(R,T)

o n to

[0, log s ] .

W e s t a r t with b a s i c p r o p e r t i e s of

t h e en t r o p y map.

6

In the sequel,

denotes t h e p a r t i t i o n of R d ef i n ed by

By t h e Kolmogorov-Sinai theorem, we g e t The mapping shift

T

1.1

, or

-+

hu(T)

d e f i n e d on

hp(T) = hp(T,[)

P(R,T)

simply, t h e e n t r o p y map.

5

= {(x,

= i ) : i C. S j ,

f o r a l l p 6 P(R,T).

i s c a l l e d t h e en t r o p y map o f t h e

I t i s well-known ( s e e B i l l i n g s l e y [ l ] ,

Denker e t a1 [31, and W a l te r s [ 7 ] ) t h a t h (T) = 1.1 according as

-

C. p. log p . 1

p

1

is the

p

logarithms a r e n a t u r a l ones.

or

pi p i j l o g p i j i,j B e r n o u l l i measure o r t h e C

(p,P) Markov measure. A l l

We s t a t e w it h o ut p r o o f t h e f o l l o w i n g lemma ( s e e

Denker e t a1 [3, P r o p o s i t i o n 10.131 o r Walters [ 7 , Theorem 8.11).

Markov Measures Lemma 5 . 1 . into P(R,T)

and

Proof.

hp(T)

i s an a f f i n e f u n c t i o n from

The e n t r o p y map

hp(T)

i s an upper semicontinuous f u n c t i o n on

o

0 5 h (T) 5 log s

n

P(R,T)

- c

=

I-ro

pk

i

...

o

P(R,T).

n- 1 Hp( v T-jg)

+

0

in

p

+

n- 1 ( v T-16)

P(R,T).

p

be any p o s i t i v e i n t e g e r .

n- 1 5 H ( v rr-jc)

Suppose

for all

lJ

Let

~

lJk


[O,m).

Lemma 5 . 2 .

ti

145

c p(x. 3 i n- 1

=

Then we h a v e , f o r e a c h 1-1 i., 1

o

5 j 5 n-1)log

(x. I

P(R,T), i. 1’

=

0 5 j 5 n-1) 5 n l o g s

.

By P r o p o s i t i o n 2 . 5 , we o b t a i n as

k

-f

m

.

Therefore, f o r each f i x e d

n Z 1, t h e

0 n-1 . tip( v ~ - 1 5 ) i s a continuous f u n c t i o n of

p . 0 n- 1 i s a n o n i n c r e a s i n g sequence o f c o n t i n u o u s Since I ~ ( vH T -~J ~ l) n 0 n-1 . f u n c t i o n s o f p , h (T) = l i m n H ( v T-IL) i s an upper semicontinuous u n l J 0 f u n c t i o n o f p . Note t h a t 0 5 h (T) 5 l o g s f o r a l l p i n P(S2,T).

mapping

p

-t

We d e n o t e by v e c t o r on

S

.

Lemma 5 . 3 .

Let

p C P(R,T).

(i)

p = A

.

( i ) = ( i i ) : Suppose

Proof.

Then t h e f o l l o w i n g a s s e r t i o n s a r e e q u i v a l e n t :

.

hp(T) = l o g s

(ii) ~

lJ t h e B e r n o u l l i measure induced by t h e uniform p r o b a b i l i t y

h

.

( i ) h o l d s , e q u i v a l e n t l y , h (T, 0 , t h e n i . c E 1 1' 1 1' 1 0 5 j 5 n, s o t h a t p . . = and p ( x . = i . 0 5 j C n t l ) = 1n 1n + l 'inin+l 1 1' V ( X . = i . 0 5 j 5 n t l ) . By i n d u c t i o n , we o b t a i n p = v . 1 1' ( i i ) = ( i ) : Suppose t h a t p i s t h e ( p , P ) Markov measure where P = (p. . ) . .

It

n .Z 0 , v(x0

Suppose t h e e q u a t i o n s h o l d f o r some

~ ( x .= i .

by

Markov measure.

pi p . . . . . p i = pi q . . . * . q i o ' 0 ~ 0 n-1 n o ' 0 ~ 1 n-1 n f o r a l l io,..., in C S . We s e e a t once t h e e q u a t i o n s h o l d f o r

If

s

S ;

= 6. f o r each ( i , j ) c F X S . 11 lj Let v denote t h e We o b t a i n e a s i l y pQ = p

9..

.

for

i s a permutation matrix with a s t a t i o n a r y d i s t r i b u t i o n

Markov Measures

.

p = (pi)i for a l l

c

i,j

Let

S

.

cp: S

be t h e b i j e c t i o n such t h a t

S

p..

f o r each

0

=

6

=

cp(i)j

11

Then we have

pij log p . .

C. 1

+

147

c

i

S ,

11

s o t h a t h (T) = Z i E ( - 1. p . . log p. . ) p . P I C E 11 11 1 T h i s completes t h e p r o o f .

=

0 where E

=

[i

c

S : p . > 01.

From t h e p r o o f o f Lemma 5 . 5 , t h e f o l l o w i n g lemma i s o b v i o u s . Lemma 5 . 6 . (i) (ii)

Let

1-1

c

M1(R,T).

Then t h e f o l l o w i n g a s s e r t i o n s a r e e q u i v a l e n t :

h (T) = 0 . P P i s induced by a unique i r r e d u c i b l e p e r m u t a t i o n m a t r i x i n

The f o l l o w i n g example i l l u s t r a t e

Assume

Example 5 . 7 .

.

s 2 3

Markov measures i n d u c e d by

u

Let

i r r e d u c i b l e permutation matrices.

P,Q

Let

c

M1(s

s)

X

w

and

P

M1(s x s ) .

P(Q,T) - M(R,T).

be d i s t i n c t

be t h e s h i f t i n v a r i a n t

Q , respectively.

and

P

u c

h (T) = 0 f o r some

I t f o l l o w s from

Lemmas 5 . 1 and 5 . 6 , t o g e t h e r with P r o p o s i t i o n 4.11, t h a t CP

( 1 - c ) ~c P(n,T) - M(Q,T)

+

Example 5 . 8 . 2 x 2

the

hcP Let

S = {O,ll.

v

c

c c

(0,l).

for a l l c c (0,l). (l-c)w (T) = 0 1~ be t h e Markov measure i n d u c e d by (0,1), i . e . ,

be t h e (q,Q)-Markov measure where

e

cp + ( 1 - c ) v

I t i s s t r a i g h t f o r w a r d t o show t h a t c

+

u n i t matrix, together with the probability vector Let

P = E

Assume

and

By Lemma 5 . 1 , we a l s o o b t a i n

P(Q,T) - M(R,T)

hclJ + ( 1 - c ) v (T) = 0

for all

for all

(0,l).

We a r e now i n a p o s i t i o n t o prove t h e main r e s u l t s o f t h i s s e c t i o n . Theorem 5 . 9 .


h ( T ) : P(R,T)

i s an a f f i n e ,

LO, l o g s ]

-f

lJ

upper semicontinuous s u r j e c t i o n .

Proof. surjective.

By Lemmas 5 . 1 and 5 . 2 , i t remains t o show t h a t t h e e n t r o p y map i s

Let

f ( t ) = htA

+

P =

Define t h e f u n c t i o n

(l-t)P(T),

0 5 t 5 1

f(t)

by

.

I t f o l l o w s from Lemmas 5 . 1 , 5 . 3 and 5 . 4 t h a t f(t) = t(l0g s ) , 0 5 t 5 1

.

u 6 ( 0 , log s ) , l e t

=

For any

( i i ) of Lemma 3 . 2 , we g e t v

c

c

u/log s

w

and

= ch + ( 1 - c ) ~ . Using p a r t

P(R,T) - M(R,T) and

u = f(c)

=

hw(T).

This

completes t h e p r o o f . We g i v e an a p p l i c a t i o n o f Theorem 4 . 9 . Theorem 5.10.


h ( T ) : M(R,T) * [O, l o g s ] P

is a continuous

surjection.

Proof. and l e t

pn

Suppose be t h e

l i m p.(n) = p i n 1

pn

+

in

p

(pn,Pn)

f o r each

M(R,T).

Let

Markov measure.

i t S

and

u

be t h e

(p,P)

Markov measure

By P r o p o s i t i o n 2 . 6 , we h a v e

l i m p . .(n) n 'J

=

p.. 11

f o r each

i,j C S

C-W. Kim

148

.

p. > 0

provided

If

p . = 0 , then

0 5 - pi(n)p. .(n)log pij(n) 5 pi(n)/e 11 j , so t h a t l i m p i ( n ) p . . ( n ) l o g p . . ( n ) = 0 11 11 n

f o r each

f o r each then

lirn h (T) = - lirn n n "n

Therefore,

h (T)

LJ

j

.

We have

s-1 s-1 s-1 s-1 . .(n) = - C 1 p.p. .log p. . i =C o j =C op . ( n ) p'1. . ( n ) l o g p 11 i=0 j=o '1 11

i s continuous on

M(R,T).

=

h (T). P

I t f o l l o w s from Theorem 4 . 9 ,

t o g e t h e r w i t h Lemmas 5 . 3 and 5 . 5 , t h a t t h e e n t r o p y map i s a s u r j e c t i o n between M(R,T) and [O, l o g s ] . We o b t a i n from Theorem 4.10, t o g e t h e r w i t h Theorem 5.10 and Lemmas 5 . 3 and 5 . 6 , t h e f o l l o w i n g theorem. Theorem 5.11.


h ( T ) : M1(R,T)

+

lJ

[O, l o g s ] i s a

continuous s u r j e c t i o n . Using Theorem 3 . 4 , t o g e t h e r with Theorem 5.10 and Lemmas 5 . 3 and 5 . 4 , we o b t a i n t h e n e x t theorem. Theorem 5.12.


h ( T ) : B(R,T)

+

v

surjection.

[O, l o g s ]

i s a continuous

Theorem 3.5 h a s t h e f o l l o w i n g a p p l i c a t i o n . Theorem 5.13.


h ( T ) : B(R,T)+

LJ

+

(0, l o g s ]

continuous

s u r j e ct i on. Proof.

By Theorem 3 . 5 , t o g e t h e r w i t h Theorem 5.12 and Lemma 5 . 3 , it i s

I _

tun}

enough t o show t h a t t h e r e i s a sequence h

(T)

as

0

+

pn p,(n) where

n h

=

(T)

-t m

.

= -

... .

Let

pn

1 (1 - x ) l o g ( l

pn as

in

B(R,T)+

be t h e -

1 -) n+l

for

pn

such t h a t

pn = ( p i ( n ) ) i n

Define t h e p r o b a b i l i t y v e c t o r s

1 1 1 - - , p.(n) = (s-l)(n+l) n+l 1

1,2,

=

n

TI by

i = 1 , 2 ,..., s-1

B e r n o u l l i measures.

1 - log n+l

1

Then we have 1

fir^ - n+l l o g X +O

n-tm. We may u s e anyone o f Theorems 5 . 1 0 , 5 . 1 1 and 5.12 t o show t h a t t h e entropy

map h p ( T ) : P(R,T) + [O, l o g s ] i s s u r j e c t i v e . We conclude t h i s s e c t i o n by o b s e r v i n g t h a t t h e e n t r o p y map i s n o t continuous on P(R,T) W e illustrate t h i s a s s e r t i o n with a m o d i f i c a t i o n o f an example i n Walters [7, p . 1841. Example 5.14. Define F(Tn) = 6 R : Tnw = W} and i.~ = 1 where

n = 1,2,

... .

Note t h a t c a r d

i s s t r a i g h t f o r w a r d t o show t h a t all

n

.

--c

n

Consequently, we have

F(Tn) = sn

and

pn

S" +

vn

6 P(R,T) - M ( ~ , T ) and

h

(T)

*

hX(T) = l o g s

as

w

A

as

h

(T) = 0

n + m .

"n n + m .

"n REFERENCES [11

B i l l i n g s l e y , P.,

E

E F(T")

E r g o d i c Theory and I n f o r m a t i o n (Wiley, 1965).

It

f o r a 11

Markov Measures [2 1

Chung, K . L . ,

149

Markov c h a i n s w i t h S t a t i o n a r y T r a n s i t i o n P r o b a b i l i t i e s ( S p r i n g e r , 2nd e d . , 1967). [31 Denker, M., G r i l l e n b e r g e r , C. and Sigmund, K . , E r g o d i c 'Theory on Compact Spaces ( S p r i n g e r , L e c t u r e Notes i n Math. 527, 1 9 7 6 ) . An I n t r o d u c t i o n t o P r o b a b i l i t y Theory and I t s A p p l i c a t i o n s , [41 F e l l e r , W . , Vol. 1 (Wiley, 3 r d e d . 1968). IS] Gantmacher, F . R . , The Theory o f M a t r i c e s , Vol. 2 ( C h e l s e a , 1 9 5 9 ) . [61 S e n e t a , E . , Non-negative M a t r i c e s and Markov Chains ( S p r i n g e r , 2nd e d . 1981). [71 W a l t e r s , P . , An I n t r o d u c t i o n t o E r g o d i c Theory ( S p r i n g e r , 1 9 8 2 ) .



151

TRANSLATION INVARIANT OPERATORS AND MULTIPLIERS FUNCTION SPACES

OF BANACH-VALUED

Hang-Chin L a i and Tsu-Kung Chang I n s t i t u t e o f Mathematics, N a t i o n a l T s i n g Hua U n i v e r s i t y , Hsinchu, Taiwan, R e p u b l i c o f China

L e t G b e a l o c a l l y compact a b e l i a n g r o u p , a n d A b e acorn m u t a t i v e B a n a c h a l g e b r a a n d X a B a n a c h A-module. In t h i s p a p e r , w e i n v e s t i g a t e t h e i n v a r i a n t o p e r a t o r s from a B a n a c k va1u.d f u n c t i o n s p a c e d e f i n e d on G i n t o a n o t h e r B a n a c k valued f u n c t i o n s p a c e , and c h a r a c t e r i z e t h e space of a l l i n v a r i a n t o p e r a t o r s as t h e f o l l o w i n g i s o m e t r i c a l l y isomorp h i c r e l a t i o n s u n d e r some a p p r o p r i a t e c o n d i t i o n s : 1 1 ( i ) (L (G,Y), L (G,X)) % E ( Y , M(G,X));

(ii)

L P ( G , x ) ) ;z ( Y , L ~ ( G , x ) ) ,1 < p < -;

(L'(G,Y),

(L'( G , A ) , L'( G , X ) ) 2 HornA(A , M ( G , X) ) ;

(iii) Horn

L (G,A)

l < p < w h e r e ( E ( G , Y ) , G ( G , X ) ) d e n o t e s t l i e s p a c e of all i n v a r i a n t bounded o p e r a t o r s of E t o F , E(Y,Z) i s t h e s p a c e o f l i n e a r o p e r a t o r s f r o m Y t o Z , a n d HOmA m e a n s t h e Am o d u l e homomorphisms. Moreover ( i ) and ( i i ) w i t h Y = A c o i n c i d e w i t h (iii) a n d (iv) r e s p e c t i v e l y i f a n d o n l y if A = (c, t h e c o m p l e x f i e l d . T h i s means t h a t any i n v a r i a n t o p e r a t o r of a B a n a c h f u n c t i o n s p a c e i s a m u l t i p l i e r A g (c i f and only i f

1.

INTRODUCTION AND PRELIMINARIES Let

dt, A

G

be a

locally

compact

abelian

g r o u p w i t h Haar

b e a commutative Banach a l g e b r a and

n o t e by

L1(G,A)

t h e space of a l l

f u n c t i o n s d e f i n e d on under convolution, and

which is

G

Lp(G,X)

a b l e f u n c t i o n s d e f i n e d on able over

G

G

Bochner

measure

a Banach space. De-

X

integrable

a commutative

A-valued

Banach

algebra

t h e s p a c e o f a l l X-valued m e a s u r -

w h o s e p-power

o f X-norm

which is a Banach s p a c e f o r e a c h

p,

are i n t e g r -

15

p

TL,

L'(G,x)

such t h a t TL(f 8 y ) = f

*

for all

L(y)

f

E

1

L (G), y

E

Y

and s a t i s f y i n g IITLII 5 IILll. This

TL

is t r a n s l a t i o n i n v a r i a n t s i n c e -rSTL(f

@

Y ) = TsT(fy)

*

= Ts(f

L(Y))

= Tsf

*

= T (T

sf y )

L

L(Y) for all

= TL7,(fy)

Hence

TL

E

1

( L ( G , Y ) , L'(G,X)).

proof, we o b t a i n

f

paragraph

(L'(G,Y),

correspondence between

is o b v i o u s .

Z(Y, M ( G , X ) )

1

E L ((3). in

the

(Ll(G,Y)

,

L1(G,X))

T h e r e f o r e we o b t a i n

1 L (G,x)) ; E(Y, M(G,x))

a n d t h e proof i s completed.

Q.E.D.

A c c o r d i n g t o Theorem C w i t h o p e r a t o r s of

By t h e f i r s t

G, Y E Y ,

IITLII = l \ L \ l .

F i n a l l y , t h e one-one and

S E

L1(X,Y)

to

A =

Lp(G,X)

(c

for

a n d Theorem 1 , t h e i n v a r i a n t

1< p
1, { x ( a ) e l a

$(b)

pping

in

A, t h a t is, a multiplica-

Define

A.

$(a) = x(a)e

Let

proof

p = 1.

property

of

Banach-valued Function Spaces

*

T a ( f C3 a ) = v , ( a )

1

Since

mation of

f.

1

6,

L (G)

is a

A = L (G,A), TN

L1(G,X)

to

L1(G,A) T a ( f C3 a )

=

161

bounded

linear

transfor-

such that

T,(fa)

*

=

uo(a)

=

L(@(a))eN

f

*

f.

Thus l i m T,(T

8 a)

=

L($(a))f.

11

We w r i t e T ( f 64 a ) = T ( f a ) = L ( $ ( a ) ) f . This

T

rt

If

t o L'(G,x). is a b o u n d e d l i n e a r o p e r a t o r f r o m L'(G,A) i s a t r a n s l a t i o n o p e a t o r f o r t E G. t h e n i t i s o b v i o u s

that =

l i m { p a a)

=

T(-rtfa)

TtT(fa)

Hence

T

is invariant.

*

T(fa

t E G.

gb) = T ( ( f

X = A, p

If

-ttf1

But

f T(fa)

REMARK 3.

*

=

*

1,

bg

then

since

Theorem

$(b) f b.

E is

redhced

to

Theorem A . I n v i e w o f Theorem 6 , w e a s k u n d e r what c o n d i t i o n s

The a n s w e r is t h a t complex f i e l d THEOREM 7. t i t y of n o r m 1

A

must b e i s o m e t r i c a l l y

isomorphic

to

the

(c.

Let and

b e a commutative Banach a l g e b r a w i t h i d e n -

A X

a n A-module.

Then e a c h i n v a r i a n t o p e r a t o r

H.-C Lai and T.-K. Chang

162 T : L1(G,A) if

A

+

Lp(G,X)

for

15p

1 , t h e n

T h i s c o n t r a d i c t s t h e assumption ( 4 . 1 ) .

Hence

A

Ic,

and

the

t h e o r e m is p r o v e d . REMARK 4 .

If

p =

a,

we t a k e

C0 ( G , X )

t h e above d i s c u s s i o n s , t h e n we c o u l d g e t

i n s t e a d of the

same

Lp(G,X) i n conclusions

above. REFERENCES [ 13 D i e s t e l , J . a n d U h l , J r . , J . J . , V e c t o r > M e a s u r e s , Ma 11. S u r v e y s , Amer. Math. S O C . N o . 1 5 , 1 9 7 7 . [ 21 D i n c u l e a n u , N . , V e c t o r ' M e a s u r e s , P e r g a m a n , O x f o r d 1 9 6 7 . [ 31 D i n c u l e a n u , N . , Integration on LocaZZy Compact S p a c e s , Noordhoff I n t e r n a t i o n a l P u b l i s h i n g , 1974. [ 41 J o h n s o n , G . P . , S p a c e s o f f u n c t i o n s w i t h v a l u e s i n a B a n a c h a l g e b r a , T r a n s . A m e r . Math. SOC. 9 2 ( 1 9 5 9 ) , 4 1 1 - 4 2 9 . [ 51 K h a l i l , R . , M u l t i p l i e r s f o r some s a p c e s o f v e c t o r - v a l u e d func t i o n s , J . U n i v . Kuwait ( S c i ) , 8 ( 1 9 8 1 ) , 1-7. [ 61 L a i , H . C . , M u l t i p l i e r s o f a B a n a c h a l g e b r a i n t h e s e c o n d conj u g a t e a l g e b r a a s a n i d e a l i z e r , Tohoku M a t h . J . , 2 6 ( 1 9 7 4 ) , 431-452. [ 71 L a i , H . C . , M u l t i p l i e r s f o r some s p a c e s o f B a n a c h a l g e b r a v a l u e d f u n c t i o n s , Rocky M o u n t a i n J . M a t h . , 1 5 ( 1 9 8 5 ) , 1 5 7 - 1 6 6 . [ 81 L a i , H . C . , M u l t i p l i e r s o f B a n a c h - v a l u e d f u n c t i o n s p a c e s , J. A u s t r a l . Math. S O C . , 3 9 ( S e r i e s A) ( 1 9 8 5 ) , 5 1 - 6 2 . [ 91 L a i , H . C . , D u a l i t y o f B a n a c h f u n c t i o n spaces a n d t h e RadonNikodym p r o p e r t y , Acta Math. (Hung. ) , 4 7 ( 1 - 2 ) ( 1 9 8 6 ) , 4 5 - 5 2 . [ l o ] Quek, T . S . , M u l t i p l i e r s o f c e r t a i n v e c t o r v a l u e d f u n c t i o n spaces, Preprint [ l l ] R i e f f e l , M . A . , M u l t i p l i e r s and tensor p r o d u c t s on LP-spaces o f l o c a l l y compact g r o u p , S t u d i a M a t h . , 3 3 ( 1 9 6 9 ) , 71-82. [12] T e w a r i , U . , D u t t a , M. and V a i d y a , D.P., M u l t i p l i e r s of g r o u p a l g e b r a s o f v e c t o r v a l u e d f u n c t i o n s , P r o c . A m e r . Math. SOC. 8 1 ( 1 9 8 1 ) , 223-229.

.

Proceedings of the Analysis Conference, Singapore 1986 S.T.L. Choy, J.P. Jesudason, P.Y. Lee(Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1988

163

A PROOF OF THE GENERALIZED DOMINATED CONVERGENCE THEOREM FOR DENJOY INTEGRALS

Lee Peng Yee We y i v e an independent p r o o f o f t h e g e n e r a l i z e d dominated convergence theorem f o r t h e Denjoy i n t e g r a l .

R e c e n t l y , Lee and Chew proved s e v e r a l convergence theorems f o r t h e Oenjoy

[7]

i n t e g r a l (see [6],

and [81).

I n t h i s note, we g i v e an independent p r o o f o f

t h e g e n e r a l i z e d dominated convergence theorem.

As a consequence, o t h e r conver-

gence theorems f o l l o w . F i r s t , we d e f i n e Denjoy i n t e g r a b l e f u n c t i o n s . AC*(X) i f f o r every

> 0 there i s

sequence o f non-overlapping i n t e r v a l s {[ai ,b.]} ail

0 such t h a t f o r every f i n i t e o r i n f i n i t e E

X satisfying

we have

l ] and i n f { f n ; n > l ] a r e D e n j o y i n t e g r a b l e on [a,b].

Hence t h e t h e o r e m r e d u c e s t o t h e

following corollary.

COROLLARY 1.

L e t fn be Denjoy i n t e g r a b l e on [a,b]

a l m o s t e v e r y w h e r e i n [a,b] Ca,bl.

as n +

m,

such t h a t f n ( x ) + f ( x )

and g, h a r e a l s o D e n j o y i n t e g r a b l e o n

If g ( x ) < f n ( x ) < h ( x ) a l m o s t e v e r y w h e r e i n [ a , b l

f o r a l l n, t h e n t h e

consequence o f t h e t h e o r e m h o l d s . F o r convenience, we w r i t e Fn(u,v)

COROLLARY 2.

= Fn(v)

E

Fn(u) i n t h e following.

w i t h p r i m i t i v e Fn such

L e t fn b e Oenjoy i n t e g r a b l e o n [ a , b l

t h a t f n ( x ) + f ( x ) a l m o s t e v e r y w h e r e i n [a,b] and

-

as n +

m.

I f f o r every 5 ~ [ a , b ]

> 0 t h e r e e x i s t a n i n t e g e r N and 6 ( 5 ) > 0 such t h a t (Fn(U,V)

whenever m,

n > N and 5

-

-

Fm(U,V)(

0 t h e r e e x i s t s

f o r e v e r y 6 E [a,b]

s ( 6 ) > 0 such t h a t IFn(u,v) whenever 5

-

-

fn(6)(v-u)I

0

Lp, where $

belong to For

f

IIf II

5

belongs to

=

HP

such that

s

$(x)dx

#

0.

I f* LP

'

then Hp = Lp with equivalent norms.

If 1 < p < between He Re Hp.

is a fixed function in

in Hp, we set

and He

Re Hp

If n = 1, the relation

is as follows: A tempered distribution f

on

E

if and only if both its real part and imaginary part belong to

Factorization Theorem for the Real Hardy Spaces For

f

0

with

h

En \

on

2

h < n, we denote by

G(A)

169

the set of those smooth functions

such that

{O}

for all multi-indices a.

En

integrable functions on We denote by

G'(n)

We shall regard the elements of and the set G(A)

G(A)

2'.

as a subset of

the set of those smooth functions f

as locally

En \

on

{O}

such that

and

If

f

G'(n),

E

then there exists a sequence

{a.} J

such that

a

>

j

0, lim

j + o aj

= 0 and the limit

f' =

+

G(n)

+ c6, where

the above way,

1x1 > aj 1

If

m

S',where

exists in

We denote by

f'

x[ I x I

lim j

x[E]

E.

denotes the characteristic function of the set

the set of all those tempered distributions of the form

f' c

is the tempered distribution arising from is a complex number, and

Proposition 1. Let

0 5- A 5- n.

Then

f

6 E

f

E

G'(n)

in

denotes Dirac's distribution.

G(A)

if and only if

f

E

G(n-A).

We can prove this proposition by elementary calculations (the integration by parts). If

m

with

G(h)

E

0

2

A < n, then an operator

T

from

3 to

s'

is

defined by

We call T

the operator associated with m, and m the multiplier correspond-1 If we set k = F m, then the operator T associated with m is

T.

ing to given by

k

Tf where

*

For

h

f E

s,

05 h < n, we denote by

with m

E

K(h)

the set of the operators associ-

G(h).

m c G(A),

We define %I

$.

f,

denotes the convolution.

ated with Let

*

by

0 5- h < n, and let T be the operator associated with m. g(c) = m ( - c ) and denote by T' the operator associated with

We call T'

the conjugate of

T. The operator T

satisfy

=

for all

f, g

E

5.

and its conjugate T'

A. Miyachi

170

Suppose T c K ( A ) ,

Proposition 2. = A/n.

2

0

Then there exists a constant C

n, p > 0, q > 0, and

A/n,

T1,

3

TN, P, q, r,

n Hr.

p 5- 1 and that

have the following properties: (a)

-A

j'

i.e.,

-A where

j = 1,

IJI

J

l/q > A/n,

(ii) In addition t o the above assumptions, assume further that the multipliers m

where

Let

+ N/n.

(i) Then, there exists a constant C1 and n for which the inequality

holds for all g

Aj

denotes the complement of

Suppose 0 < p, q, r
0.

can be decomposed as

> :ak P(T~, k=1 m

f = where

ak

, TN;

are complex numbers, gk

t

\I,

gk,

5

n Hq, hk

n Hr,

i S

and

Here

C2

and

C3

are constants depending only on

T1,

* * *

, TN, P,

q , r, and

n. g

Remark. (i) For

and

h

in

5,

the right hand side of (2) is well defined

since, by Proposition 2 and HGlder's inequality, each term in the summation in (2) belongs to

Ls

for all sufficiently large

s

and hence, a fortiori, to

S'.

(ii) In terms of the Fourier transform, the product in the theorem can be redefined as follows:

If n = 1 and

..., TN;

T1 =

g , h)

..*

= TN

is equal to

the Hilbert transform, then pi;

+

gh

(if

N

is an odd integer) o r

U Y

gh

(if

N

is an even integer) multiplied by a nonzero constant.

(iv) Theorem 1 for some special cases have been known. Coifman-Rochberg-Weiss [ 6 ] gave the theorem for the case

N = 1, A = 0, and

p = 1

(they gave (ii)

for T the Riesz transforms). Uchiyama [18], [19] treated the case N = 1, j A = 0, and p > n/(n+l). Chanillo [5] treated the case N = 1, 0 < A < n, and

p

1. Komori [ll] treated the case N = 1, 0

The author [14], [15] treated the case N 4.

21

and

n/(n+l).

= 0.

SKETCH OF THE PROOF We can prove Theorem 1 by only slightly modifying the arguments in [14] and So, we shall give only a sketch of the proof.

[15].

First we shall sketch the proof of Theorem 1 (i).

The basic idea of the

proof of this part is due to Uchiyama [ 1 9 1 . Let

0

2v

< n,

x

E

En, and

k be a nonnegative integer. We define the set

A. Miyachi

172

T'(x)

k

a s follows:

g

belongs t o

P

Tk(X)

if

g

Rn

i s a smooth f u n c t i o n on

I n o r d e r t o prove Theorem 1 ( i ) , w e u s e t h e f o l l o w i n g lemmas.

Lemma

1.

If

0 < p, q
n / p - n , then

p 5 - 1, then we can prove t h e above lemma by u s i n g t h e a t o m i c decomposiIf

p > 1, t h e n the lemma i s a c o r o l l a r y t o t h e lemma below.

Lemma 2.

0

If

5-

p < n,

0 < s n/p

fk

t

A

-

5 6 En,

'

P,M

i s a c o n s t a n t depending o n l y on

C

n/2, then every

M , n , and

f

in

He

can be

and

p.

A s f o r a proof,

Hp.

This i s a modification of t h e atomic decomposition f o r see [ 1 5 ] .

By s l i g h t l y m o d i f y i n g t h e argument i n [ 1 5 ] , we c a n p r o v e t h e f o l l o w i n g : F o r every

f

in

A

P,M

and e v e r y

0, t h e r e e x i s t

E >

g

and

h

in

5

such t h a t

and

where

CE

is a c o n s t a n t depending o n l y on

.-.,TN,

T1,

P , q , r , M, n , and

E.

Combining t h i s w i t h Lemma 3 , w e can e a s i l y p r o v e Theorem 1 ( i i ) .

5.

AN APPLICATION We s h a l l g i v e a proof o f t h e f o l l o w i n g theorem.

Theorem 2. in

Here

Let

such t h a t

He

C

0 < p

2

1.

g(5) 2- I ? ( S ) (

Then, f o r e v e r y

5

for all

i s a c o n s t a n t depending o n l y on

p

E

f

in

En

and

and

Hp, t h e r e e x i s t s a

g

n.

T h i s theorem h a s a l r e a d y been proved by s e v e r a l methods.

Proof f o r t h e c a s e

n = p = 1 can be found i n Zygmund's book 120; Chapt. V I I , Proof o f Theorem ( 8 . 7 ) , p.2871.

Coifman and Weiss [ 7 ; p.5841 used t h e a t o m i c d e c o m p o s i t i o n t o

g i v e a new proof ( a l s o f o r t h e c a s e

n = p = 1 1.

B a e r n s t e i n and Sawyer [ 2 ] ,

[ 3 ; 881, Aleksandrov [ l ] , and t h e a u t h o r [16] e x t e n d e d t h e method of Coifman

A. Miyachi

174

The a u t h o r [ 1 6 ] a l s o gave 1 two o t h e r d i f f e r e n t p r o o f s , one of which i s based on t h e d u a l i t y between H

and Weiss t o prove t h e theorem i n t h e g e n e r a l c a s e .

and

p = 1, and t h e o t h e r i s s i m i l a r t o t h e one

and is v a l i d f o r t h e c a s e

BMO

t o be given below.

Here we s h a l l g i v e a proof of Theorem 2 u s i n g our f a c t o r i z a t i o n theorem; t h i s i s an e x t e n s i o n of one of t h e p r o o f s g i v e n i n [ 1 6 ] . nology: We s a y t h a t f o r every

5

all

f

in

En

E

0 < p 5 - 2 , h a s t h e lower majorant p r o p e r t y i f g i n Hp such t h a t ^ g ( S ) 2 i ? ( S ) I f o r

H p , where

Hp,

We s h a l l i n t r o d u c e a termi-

there exists a

and

(Thus, Theorem 2 asserts t h a t

Hp

with

0 < p 5- 1 h a s t h e lower majorant

property*).) Proof of Theorem 2. t h e two f a c t s : ( a ) p

5

1,

2

l/p

In o r d e r t o prove t h e theorem, i t i s s u f f i c i e n t t o prove h a s t h e lower m a j o r a n t p r o p e r t y ; ( b ) I f

H2

l / q + 1 / 2 , and i f

Hq

h a s t h e lower m a j o r a n t p r o p e r t y , t h e n

P l a n c h e r e l ' s theorem.

We s h a l l prove ( b ) .

t h e r e and suppose

h a s t h e lower m a j o r a n t p r o p e r t y .

+

l/q

112 - x/n.

l/p < 1

l!&

N

hi

E

X

j

and

in

ri

En \

mj

p

and

q

be as mentioned

Take a p o s i t i v e i n t e g e r E

G(Xj),

j = 1,

X

Define

. a * ,

N,

N

l/p =

by

satisfying

such t h a t

A =

f and

in

Hp

{O}.

(This condition ( 4 ) i s c e r t a i n l y s a t i s f i e d

...

m = = 51, and m i s r e a l v a l u e d . ) Take an 1 j and decompose i t a s i n Theorem 1 ( i i ) ( t a k e r = 2 ) .

i s an even i n t e g e r ,

Hq

and

Let

j

5

arbitrary Since

and t a k e

Hp

m ' s s a t i s f y t h e c o n d i t i o n s i n Theorem 1 ( i i ) , and

hj,

for a l l if

+ N/n

0 5 - X < n/2.

Then

2,

The f a c t ( a ) i s obvious by v i r t u e of

a l s o h a s t h e lower majorant p r o p e r t y .

Hq

2

0 < p < q

€I2 have t h e lower m a j o r a n t p r o p e r t y , w e can f i n d

gi

E

Hq

and

2

I?(S) I .

such t h a t

H2

,

2

C llh, II 2 . H

Define

g

by

Using ( 4 ) and t h e formula i n Remark ( i i ) ( § 3 ) , we e a s i l y see t h a t

g(S)

On t h e o t h e r hand, by v i r t u e of Theorem 1 ( i ) , t h e i n e q u a l i t y ( 3 ) h o l d s . completes t h e p r o o f .

This

(To be p r e c i s e , some l i m i t i n g arguments are n e c e s s a r y

*) A s f o r t h i s property f o r

Hp = Lp

with

p > 1, see [ l o ] and [ 1 7 ] .

Factorization Tlieorervi for the Real Hardy Spaces since g i

and

hi may not belong to

5; we

175

omitted the limiting arguments.)

REFERENCES [l] A. B. Aleksandrov, The majorant property for the multi-dimensional HardyStein-Weiss classes (in Russian), Vestnik Leningrad Univ. 13 (1982), 97-98. [2] A. Baernstein I1 and E. T. Sawyer, Fourier transforms of He spaces, Abstracts Amer. Math. Sac., Val. 1, No. 5 (1980), 779-42-8, p. 444. [3] A. Baernstein I1 and E. T. Sawyer, Embedding and multiplier theorems for Hp(Rn), Mem. Amer. Math. Sac. 53 (1985), no. 318. [4] A. P. Calder6n and A . Torchinsky, Parabolic maximal functions associated with a distribution, 11, Advances in Math. 24 (1977), 101-171. [5] S . Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), 7-16. [6] R. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. 171 R. R. Coifman and G . Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Sac. 83 (1977), 569-645. [8] P. L. Duren, Theory o f Hp spaces, Academic Press, New York-San FranciscoLondon, 1970. [91 C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137-193. [lo] E. T. Y. Lee and G . Sunouchi, On the majorant properties in Lp(G), Tchoku Math. J. 31 (1979), 41-48. [ll] Y. Komori, The factorization of Hp and the commutators, Tokyo J . Math. 6 (1983), 435-445. Spaces, London Math. SOC. Lecture Note [12] P. Koosis, Introduction to Hp Series 40, Cambridge Univ. Press, Cambridge, 1980. [13] R. H. Latter, A characterization of Hp(En) in terms o f atoms, Studia Math. 62 (1978), 93-101. [I41 A. Miyachi, Products of distributions in Hp spaces, TBhoku Math. J. (2) 35 (1983), 483-498. [15] A. Miyachi, Weak factorization of distributions in Hp spaces, Pacific J. Math. 115 (1984), 165-175. [16] A . Miyachi, Majorant properties in Hardy spaces, Research Reports Dept. Math. Hitotsubashi Univ., 1983. [17] M. Rains, Majorant problems in harmonic analysis, Ph. D. dissertation, Univ. of British Columbia, Vancouver, 1976. [18] A. Uchiyama, On the compactness of operators of Hankel type, TBhoku Math. J . 30 (1978), 163-171. [19] A. Uchiyama, The factorization of Hp on the space of homogeneous type, Pacific J. Math. 92 (1981), 453-468. [ZO] A. Zygmund, Trigonometric Series, 2nd ed., Vols I, 11, Cambridge Univ. Press, Cambridge, 1959.


Proceedingsof the Analysis Conference,Singapore 1986 S.T.L. Choy,J.P.Jesudason,P.Y. Lee (Editors) 0 Elsevier Science PublishersB.V. (North-Holland),1988

177

Estimates for Pseudo-differential Operators of class hp, and bmo

Sm

P-6

in

LP,

Akihiko MIYACHI Department of Mathematics, Hitotsubashi University Kunitachi, Tokyo, 186 Japan *) The purpose of this article is to give some estimates for the operator norms of pseudo-differential operators as operators between hp, Lp, and bmo by means of certain Lipschitz norms of their symbols and to give also some negative results concerning these estimates. The negative results will show that most of our norm estimates are sharp in a sense. The results are slight generalizations of those given at the author's lecture at the Analysis Conference, Singapore, 1986.

1.

INTRODUCTION The notations used in this article will be explained in the next section. In this article, we shall consider the pseudo-differential operator of the

following form: (a(x,D)f)(x) where

= (2n)-"

J

e a(x,S)i(S)dS, ixs Rn

denotes the Fourier transform. The function a(x,E)

symbol of the pseudo-differential operator those symbols a(x,S) C

and

-S(Rn). _ constant

m.

which satisfy

a(X,D).

la(x,c)I

2

We shall consider only

C(l+lSl)m

For these symbols, the operators a(X,D)

We shall say that C

a(X,D)

is called the

for some constants

are well defined on if there exists a

is bounded in Lp

such that the inequality

holds for all

f

in

s(En); we

shall use the similar expression replacing Lp

by other function spaces. The following theorem is known.

(See the remark given at the end of this

section.) Theorem A.

If

0 5 _ 6 5_ p 5_ 1, 6 < 1, 0 < p
1).

*)Partly supported by the Japan Society for the Promotion of Science and by the Grant-in-Aid for Scientific Research (C61540088), the Ministry of Education, Japan.

178

A. Miyachi m

I t i s a l s o known t h a t t h e above c o n d i t i o n on

if

6 , p , and

p

t h e n t h e r e i s a symbol

-n(l-p)Il/p-l/21,

a

all

b u t f o r which

which s a t i s f i e s (1.1) f o r

a(x,S)

i s n o t bounded i n

a(X,D)

k

consider

k'

and

numbers

and

K

not necessarily integers.

W e s h a l l introduce a c l a s s

the followings.

f i n d t h e numbers >

or

k'

and

f o r which

More p r e c i s e l y , we s h a l l do

Sm

P,6

for arbitrary positive

(K,K')

and

K~

I B I 5-

and

K

la1

2

Then w e s h a l l

K'.

which a r e c r i t i c a l i n t h e s e n s e t h a t i f

K,!,

>

K

Lp

a r e bounded i n

symbols i n t h e c l a s s

Sm

036

i n p l a c e of

but i f

K

0. Let k and k ' -2 k t l and k' < K' -5 k ' + 1 .

denotes the set of those functions a = a(x,C)

(K,K')

on Rn x Rn -

x

i

which have the following estimates: (i) if

101

2 k and la1 zk', then the derivative aXD aaa(x,E) E

exists in the

classical sense and

2

ja:aia(x, n/2,

K'

2 n(l-p)/2

w"

then

P,6

-n(l-p)(I/p-l/Z), c

L(L' ,LP ) .

2

m

2

+ m,

(K,K') K

c

L(bmo,bmo).

> n(l/p

-l),

and

-n(l-p)/2,

( 1 - 6 ) ~ > n(l-p)/2

+ m,

-n(l-p)(1/2 - 1 / p ) , ( 1 - 6 ) ~ >

> n(l-l/p),

then

( 1 - 6 ) ~ > n(1-p)

(1) I f

0 n, then

L(bmo,bmo).

( 1 - 6 ) ~< n(1-p)/p m, t h e n ! i L(L P , L P ) ( i f 1 

L(hP,hP).

c

( Ym p , & ( ~ , ~ ' ) c) *L ( h p , h p ) .

then

m,

P,6

(K,K')

1 / 2 ) , ( 1 - 6 ) ~> n(l-p)/p

(if

2

1 n/p, then

L(hp,Lp)

c

2

1 n(l/p-l/Z),

(2) I f and

K'

L(LP,LP)

c

m,

-n(l-p)/2,

Theorem 2 . K'

and

2, - n ( l - p ) / p then

2

p 5 - 1, - n ( l - p ) / p

0

+ m + np(l/p-l),

m,

m

y P , &(K,KI)

-n(l-p)/2

f o r every < m

2

a(x,C)

< m

2

-n(l-p)(l/p-

# L(HP,LP) K'

KI

(if

o

 0.

-n(l-p)(l/Z-l/p),

# ~ ( L ~ , L P f) o r e v e r y

( 3 ) There e x i s t s a symbol *)This r e s u l t f o r

(K,K')

> 0.


p = 1 i s i n c l u d e d i n (1).

and

( 1 - 6 ) ~ < n(l-p)/Z+m,

Estimates for Pseudo-differential Operators

183

m

f o r a l l multi-indices BMO

and f o r which

a

a(X,D)

i s n o t bounded from

L

to

.

(4) I f

2

2, then ym

2)

f o r every

0 0 i t holds t h a t for 2 p < m and Ym~ , & ( K , KPOL ( L ~ , B M O ) .

0.

2

K'

>

and

to

HI

( 1 - 6 ) ~ < n(l-p)/2+m,

and

0.

- n ( l - p ) ( 1 / 2 - l / p ) , and

( ymp , 6 ( ~ , ~ ' ) ) #* _4(Lp,Lp)

then

2

< m

m

(Y

then


(1- l/p) < m

-n(l-p)

m,

K'

a and f o r which

f o r a l l multi-indices 1 L . (3) I f

< n(l/p-l),

K

f o r every

( 1 - 6 ) ~ < n(1-p)

+ m,

( 1 - 6 ) ~

K'

then

(Y:,,(K,K'))*

#

m

L(L ,BMO) f o r e v e r y -

( 6 ) If every

0

m

(7) If

m

t

(8) If

every

K

0. < n/2,

then

> 0.

# L(Lp,Lp)

(%. 6(K,K'))*

+

( ~ : , ~ ( ~ , n - n / p ) ) * &(Lp,Lp)

f o r every

m

f o r every

t

11

K

> 0.

and

> 0.

(9) W e have

(K,n))*

(Y&,:

Theorems 3 and

L(Lm,BMO)

f o r every

m

E

5

and e v e r y

4 show t h a t most o f t h e r e s u l t s i n Theorems 1 and 2 are s h a r p

i n a sense.

5.

PROBLEMS AND FURTHER RESULTS F i r s t , t h e r e i s a problem: Can o n e r e l a x t h e c o n d i t i o n on

(l)?

If

p = 0

or i f

i n Theorem 1

K

p = 1, t h e n Theorem 3 ( 1 ) shows t h a t i t c a n n o t b e

e s s e n t i a l l y r e l a x e d ; t h e problem arises i n t h e case p r e s e n t a u t h o r h a s checked t h a t i f

0 

2

0

1, m

p < 1.

and

and Ym ( K , K ' ) c L ( h p , h p ) forsome K ' > 0 , then K np(l/p-1), P,O know w h e t h e r o n e can r e l a x t h a t c o n d i t i o n i n Theorem 1 ( 1 ) . S e c o n d l y , t h e r e i s a problem c o n c e r n i n g t h e c o n d i t i o n on i n the case

m

t i o n r e a d s as

I n t h i s c a s e , t h e c o n d i t i o n on

The

-n(l-p)(l/p-l/2),

=

but does n o t

K

i n Theorem 1 (2)

K

i n t h a t asser-

=

-n(l-p)/p.

K

> 0 ; s o t h e r e arises a problem w h e t h e r o n e c a n d i s c a r d e n t i r e l y

A. Miyaclri

184

t h e c o n d i t i o n on t h e c o n t i n u i t y of t h e symbol w i t h r e s p e c t t o

x.

More p r e c i s e -

l y , does t h e c o n d i t i o n

Lp

2

0

where

p

2

2

0 < p

(if

0 < p 5- 2 , imply t h a t

1 and

1) o r

Lp

to

Lp

p a r t i a l answers t o t h i s q u e s t i o n .

i s bounded from

1< p 5 - 2)?

(if

If

a(X,D)

1

< p

2- 2 , t h e answer i s a g a i n

to

The f o l l o w i n g s a r e

p = 1, t h e answer is NO; t h i s can be s e e n

from t h e c o u n t e r example given by Coifman and Meyer [ 4 ; pp.39-401. and

he

p = 0

If

NO; t h i s can be s e e n from t h e f o l l o w i n g

c o u n t e r example:

On t h e o t h e r hand, t h e answer i s YES i f

0 5- p

0 < p

1 and

0

K

(3) with

and

i n Theorem 2 ( 2 ) w i t h

m = -n(l-p)/2,

m = -n(l-p)(l-l/p),

= 0 , i n Theorem 1

p(l/p-1)

m = -n(l-p)/2,

and i n Theorem 2 ( 4 ) w i t h

i n Theorem 2 ( 3 )

m = -n(l-p).

T h i r d l y , t h e r e i s a problem of s h a r p e n i n g Theorems 1 and 2 f u r t h e r .

1

- 4 g i v e t h o s e numbers and

Y

hold i f with

K

are

Z

inclusion

Ym P,6 K

=

n/p.

m

=

P 1 6

m

(K,K').)

+

6t

-

and

m

+ m.

n(l-;)(l/p-l/Z),

PS

5 6

-t

This

p # 1 and

-n(l-p)/p

satisfy these inequalities,

2

m

We may a n d s h a l l assume, w i t h o u t l o s s of g e n e r -

8 t - 6s

W e can f i n d

6,

( 1 - 8 ) ~> n ( l - p ) / 2 ,

w i t h c o n t i n u o u s embedding.

-(K,K')

qualities and

K'

> ( 1 - 6 ) ~> (l-p)K'

< 1,

5 2

Suppose

and

q.

tends

is arbitrary.

E

Next we s h a l l p r o v e Theorem 1 ( 2 ) f o r t h e c a s e

n(1-p)/p

1- 6

0, t h e n

1 - 6 < ~ / 2 by t a k i n g s u f f i c i e n t l y small

0; h e n c e we c a n t a k e

to

I f we let

m(p) = - n ( l - p ) ( l / p - l / Z ) .

6,

8

and

and

such t h a t

m

S p , & ( ~ , ~ c' )

(The l a s t embedding h o l d s i f t h e i n e -

hold f o r

(t,s) = (O,O),

(O,K'),

(K,O),

T h u s t h e a s s e r t i o n i n Theorem 1 (2) f o r t h e c a s e u n d e r c o n s i d e r a -

t i o n c a n b e d e r i v e d from t h a t f o r t h e c a s e

p

# 1 and m

= -n(l-p) ( l / p

- 1/2).

( 2 ) i n t h e same way as above;

W e c a n p r o v e Theorem 1 (1) and Theorem 2 ( l ) ,

t h e o n l y a d d i t i o n a l t o o l w e need i s t h e s i n g u l a r i n t e g r a l c h a r a c t e r i z a t i o n o f he

(cf.

[ l l ; t h e p a r a g r a p h j u s t below P r o p o s i t i o n 4 . 2 ] ) , by v i r t u e o f which

we can reduce t h e cf.

he

+

he

estimate t o t h e

hp

+

Lp

estimate.

For d e t a i l s ,

[ll; 141. Theorem 1 ( 3 ) , (4) a n d Theorem 2 ( 3 ) , (4) are d e r i v e d f r o m Theorem 1 ( l ) ,

(2)

A. Miyachi

186

and Theorem 2 (l), ( 2 ) by the use of the duality between Lp and l/p

+

l / q = 1, o r the duality between

h1

and

bmo.

Lq, where

As for the latter duality,

see Goldberg 181. These are the sketches of the proofs of Theorems 1 and 2. Next we shall proceed to the proofs of Theorems 3 and 4. Note that Theorem 3 ( 2 ) , (3),

(5) and Theorem 4 (4), ( 5 ) , (8), (9) are derived from the rest of the

theorems by the u s e of duality. Most of the results in these theorems can be

[lo;

proved in almost the same way as in

§5] and [ll; 151.

In the following,

we shall give a proof of Theorem 3 (1); this is the only proof which requires an extra idea which is not in those papers. Let on

Sn

p

and

m

such that

define a

t

and

at(x,E)

be as mentioned in Theorem 3 (1). supp $

{XI

c

Take a smooth function $

1 2 1x1 5- 21 and $ # 0. Let

t > 1

and

ft by =

+(tpx)e

-ixE

@(t-'t)

and

I t is easy to see that

(Hp shall be replaced by depending only on n,

P,

m,P,6,K,K where and

C' K'

Lp

if p

p, and

4.

' 5-

C't

-lli+(

>

l), where

C

is a positive constant

On the other hand, it holds that

1-6) K I

is a constant depending only on n , m,

P,

6,

K, K ' ,

and

$.

(If

K

are integers, then we can prove the above estimate by elementary calcu-

lations; in the general case, we can prove it by using the proposition in Section 3 . )

Now suppose

K,

K'

> 0

and Ym

(K,K')

c

P,6

L(Hp,Lp)

(if p

2

1)

or c L(LP ,LP ) (if p > 1). Then, from the above two inequalities, it follows that tn(l-~)/~< C,lt-mt.(1--6)K -for all t > 1 and hence n(1-p)/p 2 -m + (1--6)~.This proves Theorem 3 (1). REFERENCES

[l] A . P. CalderBn and A. Torchinsky, Parabolic maximal functions associated with a distribution, 11, Advances in Math.

2 (1977),

101-171.

[2]

A. P. CalderBn and R. Vaillancourt, On the boundedness of pseudo-differential

[3]

A. P . Calder6n and R. Vaillancourt, A class of bounded pseudo-differential

[4]

R. R. Coifman and Y. Meyer, Au-delb des operateurs pseudo-diffgrentiels,

operators, J. Math. SOC. Japan

3

(1971), 3 7 4 - 3 7 8 .

operators, Proc. Nat. Acad. Sci. USA 2nd ed., Asterisque

57,

2

(1972), 1185-1187.

Sac. Math. France, Paris, 1978.

Estimates for Pseudo-differential Operators [5]

187

H. 0. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal.

18 (1975), 115-131. [6] C. Fefferman, Lp bounds for pseudo-differential operators, Israel J. Math. 14 (1973), 413-417. [7] C. Fefferman and E. M Stein, Hp 129(1972), 137-193.

spaces of several variables, Acta Math.

[8] D. Goldberg, A local version of real Hardy spaces, Duke Math. J.

5 (1979),

2 7-42.

[9] L. Hdrmander, Pseudo-differential operators and hypoelliptic equations, Proc. Symp. Pure Math. X, pp.138-183, Arner. Math. Soc., Providence, 1967. [lo] A. Miyachi, Estimates for pseudo-differential operators of class S0,O’ to appear in Math. Nachr.. [ll] A. Miyachi, Estimates for pseudo-differential operators with exotic symbols,

preprint. [12] T. Murarnatu, Estimates for the norm of pseudo-differential operators by means of Besov spaces I, L -theory, preprint. 2

[13] L. Paivarinta and E. Somersalo, A generalization of Calder6n-Vaillancourt theorem to

Lp

and

hp, preprint.

[ 1 4 ] E. M.Stein and G. Weiss, On the interpolation of analytic families of

operators acting on HP-spaces, TGhoku Math. J. ( 2 ) 9 (1957), 318-339. [15] M. Sugirnoto, LP-boundedness of pseudo-differential operators satisfying

Besov estimates I, T I , preprints. [16] R.-H. Wang and C.-2.

Li, On the LP-boundedness of several classes of

pseudo-differential operators, Chin. Ann. of Math. Z B (2) (1984), 193-213.


Proceedings of the Analysis Conference, Singapore 1986

S.T.L.Choy, J.P. Jesudason, P.Y. Lee (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1988

189

A NOTE ON A LIFTING PROPERTY FOR CONVEX PROCESSES K . F. Ng

Chinese U n i v e r s i t y , HONG KONG. Liu

L.S.

Zhongshen U n i v e r s i t y , C H I N A . L e t E and F be l o c a l l y convex spaces, and T be an open m u l t i - v a l u e d map from E t o F such t h a t i t s graph i s a c l o s e d cone i n E

F and

x

k e r n e l k e r T complete m e t r i z a b l e , t h e n we show t h a t each compact sub-

F

s e t of

i s c o n t a i n e d i n t h e image by T o f a compact subset o f E.

F o l l o w i n g R o c k a f e l l a r [31, by a convex process i s meant a map T o f p o i n t s i n a l o c a l l y convex space E i n t o t h e subsets o f a n o t h e r l o c a l l y convex space F such t h a t o and x i n E. cone i n E

x

E

To, T(Xx) = ATx and Txl + Tx2 ~ T ( x ~ + fxo ~r a) l l

x

> 0,

xl,

x2

T h i s i s t h e case i f and o n l y i f t h e graph G(T) o f T i s a convex F.

T i s c a l l e d a c l o s e d convex process i f G(T) i s a c l o s e d convex

cone.

R e c a l l a l s o t h a t T i s s a i d t o be opened i f i t maps open s e t s t o open

sets.

The f o l l o w i n g r e s u l t was proved by Fakhoury i n t h e s p e c i a l case when T

was s i n g l e - v a l u e d ( s e e [ 2 ] o r [ l , P r o p o s i t i o n V I . 3.51. THEOREM 1.

L e t T: E

+

ZF be a c l o s e d convex process.

Suppose T i s open and

1 t h a t i t s k e r n e l T - ( 0 ) i s complete m e t r i z a b l e w i t h r e s p e c t t o t h e r e l a t i v e u n i formity.

Then e v e r y compact subset o f F i s c o n t a i n e d i n t h e image by T o f a

compact subset o f E . I f E, F a r e assumed t o be complete

Remark.

open mapping theorem ( s e e [5],

[6]),

m e t r i z a b l e then, by a g e n e r a l i z e d

t h e c o n d i t i o n "T i s open" can be r e p l a c e d

by "T i s o n t o F " . To b e g i n o u r p r o o f , we w r i t e $ f o r T-'.

Then 0 i s a l s o a c l o s e d convex p r o -

cess ( f r o m F t o E ) , and i s l o w e r semi-continuous ( 1 . s . c . )

i n t h e sense t h a t

t y E F : $ ( y ) fl w f $ } i s open f o r each open s e t w i n E, because T i s open. Note t h a t , i f y E F, x1 E $ ( y ) and x 2 E $ ( - y ) t h e n X I -+

$(O) E $(Y)

c

$ ( O ) - x2

Take c o u n t a b l y many c i r c l e d convex neighbourhoods I V i l y = l

Vitl

t

Vitl"

Vi

2

(1) o f o i n E with

such t h a t {V*: Il $ ( o ) 1 i s a f i l t e r base f o r t h e r e l a t i v e u n i -

f o r m i t y i n + ( o ) , where

1

K. F. Ng and L.S. Liu

190

v:

=' { ( X

X

1' 2

)

E

E2 :

X1-X2

E

vi].

By ( l ) , t h e r e l a t i v e u n i f o r m i t y f o r $ ( y ) i s a l s o determined by tVI

above manner f o r each y .

1 i n the

We now adopt an i d e a o f Fakhoury as p r e s e n t e d i n [ l ,

P r o p o s i t i o n V I . 3.51 t o a p p l y M i c h a e l ' s s e l e c t i o n Theorem :

For any g i v e n

compact subset K o f F t h e r e e x i s t s V 1 - s e l e c t i o n $1 o f $ on K, t h a t i s q 1 i s a continuous ( s i n g l e - v a l u e d ) f u n c t i o n f r o m K i n t o E such t h a t

D e f i n e a m u l t i v a l u e d f u n c t i o n $1 by $ 1 ( ~ )= $(Y)

n

[ J I ~ ( Y )+ V1l,

Then i t i s e a s i l y seen t h a t 91 i s 1 . s . c . and $,(y) one has V 2 - s e l e c t i o n $ 2 o f $l.

for all y I$,(y)}

E

K and a l l n.

$(K).

K. Hence

and {$,,I with

L e t $ ( y ) denote t h e l i m i t o f t h e Cauchy sequence

Do t h i s f o r a l l y i n K .

c o n t i n u i t y o f $n, $ i s c o n t i n u o u s on K and $ ( y ) Let K' =

E

i s convex f o r a l l y.

I n d u c t i v e l y we have

on t h e complete s e t $ ( y ) .

y i n K.

Y

E

$5 =) $(y)

Then K ' i s compact and T ( K ' ) 2 K.

i s a compact subset o f E, and i s mapped under

T

Then, by ( 2 ) and = T-l(y) f o r a l l

[Thus K ' fl T - l ( K )

t o t h e e x a c t image K ( n o t e t h a t

T - l ( K ) i s c e r t a i n l y c l o s e d as G(T) i s c l o s e d and K i s compact), i f T i s s i n g l e -

.

va 1ued ] REFERENCES

[l] [2] [3] [4] [5] [6]

De V i l d e , M., Closed graph theorems and webbed spaces (Pitman, 1978). Fakhoury, M., S d l e c t i o n s c o n t i n u e s dans l e s spaces u n i f o r m e s , C.R. Acad. Sc. P a r i s , 280 (1975), 213-216. R o c k a f e l l a r , R . T., Monotone processes o f convex and concave type, Mem. Amer. Math. SOC. 77 (1967). M i c h a e l , E., Continuous s e l e c t i o n s , Ann. o f Math., 63 (1956), 361-382. Ng, K . F., An open mapping theorem, Proc. Camb. P h i l . SOC., 74 (1973), 61-66. Ng, K. F., An i n e q u a l i t y i m p l i c i t - f u n c t i o n theorem, P r e p r i n t .


191

WEAK L,-SPACES AND WEIGHTED NORM INEQUALITIES FOR THE FOURIER TRANSFORM ON LOCALLY COMPACT VILENKIN GROUPS

C. W.Onneweer Department of Mathematics and Statistia University of New Mexico Albuquerque, New Mexico 87131 USA

In this paper we consider the weighted norm inequality problem for the Fourier transform for functions defined on a certain class of topologicala(kroups. We study the case in which the weight functions belong to suitable we L,,-spaces.

1. INTRODUCTION

In his 1978 survey lecture on weighted norm inequalities for certain operators, B. Muckenhoupt posed the so-called weighted norm inequality problem for the Fourier transform 181. This is the problem of characterizing, for given p and q with 1

< p,q < 60, those.

nonnegative measurable functions u and u on R or, more general, on R" so that the inequality

Iljull, ICllfullp

(1.1)

holds for all Lebesgue integrable functions f. Significant progress towards the solution of this problem has been made by, among others, Muckenhoupt himself [Q],[lo],by W. B. Jurkat and G. Sampson [7]and especially by H. P. Heinig and his co-authors J. J. Benedetto and R. Johnson [l], [2]and 15). The characterizations obtained by these authors for the weight functions u and u that are equivalent to (1.1) all impose the restrictions that both u and u are radial functions and that they satisfy certain monotonicity conditions. Conditions imposed on u and u that imply (1.1)and that deal with less restricted classes of weight functions are often rather difficult to apply and it seema likely that techniques different from those used in [2], [7]or

[Q]will be needed to solve Muckenhoupt's problem for nonradial functions u and u.

C W. Onneweer

192

In this paper we consider Muckenhoupt's problem for functions defined on certain groups different from R or R" and we study the case in which the weight functions u and u belong to certain weak L,-spaces.

In the remainder of this section we briefly describe the claes of group considered here.

In addition, we introduce most of the notation to be used and we state some of the known results that are needed afterwards. Section 2 will contain the statements and proofs of some sufficient conditions for (1.1) to hold. The final section complements Section 2 by presenting some conditions implied by inequality (1.1). Throughout this paper G will denote a locally compact Abelian topological group with a suitable family of compact open subgroups, cf. [4, 84.11. This means that there exists a

sequence (G,)?'

such that

(i) each G. is an open compact subgroup of G,

< bo,

(ii) Gn+l 5 G, and order (G./G,+l) (iii) UZmG, = G and

G, = (0).

Moreover, we shall assume that G is order-bounded, that is, (iv) sup{order(G,/Gntl) ; n E Z} < 00. Such groups are the locally compact analogue of the Vilenkiil groups [12]. Several examples of such groups are given in [4, 54.1.21. Additional examples are the padic numbers and, more general, the additive group of a local field [Ill. Let I' denote the dual group of G and for each n E Z let,'l denote the annihilator of G,, that is,

r, = ( 7 E r ; ~ ( z=) 1

for all z E G,}

Then we have, cf. [4, $4.1.43, (i)* each r, is an open compact subgroup of (ii)*

r, r,+, and order(I',+l/rn)

(iii)* nZmr. = {I} and

r,

= order(G,/G,+l),

em r, = r.

.

Weighted Norm Inequalities f o r the Fourier Transform

If we choose Haar measures p on G and X on I'

so

that p(C0) =

193 X(I'0)

= 1 then

PIG,) = (A(rn))-l for each n E Z. We set m, = X(I',).

If we define the function d : G x G

-+

R+ by

then d defines a metric on G x G and the topology on C induced by this metric is the

llzll by 1 1 ~ 1 1= d(z,O); then llzll = (mn)-I if and only if z E G. \ Gn+l. In a similar way we can define a metric 2 on 'I x I'; if we set llrll = i ( 7 ,l), then llrll = m, if and only if 7 E I?,+, \ I',. A function same as the original topology on G. For z E G we define

f :G

-+

C is called radial if f(z) = f(llzll); thus, a radial function on G is constant on

each subset G, \ G,+l in G (n E Z). A similar definition can be given for radial functions on r.

For p with 1 5 p 5 00 we denote its conjugate by p', thus p' = p/(p - 1) if 1 < p and p' = 1 or by

00

< 00

if p = 00 or 1. For a given set A we denote its characteristic function

The symbols

and

will be used to denote the Fourier transform and the inverse

Fourier transform, respectively. It is easy to see that for each n E Z we have

As usual, C will denote a constant whose value may change from one occurrence to the next. We now give the definition of the Lorenta spaces. Let ( X , A , p ) be a measure space and let f be a measurable complex-valued function on X. For y

> 0 let f.(y)

= p({z E

> y}) and define f' : R+ R+,the non-increasing rearrangement of f, by f'(t) = inf{y > 0 ; f,(y) 5 t } . The Lorents space L ( p , q ; X )is the set of all measurable functions f on X such that IlfllP,r < 00, where X ; If(z)l

IlfllP,

=

{

sup{f*(t)tl/' ; t

< p,q < 00, if o O}

The spaces L(p, 00;X)are also known as the weak P-spaces or the Marcinkiewica spacea on X.

For future reference we etate here some properties of the Lorents spaces, see [3, s51.3 and 5.31 for further details.

C W. Onneweer

194

If0 < p

< m and 0 < q1 5 qz I m then

If 0 < p < 00 then f E L(p,oo;X) if and only if

The Marcinkiewics interpolation theorem for Lorents spaces. Let ( X , R , p ) and Y , B , v ) be two o-finite measure spaces. Let 1 5 pi,qi, ri

I 00(i

= 0 , l ) with po # p1 and qo # ql.

Assume that T :L(pi, ri; X)

+ &(pi, 00;Y )

is a bounded linear operator for i = 0 , l . Let l / p = (1 - B)/po B)/qo

+ B/pl

and l/q = (1 -

+ B/qI for some B with 0 < 0 < 1. Then for each r, 1 5 r 5 00 the operator T :L(p, r;X )

+ L(q, r;

Y)

(1.6)

is a bounded linear operator.

2. SUFFICIENT CONDITIONS

Theorem 1. Let 1 < p 5 2 and 1 < p 5 q

Assume that u :

-+

R+ belongs

R+ is a function such that w - l belongs to L ( / 3 z , ~ ; G ) , and l/Bz = l/r - l / p for some r such that 1 < t 0 so that for all f E Ll(G) we have

The proof of Theorem 1 will be preceded by a lemma that is essentially the analogue on G and

r of Lemma 1 in IS].

L e m m a 1. Let 1 0 so that for all f E LI(C) we have Ilfull, 5 CllfllP,,. Proof. For f E Ll(G) define Tf : r

--*

C by Tf(7) = f(r)(u(7))", where o = -Bq'/p'

and define the measure do on J? by da(7) = (u(7))'dX(7), where b = /3

+ Bq'/p'.

We first

Weighted Norm Inequalities for the Fourier Transform

prove that T is of weak type (1,l) from L l ( G , d p ) to Ll(l',du). For each t

195

> 0 we have

'Therefore,

I

E

c n=O ( 2 V I If1I11W + ' t / I111I11

because u E L(/3,m;l")lcf. (1.5). Thus our choice of

o

9

and b implies that

a(Et) I C ~ ( 2 n t / l l f l l ~ ) - 1= cllflll/t . n=O

Next we show that if we define a by a = (p'

+ q')/p', so that a' = (p' + q')/q', then T is of

type (a,a') from &(GIdp) to Lot(r,do).

because our choice of a implies that oa'

+ b = 0 and 1 < a I 2,so that we can apply the

Hausdorff-Young inequality. Furthermore, since

it follows from the Marcinkiewicz interpolation theorem that

that is, since aq + b = qI

C W. Orineweer

196,

Lemma 1 easily yields the following sufficient conditions for the one weight norm inequality for the Fourier transform. Corollary 1. Let 1 0 so that for ail f E & ( G )

3

R+ and if

v-l

E

L(p,co;G)then there exists a C > 0 so that for all

E L1(G) we have

Ilflb 5 CllfUIl# *

(2.3)

Proof (a). Since p 5 q the first part of the corollary follows immediately from Lemma 1,

(1.3)and (1.4). (b). For any f E L1(G)n Lp(G)and any p E $(r), where S(r) is the set of all functions on I' with support in some r'I and constant on the cosete of some

rt in I',we have, cf. 16,

(31.4811,

Since u-I E L(p,ao;G)it follows from part (a), after interchanging the d l e of G and

r,

that

119~-'llo I CllPllP Thus, since S(r) is a dense subset of L,,(I'), cf. 111, Chapter II, Proposition (1.3)], we see that

Remark. If in Corollary l(a) we replace the assumption that u

E

L(p,co;I')by u

E

L,(r) and choose q = p' then it follows immediately from the Hausdoff-Young inequality that Iljull, 5 Cllfllp. Therefore, Corollary l(a), or Lemma 1, can be considered as a generaliration of the HausdorlT-Young inequality.

Weighted Norm Inequalities for the Fourier Transform

Proof of Theorem 1. Since 1 < r < q < r' and

Next, choose c so that 0

l/pl

197

= l/q- l / f , Lemma 1 implies that

< c < l/r' and define rl and p1 by l/r1

= l/r+c and l/pl = l/p+c.

Then

llfll::

=

I

If(z)~(z)l'l(~(z))-r'dCc

r

((fU)'l)*(

t)(u-'a)*( t ) dt

.

Since u-l E L(&, 00;G) we have u-'l E L(pZ/rl,00;G). Therefore,

I Ct-r'l@a

(U--'l)'(t)

=

cydP1-1

.

Thus,

lIflL,= I Ilfllri

/m

5 C(

( ( f U ) * ) ( t ) p P / P ' -1dt)+l

0

= Cllf~llPlrl

If we define rz and p2 by l/rz = l / r - c and l/pz = l/p - c, then a similar argument shows that

Ilfllrs,= I C l l f ~ l l p s r tThus it follows from (1.6), the Marcinkiewicr interpolation theorem for Lorents rpaces, that

Ilfllr,q I Cllf4lpa Icllfullp

8

because p 5 q. Combining this inequality with (2.4) we may conclude that

ICllf4lP 9

113~11q

which concludes the proof of Theorem 1. We now show that Theorem 1 implies a version of Pitt's Theorem for functions on G. This is an easy consequence of the following simple facts that will also be used in Section 3.

L e m m a 2. (a) If p(z) IC ~ ~ z ~on~C- for l ~ some a Q > 0 then (p E L(a,00;C). (b) If $ ( 7 ) 5 C ~ ~ ~on ~r for ~ some - l ~Q > a 0 then

+ E L(a,oo;I').

198

C.W. Onneweer

Roof. (a)

Fix t

> 0 and c h o w no E Z

so that C(m,o-~)l/o< t

I C(mno)l/o,where C

is the same constant as in the statement of (a). Then {z E G ; p(z) > t } c {z E G ; p(z) > C(m,o-l)l/o}

c because (p(z)

Gno

1

I C(m,)l/o for z E G, \ G,,+l(n E Z) and

the sequence (C(rn,,91/0)~m is

monotone increasing. Therefore,

due to our choice of no. Thus (1.5) implies that

(p E

L(a,00;G).

The proof of (b) is similar and will be omitted.

Corollary 2. (Pitt's Theorem on C) Let 1 0

Proof. Let r = (a + l/p)-I. Then 1

60

0, t h e r e e x i s t s g

and f,

P(L1(G)).

E

Cc(G) such t h a t

E

Ilf

-

0 there exists s

E

For every f

H(Ap(G), L l ( G ) ) .

E

o r we can say t h e r e e x i s t s a sequence xo,

+

0

T

+...+

(Tf) x1

P(Ll(G)), Tf

E

Ll(G).

G such t h a t

E

-

IITf + T s ( T f ) l l l > 2IITfII1

I I T ~( T f )

E

..., xn, ..., { x n ]

XI,

(Tf)lll

T

E,

> (n+l)iiTfiil

-

c G,

such t h a t

nE.

'n

B e s i d e s , we can e l e c t { x n } s a t i s f y i n y t h e c o n d i t i o n o f t h e lemma.

Notice that

T i s a m u l t i p l i e r , so we have (Tf)

IIT xO

+

T

( T f ) +...+ x1

I I T ~ I 0, t h e r e e x i s t s an i n t e g e r N such t h a t

vi

E

Lq(G),

215

Multipliers of Segal Algebras

Let

we h a v e l l f - g

a.

1,

Bi

E

11 AP

Cc(G),

0 such that x E D implies x E

-

Here "co"

C(D \ Be(x))

stands for closed convex hull and B,(x)

= {y E X : Ily - xII

a}

for a fixed x* E X* and any real a strictly less than supIx*(D) I. The separation theorem guarantees that a subset D of X is dentable if and only i f it has slices of arbitrarily small diameter. The next theorem amalgamates theorems of Rieffel [16]. Huff [11] and hvis-F'helps [ 5 ] .

221

Differentiation in Banach Spaces

Theorem 21. A Banach space has RNP if and onlv if every bounded subset of X is dentable. A Banach space X fails RNP if and onlv if it contains a &bush. The proof boils down to showing that if every subset is dentable, then for some difference quotient martingale (fr ) n

satisfies

f17 II = 0. m Conversely if X contains a non-dentable subset, it is possible to define a Lipschitz f : [O.l] + X such that - fT il n is bounded away from 0 for all n. It should be noted here that there are (non-dual) Banach spaces that fail RNP yet have no 6-trees. The first known example of this is the Bourgain-Rosenthal space BR [2]. Related t o dentability is the Krein-Milman property.

Definition 22. A Banach space has the Krein-Milman property (KMP) if every closed bounded convex set in X has an extreme point A separation theorem argument together with the Bishop-Phelps theorem guarantees that if X has KMP, then every closed bounded convex subset of X is the norm-closed convex hull of its extreme points. A simple and beautiful fact due to Lindenstrauss follows: Theorem 23.

If X

has RNP

then X

has KMP.

Idea of Proof. Take a closed bounded convex set D.

slice of diameter < 1

slice of diameter

(3) Keep slicing.

in X such that (a) x

n

=

"m

+

2

%n+l

0. A 6-Rademacher tree is a bounded sequence (x,)

Differen tiation in Banach Spaces

231

and (b)

llxlll 1 6 Ilx2 - x311 1 26 11x4 - x5

+ x6 - 9 1 1 1 46.

etc.

Both the trees given in Examples 1.17 and 1.18 are 1-Rademcher trees. Using a procedure similar to the discussion preceeding Definition 1.16, we can see that a Banach space X contains a 6-Rademacher tree if and only if there is a Lipschitz function f : [O,l] + X

martingale (fa ) n

such that the dyadic difference quotient

satisfies

This proves:

Fact 5. If a Banach space contains a 6-Rademacher tree, then X fails CCP. The converse is open but is known to be true in dual spaces, see Riddle-Uhl [15]. We feel that the converse is unlikely to be true in non-dual spaces but have no concrete evidence to offer. With regard to dentability, we can offer the following definition. Definition 6. A subset D of a Banach space X is not weak-norm-one dentable if there exists an E > 0 such that for all finite subsets F of

*

D there is a norm one xF in X* such

that if

x E F, then

Theorem 7. If the Banach space X contains a bocnded set that is not weak-norm-one dentable, then X fails CCP.

Proof. Refer to the proof of Huff [ll] and note that a suitable modification of Huff’s proof gives a difference quotient martingale (fa ) and a sequence n

*

(x,)

*

of norm-one members of X

such that for some

E

>

0

M. Petrakis and J.J. Whl, Jr.

238

*

lXnf*

-

*

Xnfnl

2

&

n+1 for all n = 1.2.

...

Hence for all n = 1.2,...

and this combined with Theorem 2 shows that X fails CCP. The converse is open. Another (possibly related) dentability condition is given in the next definition. Definition 8 . Let D be a bounded subset of a F3anach space X. Let E and be positive real numbers. A point x in D is called an & E,h denting point if whenever X

m

XE.h =

1

OiXi

i=1 W

with xi E: D, oi

2 0 and

1

oi = 1.

*

then for each norm-one x

E

*

X

i i=1

where the index set A

*

is defined by

X

A

*

= {i : Ix*(xi)

- X*(X)~

>

A}

X

with a little care, one can prove the following fact.

a 9. If every bounded subset of X has an E - A denting point for all s , h > 0. then x CCP. The proof is not terribly difficult. It involves taking a Lipschitz f : [O.l] + X. letting E + 0 and A + 0 and using the A ' s to define partitions rn such that

F

239

Differentiation in Banach Spaces

We'll give an alternate proof of Fact 9 a bit later. Next we come to a theorem of Bourgain's [l] which we do not believe is yet understood to the full. Theorem 10. If a Banach space X

fails CCP, then X

contains a 6-tree.

The 6-tree produced by Bourgain's argument is not the tree corresponding to a non-Dunford-Pettis operator from L1[O,l]

into X but rather the

operator corresponding to Bmrgain's tree is tree related to a Dunford-Pettis operator. We feel that Bourgain's tree has another unnoticed property that m y in fact characterize CCP. After many hours of sweat and grief we are still unable to isolate this property. Bourgain's theorem has the following immediate corollary. Corollary 11. If a Banach space X contains no 6-trees, then X has CCP. The converse of Corollary 11 is false. The dual JT* of the James Tree space has CCP and does contain 6-trees.

Still there is another condition due to Kunen and Rosenthal [17] that eliminates 6-trees. E > 0 and let D be a bounded subset of a Banach space X. A point x in D is an E-stronEly extreme point of D if there is a 6 > 0 such that if xl, x2 are in D and there is a point u on the line

Definition 12. Let

segment from x1 to x2 with the property that Ilu - x II

1

0 such t h a t

(s)

M

!d

.k

< a f o r every s

E

8

E. L e t w

E

satisfying

m

k

k

=

i s n o t g-essent-

E

i s continuous and $ (w) = 0 it f o l l o w s t h a t sup{ 15 ( v )

M

is

f?

n supp(p) # @ o r the distance

i s g - e s s e n t i a l l y bounded whereas i n t h e former c a s e E;

S i n c e J,

c

Since

I n t h e l a t t e r case i t i s clear t h a t 5

i a l l y bounded. Indeed, i f it were, t h e n t h e r e would e x i s t E u

M'

k

n Supp(p).

1 ; vcBr (w)

2 r,

f o r e v e r y r > 0 where B (w) i s t h e b a l l i n Em w i t h c e n t r e w and r a d i u s r . I n p a r t i c u l a r , i f r > a, then

15 ( v ) I

M

> CL f o r a l l v

empty. It f o l l o w s t h a t E ( B ( w ) ) = g ( E ) P ( B , ( w ) ) i s an open s e t t h e d e f i n i t i o n of SuppQ) c o n t r a d i c t i o n . Accordingly,

n Supp(&) #

M

c

@.That

1/$

E

B (w) and hence E n B r ( w )

= Q(E n B r ( w ) )

implies t h a t w

is

= 0. S i n c e B r ( w )

Supp(p) which i s a

i s n o t P - e s s e n t i a l l y bounded i f and o n l y i f

k

2 . is, q = l ( ~ j - S j ) 1 s n o t i n v e r t i b l e i n L ( X ) i f and o n l y i f

n Supp(R) # @ which i s what was t o be shown.

0

)?

So, t o complete t h e proof o f Theorem 1 it s u f f i c e s t o show t h a t t h e r e e x i s t s E

R m such t h a t

i d e n t i t y of

2

c

k

n Supp(g) # @, where

E is

( hence, a l s o of i t s s c a l a r p a r t

t h e j o i n t r esol ut i on of t h e

2

) . Let

E

Supp(g) and w r i t e

A . = a . + i b . l < j < m , w i t h a . and b . b e i n g real numbers. Then 1 I' 1 1 < m ( ~ J . - A . ) ~ = 0 i f and o n l y i f 1=1 1 1 2 9 (p.-a.)2 = b. j=1 1 1 j=1 1

k

E

IRm s a t i s f i e s

and s i m u l t a n e o u s l y

93 = 1 ( p1. - a 1. ) b 1. =

(2.2)

But, c o n s i d e r i n g centre

lRm w i t h normal

as a v a r i a b l e , (2.1) i s t h e e q u a t i o n of a s p h e r e i n IRm w i t h 2 4 and r a d i u s 11$11 = b.) and ( 2 . 2 ) i s a hyperplane i n

.,a,)

= (al,..

B

and p a s s i n g through

i s t simultaneous s o l u t i o n s

every

E

0.

of

5.

(e 1=1

1 So, i f m 2 2 , t h e n t h e r e c e r t a i n l y ex-

( 2 . 1 ) and ( 2 . 2 ) .

Actually, t h i s i s t h e case f o r

Supp(J1). The proof o f Theorem 1 i s t h e r e b y complete.

0

REMARK. The i d e a of t h e proof of Theorem 1 a p p l i e s t o o t h e r commuting

m-tuples which have an "adequate" f u n c t i o n a l c a l c u l u s . For example, l e t commuting m-tuple of

*

j'

Cm(E)

-+

be a

regular genemZized scalar operators i n a Banach space x c21. Then e a c h T . h a s a r e g u l a r s p e c t r a l d i s t r i b u t i o n

i n t h e s e n s e of C. F o i a s 0

2

L ( X ) such t h a t 0 . ( A )

1

= T. I'

1 l < j < m , where A d e n o t e s t h e i d e n t i t y

Spectral Subsets of Rn’ f o r Commuting Sets of Operators

a1

f u n c t i o n i n E. By r e g u l a r i t y , t h e t e n s o r p r o d u c t @ =

Q

. ..@

241

Qrn:

Cm(tm)

-f

L(X)

lsjsm, e x i s t s , i s a g a i n a s p e c t r a l d i s t r i b u t i o n [ 2 ; Ch.4, P r o p o s i t i o n m

of t h e O j ,

3 . 1 1 and s a t i s f i e s @ ( I T . )= T . where n . : E -+ E, lSj<m, i s t h e p r o j e c t i o n o n t o J I J m the j - t h c o - o r d i n a t e . Furthermore, @ i s a C ( E m ) - f u n c t i o n a l c a l c u l u s f o r 2 ( i n t h e s e n s e of

El1 1

w i t h compact s u p p o r t , S u p p ( 0 ) . Here S u p p ( @ ) i s t h e smallest

closed set K c Em s u c h t h a t : @ ( f )= 0 whenever f from K.

If

6

lRm and $I

m

E

C (Ern) h a s s u p p o r t d i s j o i n t

is t h e f u n c t i o n d e f i n e d i n t h e p r o o f o f Lemma 2 , t h e n

i?

i t f o l l o w s from t h e r n u l t i p l i c a t i v i t y of @ t h a t t h e o p e r a t o r @ ( $ I) =

i s i n v e r t i b l e i n L ( X ) i f and o n l y i f t h e r e e x i s t s 5 c i d e s w i t h l/$ i f li,

I?

E

9 (u

-T ) J=I j j C (Em) s u c h t h a t 5 c o i n -

2

m

2

i n a neighbourhood o f S u p p ( @ ) . But, t h i s i s t h e case i f and o n l y

k # 0 i n a neighbourhood o f Supp(4r) which, by d e f i n i t i o n o f

p a c t n e s s of S u p p ( @ ) , i s e q u i v a l e n t t o

y(2) and h e n c e , a r g u i n g a s b e f o r e ,

=

{k

6

x

k

x

k

and t h e com-

n S u p p ( 0 ) = 0. So, i t f o l l o w s t h a t

n m ;x n ‘!L

s u p p ( @ )#

@I

y ( 2 ) i s non-empty.

Examples o f r e g u l a r g e n e r a l i z e d s c a l a r o p e r a t o r s are s p e c t r a l o p e r a t o r s of finite-type, type

(B),

p r e s p e c t r a l o p erat o rs of scalar-type,

w e l l bounded o p e r a t o r s of

p o l a r o p e r a t o r s and some m u l t i p l i c a t i o n 0 p e r a t o r s ; s e e c 8 ; 531, f o r

example. ACKNOWLEDGEMENT

D i s c u s s i o n s w i t h Alan McIntosh and Alan Pryde are g r a t e f u l l y acknowledged. REFERENCES

[I1 [21 [31 [4l

[51 C6l C71

C8l

st.,

A l b r e c h t , E. and F r u n z s , Non-analytic f u n c t i o n a l c a l c u l i i n s e v e r a l v a r i a b l e s , M a n u s c r i p t a Math. 1 8 ( 1 9 7 6 ) , 327-336. C o l o j o a r g , I . and F o i a g , C . , Theory of G e n e r a l i z e d S p e c t r a l O p e r a t o r s (Math. & A p p l i c a t i o n s No.9, Gordon & Breach, N e w York-London-Paris, 1968). Dunford, N . and S c h w a r t z , J . T . , L i n e a r Operators 111: S p e c t r a l O p e r a t o r s ( W i l e y - I n t e r s c i e n c e , N e w York, 1 9 7 1 ) . K l u v L e k , I. and Kov6Fikovi, M . , P r o d u c t of s p e c t r a l m e a s u r e s , Czechoslovak Math. J. 1 7 ( 9 2 ) (19671, 248-255. McIntosh, A. and P r y d e , A . , The s o l u t i o n of s y s t e m s o f o p e r a t o r e q u a t i o n s u s i n g C l i f f o r d a l g e b r a s , Proc. C e n t r e Mathematical A n a l y s i s , C a n b e r r a , 9 (1985), 212-222. McIntosh, A . , C l i f f o r d a l g e b r a s and a p p l i c a t i o n s i n a n a l y s i s , L e c t u r e s a t t h e U n i v e r s i t y o f N.S.W.-Sydney U n i v e r s i t y j o i n t a n a l y s i s s e m i n a r , 1985. McIntosh, A. and P r y d e , A., A f u n c t i o n a l c a l c u l u s f o r s e v e r a l commuting o p e r a t o r s , I n d i a n a Univ. Math. J . ( t o a p p e a r ) . McIntosh, A . , P r y d e , A. and R i c k e r , W . , Comparison of j o i n t s p e c t r a f o r c e r t a i n classes of commuting o p e r a t o r s , S t u d i a Math. ( t o a p p e a r i n V o l . 88).

C u r r e n t a d d r e s s : C e n t r e f o r Mathematical A n a l y s i s , A u s t r a l i a n N a t i o n a l Univ e r s i t y , Canberra 2600, A u s t r a l i a .


Proceedings of the Analysis Conference,Singapore 1986 S.T.L. Choy, J.P. Jesudason, P.Y. Lee (Editors) 0 Elsevier Science Publishers B.V. (North-Holland),1988

249

THE CLASS OF MOBIUS TRANSFORMATIONS OF CONVEX MAPPINGS

Rosihan Mohamed A l i School of Mathematical and Computer S c i e n c e s U n i v e r s i t i S a i n s Malaysia, Penang, Malaysia.

I.

INTRODUCTION

Let S d e n o t e t h e c l a s s of a n a l y t i c u n i v a l e n t f u n c t i o n s f d e f i n e d i n t h e u n i t disk D If f

=

E

{ z : IzI < 11 and normalized so t h a t f ( 0 ) = f ' ( 0 ) S and w k! f ( D ) , t h e n t h e f u n c t i o n

i

=

wf

1 (w

-

-

1 = 0.

f) A

belongs again t o S.

T h i s MGbius t r a n s f o r m a t i o n f

+

f i s important i n the

a n a l y s i s of t h e c l a s s S and o t h e r r e l a t e d c l a s s e s . I f F i s s u b s e t of S , l e t A

h

F = {f : f

E

F, w

f(D)}.

C*

E

Here C* i s t h e extended complex p l a n e which i s C U

.

{m}

S i n c e we a l l o w w =

m,

h

i t f o l l o w s t h a t F c F c S , and s i n c e t h e composition of normalized Mobius t r a n s f o r m a t i o n s i s a g a i n a normalized Msbius t r a n s f o r m a t i o n , i t f o l l o w s t h a t a

h

F = F.

I n t h i s a r t i c l e , we s h a l l c o n s i d e r t h e s u b c l a s s K o f S c o n s i s t i n g of t h o s e f u n c t i o n s f i n S which map t h e u n i t d i s k D conformally o n t o convex domains. A

Simple examples show t h a t K i s a p r o p e r s u b s e t of K , and so one e x p e c t s t h a t h

some i n t e r e s t i n g p r o p e r t i e s o f K a r e n o t i n h e r i t e d by K. h

We d e t e r m i n e t h e r a d i u s o f c o n v e x i t y f o r K , t h a t i s , t h e r a d i u s of t h e l a r g e s t d i s k c e n t e r e d a t t h e o r i g i n which i s mapped o n t o a convex domain by h

each f u n c t i o n i n K.

We a l s o f i n d t h e l a r g e s t d i s k c e n t e r e d a t t h e o r i g i n which T h i s d i s k i s c a l l e d t h e Koebe

i s c o n t a i n e d i n t h e range of each f u n c t i o n i n K. h

d i s k f o r K.

h

The s i z e of t h e Koebe d i s k and t h e r a d i u s of c o n v e x i t y f o r K have

a l s o been i n d e p e n d e n t l y determined by Barnard and Schober [ 2 1 . A

d e r i v e s h a r p upper and lower bounds f o r l i ( z ) I , f

F i n a l l y we

h

E

K.

In a l l cases,the results

o b t a i n e d a r e d i f f e r e n t from t h o s e of t h e c l a s s K. We w i l l need t h e r e s u l t s o b t a i n e d by Barnard and Schober [ I ] ,

and t h e follow-

i n g theorems summarized t h e i r r e s u l t s . h

Theorem A .

If

X

: K

assumes i t s maximum o v e r

+

R i s an a d m i s s i b l e c o n t i n u o u s f u n c t i o n a l , t h e n 1

k

h

a t a f u n c t i o n f = wf

/ (w

-

h

f ) where e i t h e r f i s a

h a l f - p l a n e mapping o r e l s e f i s a s t r i p mapping and w i s a f i n i t e p o i n t of i t s boundary af ( D )

.

M.A . Rosihan

250 Barnard and Schober [ I ]

a l s o observed t h e f o l l o w i n g a p p l i c a t i o n of

Theorem A: Let A be d e f i n e d by

A(?)

= Re { O ( l o g [ ? ( z ) /z1)

1, { O } i s fixed.

where 0 is a nonconstant e n t i r e f u n c t i o n , and z E D

By a r e s u l t

So by choosing

of Kirwan [ 5 ] , A i s a c o n t i n u o u s f u n c t i o n a l as d e f i n e d i n [ I ] .

O(w) = +w, Theorem A i m p l i e s t h a t an e x t r e m a l f u n c t i o n t o t h e problems of maximum and minimum modulus i s e i t h e r a h a l f - p l a n e mapping o r i s g e n e r a t e d by a s t r i p mapping.

Notice t h a t t h e e x t r e m a l s t r i p domains f ( D ) need n o t be

symmetric about t h e o r i g i n . Theorem B.

2 I f i ( z ) = z + a2z + la21 c 2 x-'

where x

sin x

belongs t o

k,

-

1.3270,

cos x

M

o =

then

= 2.0816 is t h e unique s o l u t i o n of t h e e q u a t i o n cot x = l/x

i n t h e i n t e r v a l (0,

-

x/2

Equality occurs for the functions e

TI).

-ia-

f(e

ia

z),

a

f R,

where ? ( z ) = f ( l ) f ( z ) / [ f ( l ) - f ( z ) ] and f i s t h e v e r t i c a l s t r i p mapping d e f i n e d

2.

THE RADIUS OF CONVEXITY

Since w e s h a l l be concerned w i t h t h e f a m i l y

k,

i t w i l l be convenient t o drop

h

the

i n r e f e r e n c e t o f u n c t i o n s i n K. Observe t h a t f maps I z I = r o n t o a convex curve y ( r ) i f t h e t a n g e n t t o y ( r )

a t t h e p o i n t f ( r e i e ) t u r n s c o n t i n u o u s l y i n an a n t i c l o c k w i s e d i r e c t i o n as 8 increases. z = reie,

Since t h e tangent vector t o y ( r ) a t w = f(reie)

i s given by i z f ' ( z ) ,

a n a l y t i c a l l y , t h e c i r c l e I z I = r i s mapped o n t o a convex curve i f and

only i f

a

>

arg tizf'(z)}

0,

z = re

i0

,

o r , e q u i v a l e n t l y , i f and o n l y i f

I + Re{zf"(z)/f'(z)} 2 0

(2.1) on t h a t c i r c l e .

Since a convex curve bounds a convex domain, t h e r a d i u s of

convexity i s t h e least upper bound of r f o r which (2.1) h o l d s .

L e t u s d e n o t e t h e s h a r p bound f o r t h e second c o e f f i c i e n t i n A

=

maxtla

I 2

: f ( z ) = z + a z2

2 Theorem B g i v e s t h e e x p l i c i t v a l u e of A h

Lemma 2.1.

2

+

...

E

by A2, t h a t i s ,

;I.

2'

For each f E K , Re{zf"(z)/f'(z))

>

2r(r

-

2 A 2 ) / ( 1 - r ),

(21

=r< I.

M6bius Transformations

25 1

h

Proof. -

Given f

K , f i x 5 i n D.

E

g(z)

=

[f((z

+

Let

5)/(1

-

?z))

+

f(c)l/f~( O ) be a l o c a l l y i n t e g r a b l e s e m i -

group of bounded l i n e a r o p e r a t o r s on X .

That i s , T ( s + t ) = T ( s ) T ( t ) f o r a l l s , t >

ti- and A X . t L e t S ( t ) , t > O , be t h e o p e r a t o r d e f i n e d by S ( t ) x : = / T ( s ) x d s (x€X), and l e t

0 , and T ( . ) x i s Bochner i n t e g r a b l e o v e r ( 0 , t ) f o r a l l

(x€X) f o r t h o s e A

R (A) be t h e o p e r a t o r d e f i n e d by R s ( h ) x : = & S I:e-xut(u)xdu

f o r which t h e l i m i t e x i s t s f o r a l l A X .

A s i n [6],

we denote o:=inf{u€(--,-);

R (u) e x i s t s ) ,

a :=inf{uE(--,m);

R (A) i s a n a l y t i c f o r a l l h w i t h Keh>u),

w o : = i n f { t - l log1 I T ( t )

I I;

t>O).

-

I t i s c l e a r t h a t u ~ u a ~ w o .For t h o s e X w i t h R e h > w o t h e Bochner i n t e g r a l s R(X):= -xu AJoe S(u)du e x i s t and form a pseudo-resolvent ( c f . L1, p.510]), which h a s R ( - ) as i t s e x t e n s i o n t o t h e s e t { A ; s o l v e n t on

Reh>a].

Thus R

( a )

i s a l s o a pseudo-re-

{A; Reh>oa).

Uniform e r g o d i c theorems a r e concerned w i t h t h e uniform o p e r a t o r convergence of t h e Cesaro mean t - ' S ( t )

as t+- and of t h e Abel mean ARs(X) a s X 4 ' .

one assumes ' ' ~ 0 5 0 " o r t h e even s t r o n g e r c o n d i t i o n

[l,

Theorem 18.8.41, L2

T(*) i s uniformly ergodic. O, and T(

a )

i s unifromly Abel-ergo-

dic.

(5)

I /T(t)S(u)I / = o ( t ) ,

The e q u i v a l e n c e of (1)

f o r a l l u>O, and R ( R s ( l ) - I )

(t-)

is closed.

and ( 3 ) h a s been proved i n 1 6 , Theorem 41

(2),

.

To

s e e t h a t (4) and (5) a r e also e q u i v a l e n t c o n d i t i o n s , we prove t h e f o l l o w i n g lemma. ( i ) I f T(*) i s uniformly Cesaro-ergodic,

Lemma 2 . (t-)

then

I IT(t)S(u) I I=o(t)

f o r a l l u>O. ( i i ) If

o ( t ) (t-t-)

roof. ( i ) Let 2.3]),

I IT(t)S(u) I I=o(t)

I / T ( t ) R s ( p ) I I=

f o r a l l u>O, then

(t-)

f o r a l l v>oa. P:=u,-

-1

im t L=

we have t h a t u,-$Q

s(t).

S i n c e T ( t ) S ( u ) = S ( t + u ) - S ( t ) ( s e e L4, Lemma t - l S ( t + u ) - u o - $ g t-'S(t)=P-P=O.

t-'T(t)S(u)=u,-$g

( i i ) For h>wo we have t h a t T ( t ) R( A ) =T( t ) /:Ae-AUS

so t h a t

1 IT(t)R(h) I I/t 5 A/;e-AU(I

minated convergence theorem.

( u ) du=A/:e-AUT(t)

jl+/A-vI

A l o c a l l y i n t e g r a b l e semigroup T(.)

converges s t r o n g l y t o I as A-.

g e n e r a t o r of T(.). t+O

+.

T(.)

on { E C ; R e k >

and s o

I IRs(v) 1111 IT(t)R(X) I I/t

d e n s e l y d e f i n e d and c l o s a b l e .

by Lebesgue's do-

Since R ( - ) i s a pseudo-resolvent

oa}, w e have R s ( v ) = R ( X ) + ( A - v ) R ( A ) R s ( ~ )

I IT(t)Rs(v) I I / t 5

S ( u ) du

I

I T ( t ) S ( u ) I / t ) d u + O a s t-,

-.O

a s t-.

i s s a i d t o b e of c l a s s ( 0 , A )

The o p e r a t o r

Ao:x +

i$irn+

i f ARIA)

t -1 ( T ( t ) - I ) x

is

The c l o s u r e A of Ao i s c a l l e d t h e i n f i n i t e s i m a l

i s of c l a s s (C,)

i f i t i s s t r o n g l y convergent t o I a s

I n t h i s c a s e we have A=Ao.

I t i s known 16, P r o p o s i t i o n 7 1 t h a t i f T ( - ) i s of c l a s s (O,A) t h e n u=u and f o r Am. I n p a r t i c u l a r w e see t h a t R(ARs(A)-I)=R(A(A-A) - l a)=R(A).

Rs(A)=(A-A)-'

Since t h e C e s 5 r o - e r g o d i c i t y i m p l i e s

I I S ( t ) I I=O(t)

(t-),

and s i n c e t h e l a t t e r

c o n d i t i o n i n t u r n i m p l i e s 020 ( s e e t h e proof of P r o p o s i t i o n 8 i n [ b ] ) ,

one can

e a s i l y deduce from Theorem 1 a complete c h a t a c t e r i z a t i o n of t h e uniform Cesaro-

Uniforni Ergodic Theorems for Operator Semigroups e r g o d i c i t y of (O,A)

263

semigroups.

rheurm 3. L e t T(.)

b e a s e m i g r o u p o f c l a s s (O,A).

The f o l l o w i n g s t a t e m e n t s

are equivalent:

(1) 'l'( - ) i s u n i f o r m l y C e s h r o - e r g o d i c . (2)

(3)

1 (S(t) j I=O(t) I / S ( t ) 1 I=O(t)

(tm),

(t-1,

1 ( T ( t ) R s ( I ) 1 ( = o ( t ) (t-), I / I ' ( t ) S ( u ) I I = o ( t ) (t-)

f o r a l l u>O, and R(A) i s

I lT(t)S(u) I I=o(t)

f o r a l l u>O, a n d T(.)

and R(A)

is c l o s e d .

closed.

(4) I I S ( t ) 1 I = O ( t ) ( t * ) ,

(t-)

is

uniformly Abel-ergodic.

3.

GKOTHENDIECK SPACE WITH THE DUNFORD-PETTIS PROPERTY A Banncti s p a c e X i s c a l l e d a G r o t h e n d i e c k s p a c e i f i t h a s t h e p r o p e r t y t h a t

e v e r y weakly* c o n v e r g e n t s e q u e n c e i n t h e d u a l s p a c e X" i s w e a k l y c o n v e r g e n t . The f o l l o w i n g s t r o n g e r g o d i c t h e o r e m i s a c o m b i n a t i o n of P r o p o s i t i o n 4 . 2 o f

[4] and Theorems 1 and 2 o f Theorem 4 .

L e t T(.)

El. We

s t a t e i t h e r e f o r u s e i n Theorem 5 .

b e a l o c a l l y i n t e g r a b l e s e m i g r o u p o f o p e r a t o r s on a

G r o t h e n d i e c k s p a c e X.

I IS(t) I I=O(t)

(i) I f

P:x

+

__

s-

L 1111

t-'S(t)x

=span{R(T(t)-I); (ii) T(.) tisfies:

(t-)

and

I

[ T ( t ) S ( u ) ( = o ( t ))..-t(

i s a bounded p r o j e c t i o i i w i t h

t > O ) , and

s-&&

f o r 3 1 1 u>O, t h e n

R ( P ) = Ti N ( ' l ( t ) - I ) t >O

and N(P)

e x i s t s f o r a l l x"EX*.

t-'S"(t)x*

i s s t r o n g l y C e s a r o - e r g o d i c ( i . e . D(P)=X) i f and o n l y i f i t s a -

1 I S ( t ) 1 I=O(t)

(t-);

I IT(t)S(u)xI I=o(t)

c l ( R * ) - p (or w " - c l ( R ( A " ) ) = m

i n case T(.)

(t-)

f o r a l l xEX, u>O;

w*-

i s o f c l a s s ( C o ) ) , w h e r e R*:=

s p a n { R ( T " ( t ) - I ) ; t>Ll}. x * > 4 whenever x X is s a i d t o have t h e Dunford-Pettis p r o p e r t y i f <x n' n w e a k l y i n X and x * 4 w e a k l y i n X*. Lm i s a G r o t h e n d i e c k s p a c e w i t h t h e F o r o t h e r e x a m p l e s o f s u c h spaces see L3].

Dunford-Pettis property.

I t was r e c e n t l y p r o v e d by L o t z [3]

t h a t , on a G r o t h e n d i e c k s p a c e w i t h t h e

D u n f o r d - P e t t i s p r o p e r t y , e v e r y (C,)-semieroup

i s u n i f o r m l y continuoils and e v e r y

s t r o n g l v e r g o d i c d i s c r e t e semigroup ITn] i s uni forml y e r g o d i c . shown i n [7]

-to

It has been

t h a t t h e same a s s e r t i o n s a r e t r u e f o r c o s i n e o p e r a t o r f u n c t i o n s .

The f o l l o w i n g t h e o r e m a b o u t t h e e r g o d i c i t y of l o c a l l y i n t e g r a b l e s e m i g r o i i p s i s o f t h e same n a t u r e . Theorem 5 .

Let T(.)

b e a l o c a l l y i n t e g r a b l e s e m i g r o u p of o p e r a t o r s on a

Grothendieck space X w i t h t h e Dunford-Pettis p r o p e r t y .

I I=o(t)

(t-)

f o r a l l u>O.

Suppose t h a t

Then T ( - ) i s u n i f o r m l y C e s a r o - e r g o d i c

I /T(t)S(u)

i f and o n l y

i f i t i s s t r o n g l y Ces3ro-ergodic. For t h e proof of t h i s theorem w e need Lemma 6 ([3]).

Let V

b e a s e q u e n c e of bounded l i n e a r o p e r a t o r s on a Banach

space X with t h e Dunford-Pettis property.

Suppose t h a t

S.-Y. Shaw

264 (1) w - l i m V x =O n+- n n

whenever [x } i s bounded i n X;

( 2 ) w - l i m V* x*=O nn n

whenever {x*} i s bounded i n X".

I IViI 14.In p a r t i c u l a r ,

Then

proof of Theorem 5 .

V -I and V +I a r e i n v e r t i b l e f o r l a r g e n .

I f T( .) i s s t r o n l g y e r g o d i c , then

n

Since R(P) i s f i x e d by t - l S ( t ) N(T(t)-I). t >o assume t h a t P=O w i t h o u t l o s s of g e n e r a l i t y .

w i t h R(P)=

PEB(X), X=R(P)@N(P) f o r a l l t > O , we may

s - l i m V x=Px=O f o r a l l xEX, so t h a t f o r any bounded n+a n t h e sequence {V* x*] converges weakly* and hence weakly t o n n 0 . By ( i ) of Theorem 4 w e have t h a t s - l i m V* X* e x i s t s and i s e q u a l t o w*n+- n l i m V* x*=P"x*=O f o r a l l x*EX*. Hence {V x ) converges weakly t o 0 whenever

Let

V =n-lS(n).

Then

sequence {x*) i n X*,

n-

n

n

Using t h e i d e n t i t y o ( t ) (t-)

n

I t f o l l o w s from Lemma 6 t h a t V -I i s i n v e r t i b l e f o r l a r g e n .

{ x n ) i s bounded.

vu>O,

S(tXT(u)-I)=(T(t)-I)S(u),

t h e assumption

I I T ( t ) S ( u ) I I=

and Lebesgue's dominated convergence theorem, w e o b t a i n

-1 -1 -1 ~ ~ t - l s ( t ) ~ ~ = ~ ~t ( ~s (nt )- (ln) s ( n ) - I ) I j

(1 I (Vn-I) -1 I In-1 lon I It -1 s ( t ) ( T ( d - 1 ) I Idu -1 n -1 = ~ ~ ( V n - I ) - l ~/ o~ tn I I ( T ( t ) - I ) S ( u ) l l d u -+

0

as

t-.

Hence T(.) i s uniformly Cesgro-ergodic. The f o l l o w i n g C o r o l l a r y i s deduced from Lemma 2 ( i ) , Theorems 4 ( i i ) and 5 . Corollary 7.

Let T(.) and X be as assumed i n Theorem 5 .

1

Then T ( - ) i s uni-

I

I IT(t).

formly Ceshro-ergodic i f and o n l y i f i t s a t i s f i e s : I S ( t ) I=O(t) (t-);

I

S ( u ) I = o ( t ) (t-)

f o r a l l u > O ; and

w*-cl(R*)=p (or

w*-cl(R(A*)=R(A")

in

c a s e T(.) i s s t r o n g l y c o n t i n u o u s and hence uniformly c o n t i n u o u s ) . T h i s and Theorem 3 y i e l d t h e f o l l o w i n g C o r o l l a r y 8.

L e t T(.) be a uniformly c o n t i n u o u s semigroup on a Grothendieck

space with t h e D u n f o r d - P e t t i s p r o p e r t y .

I IT(t)S(u) I I = o ( t )

(t+-)

f o r all u>O.

Suppose t h a t

I I S ( t ) / I=O(t)

(t-)

and

Then t h e f o l l o w i n g c o n d i t i o n s a r e e q u i -

valent: (1) T(.) i s s t r o n g l y Cesaro-ergodic.

(2) T(.)

i s uniformly CesAro-ergodic.

( 3 ) T(.) i s uniformly Abel-ergodic.

(4) R(A) i s c l o s e d . (5) w * - c ~ ( R ( A * ) ) = R ( A ~ ) . This i s i l l u s t r a t e d by t h e f o l l o w i n g example. For 0 5 A i l l e t g be t h e f u n c t i o n g ( s ) : =i s . 1 ( s ) , where Icx,13i s t h e A A 9 11 c h a r a c t e r i s t i c f u n c t i o n of t h e i n t e r v a l [A, 11. The m u l t i p l i c a t i o n o p e r a t o r s

cx

T (t):f

-+ e x p ( t g x ) f ( ~ C L ~ [ O , ~ - I< t)

a

1

5

C

a

1

I f ( x ) l P o W(x) dx

(a

' 01,

wn C b e i n g a c o n s t a n t independent o f f . Theorems i and 2 a r e proved i n $ 2 and 53. I n comparison w i t h w e i g h t e d norm i n e q u a l i t i e s f o r H a r d y - L i t t l e w o o d maximal " and "W e All' P/ Po i n Theorem l ( i i ) and Theorem 2 can be r e p l a c e d by more weaker c o n d i t i o n s

f u n c t i o n s , i t i s n a t u r a l t o ask whether t h e c o n d i t i o n s " W e A

"W E Ap" and " W e A

"

respectively.

The r e s u l t s o b t a i n e d i n 54 a r e as

PO

follows. THEOREM 3. (i)

I n g e n e r a l , i n e q u a l i t y ( 1 . 6 ) does n o t h o l d f o r W e A

P' Then (1.6) h o l d s t r u e i f and o n l y i f

( i i ) Suppose t h a t W(x) = I x l a . WeA P/ Po

.

THEOREM 4. (i)

.

I n g e n e r a l , ( 1 . 7 ) does n o t h o l d f o r W e A PO

(ii)

I f W(x) = I x I o y t h e n t h e c o n d i t i o n W e A1 i s a l s o necessary and

s u f f i c i e n t f o r the i n e q u a l i t y (1.7). Some f u r t h e r p r o p e r t i e s a r e i n v e s t i g a t e d i n 55. weak t y p e e s t i m a t e s f o r mapping f e s t i m a t e s i n t h e case p = p,

+

= y/h.

F i r s t , we p r o v e t h a t t h e

uA,y ( f ) cannot be s t r e n g t h e n e d t o s t r o n g More p r e c i s e l y t h e f o l l o w i n g theorem i s

true. THEOREM 5.

Under t h e assumptions o f Theorem 2, t h e weak i n e q u a l i t y cannot

be s t r e n g t h e n e d by a s t r o n g t y p e i n e q u a l i t y . I n f a c t , t h e r e e x i s t s a f u n c t i o n P g e L '(Wdx) w i t h W e A1 such t h a t t h e s t r o n g t y p e i n e q u a l i t y (1.6) f o r g does not hold true.

S. L. Wang

270

As f o r p < y/A, we have THEOREM 6.

L e t 1 5 p < po = y/X.

f E LP(Wdx) so t h a t

,Y

(f)(x) E

m

Then t h e r e e x i s t a w e i g h t W(x) e A1 and

everywhere.

The f o l l o w i n g lemmas w i l l be used i n t h e p r o o f o f Theorem 1 and 2.

$2.

LEMMA 1. L e t p 2

p 2

Poisson i n t e g r a l u(x,y)

(2.1)

lu(x-t,y)I

1.

I f M f ( x ) i s f i n i t e f o r some x E R n , t h e n t h e P o f f i s f i n i t e everywhere i n Rytl, and

5 Cn (1 +

+In M f ( x ) ,

and more g e n e r a l l y (2.2)

lu(x-t,Y)I

5 CnlP(l

+

v)n’’

M,f(x),

where (2.3)

M y f ( x ) = (M(MlfiP)(x))l/P

PROOF.

F i r s t we n o t e t h a t t h e f i n i t e n e s s o f M f ( x ) f o r some x l e a d s t o t h e P

existence o f the i n t e g r a l

( A > 1).

T h i s c o n d i t i o n i s e q u i v a l e n t t o t h e f i n i t e n e s s o f t h e Poisson i n t e g r a l u ( x , y ) o f f a t a l l p o i n t s (x,y) e R:+~. Thus,

Now, by v i r t u e o f i n e q u a l i t y ( 2 . 5 ) , t h e argument used t o prove Lemma 4 i n [ S l , p.921 a l s o works f o r t h e p r o o f o f (2.1) and ( 2 . 2 ) . If 1 1.

PROOF OF LEMMA 2 L e t 2Q be a cube w i t h t h e same c e n t e r as Q, b u t w i t h t w i c e as l a r g e a s i d e .

say.

Suppose t h a t t h e l e n g t h o f Q i s d; t h e n c l e a r l y f o r x e Qc, ( Q c b e i n g t h e

complement o f Q ) lx-xol ? d

so I x - x o l n p -> ( Q I P

.

Hence i t f o l l o w s t h a t (2.8)

I1 5

Q

2Q-QC since the A

c o n d i t i o n implies t h e "doubling condition". P As f o r I , we c o n s i d e r a s u i t a b l e maximal f u n c t i o n as f o l l o w s .

l e t Qx be a cube whose c e n t e r i s x and l e n g t h d i s equal t o

and x E (ZQ)', 2 sup I x - y l .

Y ~ Q

Then i t i s easy t o v e r i f y t h a t Q c Qx and

IQxI

s i n c e Cllx-yI

For a given Q

= (2dIn 5 C Ix-x0I

-< I x - x 0 1

(being t h e center o f Q), C

5 C21x-y( i f x E ( 2 Q )

Now, suppose t h a t x E (2Q)'.

A s an a p p l i c a t i o n o f ( 1 . 2 ) ,

and y i s any p o i n t o f Q.

By d e f i n i t i o n o f t h e H a r d y - L i t t l e w o o d maximal

i t follows t h a t

nP (2QIC

n

dx 5

1

(MXQ(x))' W(x) dx

(2QF

S. L. Wang

272

that i s

I 2 C

J

W(x)dx

Q Combining (2.8) w i t h ( 2 . 9 ) ,

t h i s completes t h e proof o f Lemma 2.

PROOF OF THEOREM 1. Before we prove (i), we make some comments.

We n o t e t h a t (1.2) shows t h a t

under t h e assumption of Theorem 1. I n p a r t i c u l a r , f e LP(Wdx) i m p l i e s M p f ( x ) P and M((Mf) ‘ ( x ) a r e f i n i t e f o r a l m o s t a l l x. Now we come t o ( i ) . Using ( 2 . 2 ) ,

we g e t

Now t h e same method used i n p r o v i n g ( 2 . 5 ) g i v e s t h e e s t i m a t e o f t h e l a s t integral :

Weighted Norm Inequalities for some Maximal Functions

s i n c e t h e assumption IJ > p

0

-

implies the condition A

(y-p0)/,,

213 > 1.

Combining

( 2 . 6 ) and ( 2 . 1 1 ) , we g e t ( 1 . 5 ) .

(ii)

F o r p > po, choose u such t h a t p >

p o s s i b l e , s i n c e (1.4) h o l d s .

For p >

11 >

po and W B A

PIP.

This i s

po, we a p p l y (1.5) and H o l d e r ' s i n -

p >

equal ity and we g e t

!

(%,Y ( f ) ( x ) ) p * W ( x ) d x L C

Rn 5

C (

(Mvf(x))

P h - Po) I Y

(M(Mf)

x) )

PIY

*W(x ) d x

Rn

I

(Muf(x))p.W(x)dx))

(Y-Po) I Y

(

Rn

I

(M(Mf)

PIP,

P

PolY

(x)'W(x)dx)

Rn

i n which we have used p r o p e r t y (1.2) and c o n d i t i o n W e A

T h i s completes

P/U*

t h e p r o o f o f Theorem 1.

83. DECOMPOSITION LEMMA.

P L e t f e L O(Wdx), W e A

, and l e t

a >

0 be g i v e n .

PO

There i s a c o l l e c t i o n { Q . ) o f p a i r w i s e d i s j o i n t cubes w i t h t h e f o l l o w i n g J properties:

1 W{QjI

(3.1)

-
po.

Then P r o p e r t y ( 3 . 7 ) c l e a r l y i m p l i e s t h a t

( j = l , 2 , ...).


(3.11)

1

P-Po I g ( x ) I p W(x)dx -< C - a

Rn

275

W(x)dx. Rn

By Theorem l ( i i ) , uA ( f ) i s a bounded o p e r a t o r on Lp(Wdx)(p>po), so t h a t by S Y

t h e Chebyshev i n e q u a l i t y and (3.11),

On t h e o t h e r hand, uA ( f ) ( x )

5

~ ~ ~ , ~ ( g t) (~ x~ ) , ~ ( b ) ( x So ) . i n order t o prove

3Y

Theorem 2, i t i s s u f f i c i e n t t o p r o v e t h a t

I f x e R n and Q . i s a cube f r o m t h e c o l l e c t i o n , by x Q Q . we mean t h a t x J J belongs t o a cube Q, ( a l s o from t h e c o l l e c t i o n ) which touches or c o i n c i d e s w i t h Note t h a t f o r f i x e d x, x

4.. J x

R then x (3.14)

where X,(x)

I

-

Q . never h o l d s . J

Q . h o l d s a t most N Whitney cubes; and t h a t i f J

Now, l e t

b j ( X ) = b(X) .XQ ( X ) , j denotes t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e s e t E, and l e t uj(x,y)

denote t h e Poisson i n t e g r a l o f b . ( x ) . J

where

By d e f i n i t i o n ,

S.L. Wang

216

Now

By H o l d e r ' s

inequality, the A

c o n d i t i o n and ( 3 . 9 ) , PO

(3.19)

From (3.18) and (3.19),

one v e r i f i e s t h a t

since

f o r any cube Q . s a t i s f y i n g t IC Q . by p r o p e r t y ( 3 . 5 ) . J J (3.20), we g e t

(3.21)

(pA(')(b)(x))' S Y

-< c.aY sup Y>O

J Rn

(+)"I

x- t I t y

Therefore, by (3.16) and

y-" d t

-< c - ~ Y .


So, by v i r t u e of (3.15) and ( 3 . 2 1 ) , w { % , (y2 ) ( b ) ( x ) >

C a 3

5

211

t o p r o v e (3.13) we need o n l y show t h a t C.a-”

/ f ( x ) l p o W(x)dx. Rn

Let R

*

= U(2Qj) and r e c a l l t h a t

J

J

Rn

so i t w i l l be enough t o p r o v e t h a t

Now t

4

R implies t h a t

since Cllx-t.l

J

1

,

u,(t,y) tQQ

i s an empty sum, so

-< I x - t l 5 C 2 ( x - tJ. l where tj i s t h e c e n t e r o f Q J. and t i s any

point i n Q

j’ We w i l l i n v o k e two e s t i m a t e s on u,(t,y)

= b,*P(t,y)

w i t h b,

= b-x Q,

(3.14)) : (3.24)

lu,(t,y)l

-
0, there exists a natural number

IIPnxo-xoII

We conclude that natural number

S

n > N.

* *

*

Hence

limllPnxOII = IIxoII = 1 ,

N1,

*

E

whenever

e

such that

and s o there exists a

* *

IIPnxOII > 0

whenever

n

X*(P for

n > N1,

such

O

and

>

N1.

x )

n * o* 4 IIPnXO II

Prx;

(n + a ) .

Now

X

is weakly very smooth, s o

{-}IIp*x* II

has a

1

A New Smoothness of Banach Spaces

subsequence

* * { "n x~* } IIpn xo II

P; x*

x E X,

* xo* (x) Pn i

w

* A y*

1 0

such that

E X*.

lip, XOII

But for

i

i

any

303

= x

* (P 0

n ix)

-*

*

xo(x),

hence

*

w

*

* xO.

*

- x

0

that

span{fn}

is weakly closed, we have

{fn}

is a basis of

X*.

If X* is locally uniformly convex, then hence weakly very smooth; thus we have COROLLARY 1. Let

*

x0 E span{fn}.

Hence Q.E.D.

X

is very smooth,

X

be a Banach space with a monotone basis, if is locally uniformly convex, then the basis i s shrinking. REMARK. There exists a Banach space X with a basis, but X* For such a Banach is separable and X* does not have basis [ 3 ] . space X , we can define a new equivalent norm on X under which

* x

X* X*

is locally uniformly convex [2, p. 1181. Under this new norm also does not have basis, hence any basis of X under this

norm is not monotone. Thus there are weakly very smooth Banach spaces which have bases but which do not have a monotone basis.

For other examples of Banach spaces with a basis which do not have monotone basis, see [5, p. 2 4 8 1 .

It is known that if X is very smooth, then X* has RadonNikodym property. Since every separable dual space has RadonNikodym property, we have COROLLARY 2. Let X be a Banach space with a monotone basis, if X is weakly very smooth, then X* has Radon-Nikodym property. The author would like to thank his adviser, Professor Bor-Luh Lin, for valuable guidance in revising this paper.

W.Zhang

304

REFERENCES [l] [Z] [3] [4] [5] [6]

Sullivan, F . , Geometrical properties determined by the higher duals of a Banach space. Illinois J. Math. 21 (1977) 315-331. Diestel, J . , Geometry of Banach S p a c e s 4 e l e c t e d Topics. Lecture Notes in Math. 485 (Springer-Verlag, 1975). Lindenstrass, J. and Tzafriri, L., Classical Banach spaces I (Springer-Verlag, 1977). Bishop, E. and Phelps, R.R., The support functionals o f a convex set, Convexity, Proceedings of Symposia in Pure Math., AMS 7 (1963). Singer, I., Bases in Banach spaces I (Springer-Verlag, 1970). James. J . , Characterizations o f reflexivity, Studia Math. 23 (1964) 205-216.

Analysis: Conference Proceedings

Proceedings of the Analysis conference, Singapore 1986

Enumerative Geometry. Conference Proceedings

Thermodynamics: Conference Proceedings

Graph Theory: Conference Proceedings

p-adic Functional Analysis.. Proceedings of the Sixth International Conference

Global Differential Geometry and Global Analysis. Proceedings Conference, Berlin, 1990

P-adic functional analysis: proceedings of the fourth international conference

Conference on Harmonic Analysis

Complex analysis and geometry: proceedings of the conference at Trento

International Mathematical Conference 1982: Proceedings

Recent Advances in Nonlinear Analysis (Proceedings of the International Conference on Nonlinear Analysis)

Recent Advances in Nonlinear Analysis (Proceedings of the International Conference on Nonlinear Analysis)

Recursion Theory Week. Proceedings Conference Oberwolfach, 1989

Proceedings of Gokova geometry-topology conference 1994

Recursion Theory Week. Proceedings conference Oberwolfach, 1984

Proceedings of Gokova Geometry-Topology Conference 1994

Proceedings of the European Computing Conference 2

Remembering the Space Age: Proceedings of the 50th Anniversary Conference: Proceedings on the 50th Anniversary Conference

Proceedings of Gökova Geometry-Topology Conference 2005

International Mathematical Conference 1982: Proceedings (Mathematics Studies)

Proceedings of the LFG 06 Conference

Proceedings of International Science Education Conference 2009

Systematic Biology: Proceedings of an International Conference

Proceedings of a Conference on Operator Theory

Complete Proceedings of the NordiCHI 2010 Conference

Proceedings of Gokova Geometry-Topology Conference 1996

Salam + 50: Proceedings of the Conference

Intuitionism and Proof Theory: Conference Proceedings

7th International Conference on Automated Deduction: Proceedings

Dynamics in logistics first international conference; proceedings

Analysis: Conference Proceedings

Proceedings of the Analysis conference, Singapore 1986

Enumerative Geometry. Conference Proceedings

Thermodynamics: Conference Proceedings

Graph Theory: Conference Proceedings

p-adic Functional Analysis.. Proceedings of the Sixth International Conference

Global Differential Geometry and Global Analysis. Proceedings Conference, Berlin, 1990

P-adic functional analysis: proceedings of the fourth international conference

Conference on Harmonic Analysis

Complex analysis and geometry: proceedings of the conference at Trento

International Mathematical Conference 1982: Proceedings

Recent Advances in Nonlinear Analysis (Proceedings of the International Conference on Nonlinear Analysis)

Recent Advances in Nonlinear Analysis (Proceedings of the International Conference on Nonlinear Analysis)

Recursion Theory Week. Proceedings Conference Oberwolfach, 1989

Proceedings of Gokova geometry-topology conference 1994

Recursion Theory Week. Proceedings conference Oberwolfach, 1984

Proceedings of Gokova Geometry-Topology Conference 1994

Proceedings of the European Computing Conference 2

Remembering the Space Age: Proceedings of the 50th Anniversary Conference: Proceedings on the 50th Anniversary Conference

Proceedings of Gökova Geometry-Topology Conference 2005

International Mathematical Conference 1982: Proceedings (Mathematics Studies)

Proceedings of the LFG 06 Conference

Proceedings of International Science Education Conference 2009

Systematic Biology: Proceedings of an International Conference

Proceedings of a Conference on Operator Theory

Complete Proceedings of the NordiCHI 2010 Conference

Proceedings of Gokova Geometry-Topology Conference 1996

Salam + 50: Proceedings of the Conference

Intuitionism and Proof Theory: Conference Proceedings

7th International Conference on Automated Deduction: Proceedings

Dynamics in logistics first international conference; proceedings

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