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
N. Asmar and E. Hewitt
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.
N. Asmar and E. Hewitt
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
N. Asmar and E. Hewitt
(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.
N. Asmar and E. Hewitt
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.
N. Asmar and E. Hewitt
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).
Marcel Riesz's Theorem on Conjugate Fourier Series
$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.
N. Asmar and E. Hewitt
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.
N. Asmar and E. Hewitt
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
Marcel RieszS Theorem on Conjugate Fourier Series
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).
Marcel Riesz's Theorem on Conjugate Fourier Series
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
Marcel Riesz's Theorem on Conjugate Fourier Series
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<0
22
N. Asmar and E. Hewitt
for every y in S n (-P). This establishes (1) and (2), and completes the proof.
0
(4.13) Theorem. Let P be an order in Z '. There is a sequence of Archimedean orders PI,Pa,. . . such tbat
Proof. We need only show that KP, c P c h P , . For n 2 1, let C, denote the parallelepiped in Z a defined by
Apply Theorem (4.11) with S = C, to obtain an Archimedean order P, such that (3)
P,
n C,
= P n C,.
For positive integers n and N with n 2 N, use (3) to infer that P, n CN = PNn C N . For x in P, let N be a positive integer such that x is in CN n P. Then x is in Pn for all n 2 N. Therefore, we have P c h P , . Replacing Pn by -P, and P by -PI we find that -P c b ( - P n ) . Suppose that y does not belong to P. Then y belongs to -P, and so y is in all but finitely many -Pn's. Hence y is in only finitely many Pn's. Therefore y is not in KP,,and so GP, c P. 0
Marcel Riesz's Theorem on Conjugate Fourier Series
23
$5. Orders on locally c o m p a c t A b e l i a n groups.
Like $41 this section is of an ancillary character. It is essential for our goal of extending M. Riesz's Theorem (1.6), as adumbrated in $3. We will use the results of $4 and of Hewitt and Koshi 119831 to give an analogue of Theorem (4.11) for arbitrary measurable orders on groups. Hewitt and Koshi 119831 have identified the measurable orders on certain locally compact Abelian groups. For the reader's convenience, we quote one of their results. (6.1) Theorem. Let B be a torsion-free locally compact Abelian group tbat is the union
of its compact open subgroups. Let P be any nondense order in tbe group R"x B, wbere a is a positive integer. There are a nonzero real-valued linear function L on Raand an order Po in L-l({O}) x B sucb tbat
(4
P = (L-'(]O, w[) x B ) u Po.
Tbe mapping Ll defined on
R" x B by
is a continuoris real-valued bomomorpbism sucb tbat
Proof. See Hewitt and Koshi 119831, Theorem (3.12).
0
Throughout the present section, H will denote a torsion-free locally compact Abelian group, F will denote a torsion-free compact Abelian group, and o will denote a positive integer. Unless otherwise stated, H is noncompact and F is infinite. (5.2) Deflnition. A nondense order P on the group R"x FXH is called strongly nondenee
if P is nondense in R" x F x {g} for all g in H. The next theorem extends Theorem (3.7) of Hewitt and Koshi [1983]. (5.3) Theorem. Suppose tbat P is a strongly nondense order on R" x F x H. Tben tbere is a linear function L mapping R" onto R sucb tbat
(i)
L-'(]O, w[) x F x (0)
5 P n (Ra x F x (0)) 5 L-'([O, w[) x F x (0).
For every g in H tbere is a real number a(g)sucb that (ii)
and
L-71
-
0 0 1
a(v)I)x F x { v } 5 - p
N. Asmar and E. Hewitt
24 (iu)
The mapping defined by g
H
a(g) is a continuous real-valued homomorphism of
H. Proof. The set P n (R" x F x (0)) is nondense in R" x F . Theorem (5.1) yields a homomorphism L of B" onto B such that (i)holds. For every g in H, there are nonvoid open subsets U of R" and V of F such that U x V x {g} C P. Thus P contains the set U x V x { g } + ( { O } x F x {o})nP. Since the set ( { O } x F x {o})nP is dense in { O } X F X {0} (Theorem (3.2)) of Hewitt and Koshi [1983]),it follows that U x F x {y} c P. The same argument, with P replaced by -P, shows that for every g in H there is a nonvoid open subset U' of R" such that U' x F x {g} C -P. Now suppose that (xl,f , g) is in P and that = L(x2) > 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').
Marcel Riesz's Theorem on Conjugate Fourier Series
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.
N. Asmar and E. Hewitt
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 [ ) .
Marcel Riesz's Theorem on Conjugate Fourier Series
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
N Asmar and E. Hewitt
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
Marcel Riesz's Theorem on Conjugate Fourier Series
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
Marcel Riesz's Theorem on Conjugate Fourier Series
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
N. Asmar and E. Hewitt
(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
Marcel Riesz's Theorem on Conjugate Fourier Series
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
N Asmar and E. Hewitt
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.
Marcel Riesz 's Theorem on Conjugate Fourier Series
35
(8.5) Theorem. Notation is as in (6.4). For every function f in f,(G)( I < p < co),the inequalities
(C, is m in ( 6 . 3 . i ~ ) obtain. ) For f in L,(G)(15 p < co) and every positive real number I , we have (iii)
L(({z E G : M'f(2)
>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 < p < 00. Moreover for 1 < p < co and f E L,(G),we have ( i v ) For every f in L,(G)(I 5 p
(4
l/H'fll~,~
5 M~llfll~,G*
For 1 _< p < oo and f E L,(G)we have
for all y > 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
N. Asmar and E. Hewitt
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
Marcel Riesz 's Theorem on Conjugate Fourier Series
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
N. Asmar and E. Hewitt
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,
N. Asmar and E. Hewitt
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
<-+= 6.
We now eshblish a pure existence theorem, our first step in proving the analogue of
M. Riesz's theorem. (7.2) Theorem. Let G be a locally compact Abelian group with character group X such
that X admits a Haafimeasurable order P. There is a linear operator f E,(G) for 1 < p < co with the following properties:
I-+
f" defined on
for all f in l , ( G ) (1 < p < co); (ii)
* ,
!(XI
= - i sgnp(x)f^(x)
for almost all X in X and all f in l , ( G ) (1 < p < 00 if G is compact, 1 < p 5 2 if G is not compact). Proof. We define the linear operator f ++ f on a dense subset of E,(G), show that it is continuous and that it satisfies ( i ) and ( i i ) . We then extend the operator f I+ f" uniquely to all of l,(G). I t is clear then that the extended operator satisfies ( i ) and ( i i ) . By Theorem (7.1), it is enough to consider functions with compactly supported Fourier transforms. We suppose that f is in E P ( G )and that f^ vanishes off of a compact subset K of X. We denote by p and p x Haar measures on G and X, respectively. Apply Theorem (5.14) to obtain a continuous real-valued homomorphism t,b on X such that
Marcel Riesz's Theorern on Conjugate Fourier Series
41
for px-almost all X in K n P\{O}, and
$(XI < 0 for px-almost all X in K n (-P)\{O}. Equivalently, the equality (1)
sgn&)
= sg+(X)
holds for px-almost all X in K . Form the function (6.4,i) with r replaced by $ and apply Theorem (6.5,i, i u , u) to obtain the Hilbert transform H$f.Denote this function by f . Then ( i ) follows from (6.5,~).I t remains to show that ( i i ) holds. From (6.7,i) we have " ,
(2) Since
!(XI
= - i sgn*(X)f^(x).
f =0 off of K, (I) shows that the equality sgn*(X)f(X) = sgnp(X)f^(X)
holds for px-almost all X in X. Putting this in (2) we see that (ii)holds.
(7.3) Remarks. (a) For compact G, Theorem (7.2) is the Bochner-Helson theorem (2.6), with the unknown constant A, replaced by M. Riesz,'s constant M,. Our proof is totally different from previously published proofs. (b) We owe to Berkson and Gillespie 119851 the proof that the constant in (7.2.i) is actually M p , in the case that G is compact. (7.4) Construction. In the remainder of this section, we present a summability method for that yields pointwise convergence for f in &(G) 1 < p < 00. We will also obtain analogues of Kolmogorw's theorem (1.4) for giving weak (1,1)-type inequalities for the conjugate function, and of Zygmund's theorem (1.11).
Let X be a torsion-free locally compact Abelian group admitting a Haar-measurable order P, let G be the dual group of X, and let p and px be Haar measures on G and X respectively. Let I? be any a-compact open subgroup of X. Because I? n P is a Haarmeasurable order on I?, the subgroup I? has the form Ra x HI where either a is a positive integer and H is a a-compact torsion-free Abelian group, or a is zero and H is countable and discrete. (See Hewitt and Koshi [1983], Remark (4.4.e), p. 453.) Write 00
I'=UKn n=l
where each K , is compact with nonvoid interior, and K , C K , if m 5 n. For each n, apply Theorem (5.14) and obtain a continuous real-valued homomorphism $, on X and a null subset N of X (independent of n) such that $n(x)
>0
42
N Asniar and E. Hewitt
for all X in (P n K,)\(N u (0)) and $n
(x) < 0
for all X in ((-P)n K,)\(NU ( 0 ) ) . Let Go denote the annihilator of I' in G. According to (23.24.e) of Hewitt and Ross 119791, p. 365, the group Go is compact. Let PO be normalized Haar measure on Go. We have
20 = l r (see Hewitt and Ross [1970], p. 214, (31.7.i)). A function f in L,(G) (1 constant on the cosets of Go if and only if the equality
5
p < co) is
f = f *Po holds p-almost everywhere on G (see Hewitt and Ross [1970], p. 105, (28.68)).
For a subset S of &(G) (1 5 p < 00) we will write S * po to denote the subset of L,(G) consisting of functions of the form f * po where f is in S. Finally, we let hoo(I') denote the subset of &,(G) * PO (1 5 p 5 2) consisting of functions constant on cosets of GOwith Fourier transforms vanishing off of compact subsets of I'. For f in L,(G) (1 5 p 5 2), one can recognize the elements of f in L,(G) * PO by looking at their Fourier transforms. Indeed, the equality (i)and the uniqueness of the Fourier transform show that f is in L,(G) * po if and only if f vanishes off of I'. (7.5) Lemma. Let p be a nurriber in 11, co[. The linear space L,(G) n Ll(G') n &o(I') dense in 1,(G) * PO.
is
Proof. The set L,(G) n Ll(G)* PO is plainly dense in L,(G) * po. Let f be in L,(G) * P O . Given E > 0, use the Theorem (7.1) to choose g in Ll(G) n L,(G)such that has compact support and
Ilf - BllP < 8 . The function g * P O is in &(G) n L,(G) n &oo(I'),and since f = f * po, we have
Ilf - ff * P o l l p
= Ilf*PO - ff *Poll,
5 Ilf - UllPllPOll < E. 0 (7.6) Notation is as in (7.3). For f in L,(G) (I 5 p < m), let Hnf denote the Hilbert transform of f with respect to the homomorphism &. The function H n f is given by (6.5.1'~) and has the following properties:
(4
IIHnfll,,G
5 M~llflh
for all f in L,(G), 1 < p < co; D U
(ii)
where B, is as in (6.7.iii) and f is in &(G) (15 p < co).
Marcel Riesz's Theorem on Conjugate Fourier Series
43
(7.7) Theorem. Notation is its in (7.4]-(7.6). Let f be in l , ( G ) * p o (15 p < m). There is a function H f deffned p-almost everywhere on G with the following properties: ( i ) The function Hnf converges in measure to Hf; ( i i ) for all g > 0 and all f in l , ( G ) , 1 5 p
< 00, tbe inequality
bolds.
Proof. Note that f * po = f . Suppose that g is in Ll(G) n l , ( G ) and that 3 vanishes off of a compact subset K of r. Let no be a positive integer such that K c K, for all n 2 n o . It is easy to check that the equality (11
Hng = Hmg
holds p-almost everywhere for all n4, n 2 n o . (Compute the Fourier transforms of both sides of (I) and use (7.2.1).) We define H g to be H'og. For g, property ( i ) is trivial and ( i i ) is (7.6.ii). Given f in L,(G), 1 5 p < and such that
and an arbitrary
00,
Ilf - 911,
E
> 0, choose a function
g as above
< ~-'-'/PE'+'/PB P
(use Lemma (7.5)). Choose a positive integer no so that (I) holds for g. For m, n 2 n o we have C(({z: IH"f(z) - Hrnf(z)l>
+
€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)
N. Asmar and E. Hewitt
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< p < co),and I > 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.
Marcel Riesz's Theorem on Conjugate Fourier Series
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
Marcel Riesz's Theorem on Conjugate Fourier Series
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)
N. Asmar and E. Hewitt
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
Marcel RieszS Theorem on Conjugate Fourier Series
( 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
<-+= E.
(p)p=.=,
I t follows from (8) that the sequence of functions is Cauchy in lp(T"). By the there is a function f' in l,(?r")such that completeness of lp(Ta), (9)
r""
.--t
f'
in the l p ( T anorm. ) From (9) we infer that (10)
F(x)
.--t
for every X in 23". If X is any character of there is a positive integer n(X) such that
"",
?(X)
we see from (1) and (2) with S = {X} that
P, n {x} = P n {x} and
for every n (12)
1 n(X). Plainly, the equalities (11) imply that sgn,X = sgnp X
for every n 2 n(X). The equality (12) and the defining property of the conjugate function o f f with respect to P yield the equalities
= -i sgn,(X)f?X) = -i sgn =
(x).
h
(x)f(x)
Marcel Riesz's Theorem on Conjugate Fourier Series
51
Uniqueness of the Fourier transform, (lo), and (13) show that
p-almost everywhere on
Ta.
For every positive integer n, we have
The inequality (u) follows on combining (9), (la), and (15).
For f in ,!!,(lr')(1 < p < co),the function is in .&,('ITa) for every positive integer n. Thus the convolution K , * is a well-defined element of &(a"). Let us compute the and K , * f . If X is in Cn, we see from Fourier transforms of the functions Kn * f,Kn * (4.12.3) that
F
N
Hence, for every X in C,, (16) shows that
(17)
Combining (17) and ( 8 . 2 . ~we ) ~ obtain
Also from (8.2.v), we have
For every X in Z a ,we have
Uniqueness of the Fourier transform, (18), (19), and (20) show that for every positive integer n, the equalities
52
N. Asmar and E. Hewitt
hold p-almost everywhere on TI!". The second equality in (21) shows that the functions * K , converges to the same limit function as f *K,, p-almost everywhere on TI!", and in the &,(T")-norm. The properties (8.2.ii) and (8.2.wii) of the FejCr kernel show that this limit function is f. This proves ( i i i ) and ( i u ) . 0 Single sequences of functions that converge to .f are described in Asmar [1986],section 6, for G = Taand f in .CP(T4), 1 5 p < 00, where a is a positive integer. Also single sequences of functions that yield pointwise convergence to f are described by Theorem (6.9) and Theorem (6.8). We were unable to determine whether the iterated sequences used for pointwise convergence to f in the general case treated in Theorem (7.10) are actually necessary. We conjecture that they are. (8.4) Remark.
Marcel Riesz’s Theorem on Conjugate Fourier Series
53
Footnotes.
’ This essay is based on the first-named author’s doctoral dissertation (Asmar [1986]) and on three lectures given by the second-named author at the Analysis Conference Singapore 1986. The authors are grateful to the editor, Professor s. T. L. Choy, for the opportunity he has graciously given us to present this extended essay for publication in the Proceedings of the Conference. They are also grateful to Professors Gerald B. Folland and Paul Pascal for valuable counsel, and to Ms. Laura Plaut for typesetting our MS. There is a bibliography a t the end of the essay. Authors are listed alphabetically by surname and then by year of publication with superscripts where needed. a
a This is the Austrian mathematician Alfred Tauber who published the first Tauberian fheorem (Tauber [1897]). He lived from 1866 to about 1942. On 28 June 1942 he was reported as being in the Konzentrationslager Theresienstadt and was beard of no more.
‘ The spoken form for the symbol varies widely from language to language. Here is a small sample: eff wiggle (U. S. A.); eff twiddle (Great Britain); eff tilde (French); eff tilde (Spanish); eff Schlange (German); eff slang (Dutch); eff mato (Finnish); eff hulliLm (Hungarian); eff fala (Polish); eff s vol’noi (Russian); eff aakfah (Arabic); eff nami (Japanese); eff bolang (Mandarin). Hardy and Littlewood are referring to Privalov 11916(’)]and Privalov 119171. This view was held by Riesz himself. See the footnote to page 218 of Riesz [1928], in which Riesz’s irritation is partly concealed. In 1954, Riesz expressed himself vigorously to the second-named author about Titchmarsh and in particular about his methods of proof.
’ It
wag proved only recently (Pichorides 11972)) that the smallest value of Mp is
tan($) for 1 < p 5 2 and ctn($) for 2
5 p < 00.
So christened in Hardy 119251, where historical reasons for the name are given.
For a positive real number t , log+t = max(logt, 0). See also Bochner 119593. Bochner retained a life-long interest in M. Riesz’s Theorem (1.6) and its congeners. In the mid 1960’s he told the second-named author of his admiration Cp(-n, T ) for this theorem. The mapping f H !is a linear mapping of the linear space
u
P>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
N. Asmar and E. Hewitt
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 <
TI,I
si( 6
M
E
+ g(s,12) whenever I1and I2 are disjoint; > 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
-
<
...
x1 <
&(ti)
xi-l
<
xn = b w i t h t l , t2,
<
s ti s x i
n
I 1+(f(ti),
-
<
..., tn, f o r each i we have
ti t 6 ( t i ) AI <
E
i=1
where Ii = [ti-1, ti] f o r i = 1 , 2 ,
..., n .
The v a l u e A i s d e f i n e d t o be t h e
H e n s t o c k - K u r z w e i l i n t e g r a l o f f o v e r [a,b]. DEFINITION 2.
L e t F be a f u n c t i o n d e f i n e d on [a,b].
Write F(I) = F(v) -F(u)
i f u and v a r e t h e e n d p o i n t s o f an i n t e r v a l I. L e t x e [a,b]
i n t e r v a l o f t h e f o r m ( x , x+h) o r ( x - h , x ) , where h > 0.
and 1, b e an
Then F i s s a i d t o have
derivative F i ( x ) a t x with respect t o @ i f
A f u n c t i o n f i s s a i d t o be D e n j o y i n t e g r a b l e w i t h r e s p e c t t o t h e r e e x i s t s a f u n c t i o n F w h i c h i s c o n t i n u o u s o n [a,b] derivative F i ( x ) with respect t o
+
+
and ACG,
o n [a,b]
if
such t h a t i t s
i s e q u a l t o f ( x ) a l m o s t e v e r y w h e r e on [a,b].
The v a l u e F ( a , b ) i s s a i d t o be t h e D e n j o y i n t e g r a l o f f o v e r [a,b]. DEFINITION 3. f i n [a,b]
where
p
A f u n c t i o n H i s s a i d t o be a P e r r o n m a j o r f u n c t i o n o f a f u n c t i o n
with respect t o $ i f
d e n o t e s t h e l o w e r d e r i v a t i v e , I an i n t e r v a l c o n t a i n i n g x and 111 t h e
length o f I .
A f u n c t i o n G i s s a i d t o be a P e r r o r m i n o r f u n c t i o n o f f i n [a,b]
w i t h r e s p e c t t o 0 i f -G i s a P e r r o r m a j o r f u n c t i o n o f f i n [a,b]
-0
,
i.e.,
w i t h respect t o
On Nonlinear Integrals
59
A f u n c t i o n f i s s a i d t o be P e r r o n i n t e g r a b l e on [a,b]
-
infIH(b)
H ( a ) } = supIG(b)
-
w i t h respect t o 0 i f
G ( a ) } # _c,
where t h e infimum i s o v e r a l l P e r r o n major f u n c t i o n s H o f f i n [a,b] r e s p e c t t o @ , s i m i l a r l y f o r t h e supremum.
with
The common v a l u e i s d e f i n e d t o be
t h e P e r r o n i n t e g r a l o f f on [a,b]. THEOREM 1.
The above t h r e e n o n l i n e a r i n t e g r a l s a r e e q u i v a l e n t .
The p r o o f f o l l o w s t h e same argument as i n t h e p r o o f o f t h e l i n e a r case [6, pp. 123-126; 9; 14, pp. 250-2511.
We s h a l l n o t e l a b o r a t e here.
We remark t h a t t h e t e c h n i q u e o f t h e p r o o f employed i n t h e g e n e r a l i z e d dominated convergence theorem
[lo]
can be m o d i f i e d t o g i v e a s i m i l a r conver-
gence theorem f o r a n o n l i n e a r P e r r o n i n t e g r a l .
2 , CHARACTERIZATION OF A NONLINEAR MEASURE DEFINITION 4 .
A f u n c t i o n k : [a,b]
L e t R be t h e r e a l l i n e .
x R
-f
R i s called
a Caratheodory f u n c t i o n if k ( x , - ) i s c o n t i n u o u s , f o r almost a l l x e [a,b] k( - , s )
and
i s measurable f o r e v e r y s e R.
I n what f o l l o w s , L,
denotes t h e space o f a l l e s s e n t i a l l y bounded measurable
f u n c t i o n s on [a,b]. DEFINITION 5.
L e t { f n }c ,L,
t h e n ( f n } i s s a i d t o be boundedly convergent t o
f, i f { f n ) converges t o f p o i n t w i s e almost everywhere and { f n } i s u n i f o r m l y bounded almost everywhere by some c o n s t a n t .
A f u n c t i o n a l F d e f i n e d on
DEFINITION 6. i f F(fn)
F ( f ) as n
-f
DEFINITION 7. d e f i n e d on L, E
L,
i s s a i d t o be boundedly c o n t i n u o u s
whenever { f n } i s boundedly convergent t o f .
* m
L e t h e L,
and Nh = I g ; I g j 2 h, g e L,}.
i s s a i d t o be u n i f o r m l y
11
II,-continuous
A functional F on Nh i f f o r e v e r y
> 0, t h e r e i s 6 > 0 such t h a t
IF(f) whenever I l f - g l / , DEFINITION 8.
< 6
w i t h f, g
E
-
F(g)I <
E
N h y where
11
A f u n c t i o n a l F d e f i n e d on L,
ll,is
t h e usual norm o f .L,
i s s a i d t o be o r t h o g o n a l l y a d d i t i v e
if F(f+g) = F ( f ) whenever f,g e L, LEMMA 1.
+
F(g)
w i t h f ( x ) g ( x ) = 0 f o r almost a l l x e [a,b].
[4, Theorem 2 - 6 1 I f F i s a boundedly c o n t i n u o u s and 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 on
L,
then F i s uniformly
11
Il,-continuous
on Nh, f o r
e v e r y h e L., THEOREM 2 . L, i f f
A f u n c t i o n a l F i s boundedly c o n t i n u o u s and o r t h o g o n a l l y a d d i t i v e on
TS.Chew
60
F(f) =
,”
k(x, f ( x ) ) d x
for
f e L,
where k i s a Caratheodory f u n c t i o n from [a,b] x R t o R w i t h k(x,O) = 0 f o r almost a l l x and s a t i s f i e s t h e f o l l o w i n g p r o p e r t y :
( * ) f o r e v e r y n a t u r a l number m y t h e r e e x i s t s a Lebesgue i n t e g r a b l e f u n c t i o n g, such t h a t Ik(X,SIl 5 gm(x) f o r a l l Is1
5
m and almost a l l x .
T h i s theorem i s a s p e c i a l case o f t h e r e p r e s e n t a t i o n theorem i n [4, Theorem 3-21.
We remark t h a t t h e c o n d i t i o n t h a t a f u n c t i o n k ( x , f ( x ) )
i n t e g r a b l e f o r e v e r y f E L,
i s Lebesgue
i s e q u i v a l e n t t o t h e c o n d i t i o n ( * ) . T h i s can be
seen e a s i l y by o b s e r v i n g t h e f o l l o w i n g f a c t : f o r e v e r y m, t h e r e e x i s t s u(X) E L,
such t h a t Ik(x,s)l 5 Ik(x, u ( x ) ) I
f o r a l l Is1
4
m and almost a l l x.
I t i s so s i n c e k(x,.)
i s continuous, f o r
almost a l l x . We remark t h a t t h e above r e p r e s e n t a t i o n theorem was a l s o proved by A. D . M a r t i n and V . J . M i z e l [12], THEOREM 3.
b u t w i t h a s t r o n g e r assumption.
A f u n c t i o n 0 i s a n o n l i n e a r measure d e f i n e d as i n s e c t i o n one i f f
$(s,I) =
1,
f o r a l l s and a l l I
k(x,s)dx
where k i s a Caratheodory f u n c t i o n f r o m [a,b] x R t o R w i t h k(x,O) = 0 f o r almost a l l x and s a t i s f i e s t h e c o n d i t i o n ( * ) . The s u f f i c i e n c y o f t h e theorem f o l l o w s immediately f r o m Theorem 2 and Lemma
1. To prove t h e n e c e s s i t y , we need t h e f o l l o w i n g theorem and Lemmas. THEOREM 4 .
L e t { f n l be a sequence o f measurable and Henstock-Kurzweil i n t e -
g r a b l e f u n c t i o n s w i t h r e s p e c t t o 9 on [ a , b l .
I f ( i ) fn(x)
+
f ( x ) almost every-
where i n [a,b], as n m and ( i i ) t h e p r i m i t i v e s F w i t h r e s p e c t t o @ o f fn 9,n a r e a b s o l u t e l y continuous u n i f o r m l y i n n, t h e n f i s Henstock-Kurzweil i n t e g r a b l e +
w i t h r e s p e c t t o 9 o v e r [a,b]
I,
and we have
b
1
b
fnd9
+
a
The p r o o f i s standard [7; pp. 86-87], LEMMA 2.
fd9
as
n
+-.
and t h e r e f o r e o m i t t e d here.
L e t { f n l be a sequence o f f u n c t i o n s which a r e Henstock-Kurzweil i n t e g r a b l e w i t h r e s p e c t t o 9 . I f a l l f u n c t i o n s fn a r e bounded by some c o n s t a n t M y then the functions F d e f i n e d by 9,n
On Nonlinear Integrals
61
a r e a b s o l u t e l y c o n t i n u o u s u n i f o r m l y i n n. PROOF.
F i r s t , by ( N 5 ) , f o r e v e r y
whenever Ii, f o r i = 1,2, length less than
q
E >
...,n y a r e
0, t h e r e e x i s t s
rl >
0 such t h a t
pairwise d i s j o i n t intervals w i t h the t o t a l
and / s i I 2 M, f o r i = 1,2,
...,n.
Since each fn i s Henstock-
K u r z w e i l i n t e g r a b l e w i t h r e s p e c t t o 4 , t h e r e f o r e , t h e r e e x i s t s 6 n ( 5 ) > 0 such t h a t f o r any &,,-fine
d i v i s i o n o f [a,b],
11,” L e t J j , j = 1,2,.,.,m
fnd$
-
we have
1 4 ( f n ( c ) , Cu,vl)l
5
E.
be p a i r w i s e d i s j o i n t i n t e r v a l s w i t h t h e t o t a l l e n g t h l e s s
t h a n n , then, by H e n s t o c k ’ s Lemma [7, Theorem 51,
Thus
I j=1
fnd+
5
6 ~ .
Jj
are absolutely continuous u n i f o r m l y i n n. ,n Every f u n c t i o n i n La i s Henstock-Kurzweil i n t e g r a b l e w i t h r e s p e c t t o
Consequently, t h e f u n c t i o n s F LEMMA 3. $*
PROOF.
F i r s t , i t i s easy t o see t h a t e v e r y s t e p f u n c t i o n i s Henstock-Kurzweil Then, by Theorem 4 and L e m a 2 , we can
i n t e g r a b l e w i t h r e s p e c t t o $ on [a,b].
show t h a t e v e r y s i m p l e f u n c t i o n and, t h e r e f o r e , e v e r y f u n c t i o n i n L,
are
Henstock-Kurzweil i n t e g r a b l e w i t h r e s p e c t t o $ on [a,b]. LEMMA 4 .
A f u n c t i o n a l G d e f i n e d by b fd$ G(f) = a
I
for all
f
E
L,
i s o r t h o g o n a l l y a d d i t i v e and boundedly continuous on Lm. PROOF.
I t f o l l o w s f r o m Lemma 3, Lemma 2 and Theorem 4.
PROOF OF NECESSITY.
I t f o l l o w s f r o m Lemma 4 t h a t a f u n c t i o n a l G d e f i n e d by
G(f) =
1;
fd$
f o r a l l f e L,
T.S. Chew
62
i s boundedly continuous on Lm.
Hence, by Theorem 2, t h e r e e x i s t s a Caratheo-
dory f u n c t i o n k s a t i s f y i n g t h e r e q u i r e d c o n d i t i o n s such t h a t ,b
G(f)
=
1,
L e t I be a s u b i n t e r v a l o f [a,b]
G(sxI) where
xI
=
f o r a l l f E L_.
k(x,f(x))dx
1,
and s be a r e a l number. k(x,s)dx =
1,
Then
sdO = $ ( s , I )
i s the characteristic function o f I .
The p r o o f o f t h e n e c e s s i t y i s complete. REFERENCES
1. J . B a t t , N o n l i n e a r i n t e g r a l o p e r a t o r s on C(S,E), S t u d i a Math. 48( 1973), 145-177. 2. R. V . Chacon and N. Friedman, A d d i t i v e f u n c t i o n a l s , Arch. R a t i o n a l Mech. A n a l . 18( 1965), 230-240. 3. T. S. Chew and P . Y . Lee, N o n l i n e a r Henstock-Kurzweil i n t e g r a l s and r e p r e s e n t a t i o n theorems, s u b m i t t e d . 4. L. Drewnonwski and W. O r l i c z , C o n t i n u i t y and r e p r e s e n t a t i o n o f 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 , B u l l . Acad. Polon, S c i . , Ser. Math. 17( 1969), 647653. 5. N . Friedman and A . E . Tong, On a d d i t i v e o p e r a t o r s , Canad. J. Math. 23(1971), 468-480. 6. R . Henstock, Theory o f i n t e g r a t i o n , B e t t e r w o r t h s , London, 1963. 7. R. Henstock, A Riemann-type i n t e g r a l o f Lebesgue power, Canadian J . Math. 20(1968), 79-87. 8. P:Y. Lee, Riesz r e p r e s e n t a t i o n theorems, SEA B u l l . Math. 10(1986), no.2. 9. P . Y . Lee and W i t t a y a Naak-in, A d i r e c t p r o o f t h a t Henstock and Denjoy i n t e g r a l s a r e e q u i v a l e n t , B u l l . Malaysian Math. SOC. ( 2 ) 5(1982), 43-47. 10 . P . Y . Lee and T. S. Chew, On convergence theorems f o r nonabsolute i n t e g r a l s , B u l l . A u s t r a l . Math. SOC., v o l . 34(1986), 133-140. l l . M . Marcus and V. J. M i z e l , A Radon-Nikodym theorem f o r f u n c t i o n a l s , J . F u n c t i o n a l A n a l y s i s 23( 1976), 285-309. 12.A. 0. M a r t i n and V . J . M i z e l , A r e p r e s e n t a t i o n theorem f o r c e r t a i n nonl i n e a r f u n c t i o n a l s , Arch. R a t i o n a l Mech. A n a l . 15( 1964), 353-367. 13.V. J . M i z e l and K. Sundaresan, R e p r e s e n t a t i o n of a d d i t i v e and b i a d d i t i v e f u n c t i o n s , Arch. R a t i o n a l Mech. Anal 30( 1968), 102-126. 14.S. Saks, Theory of t h e i n t e g r a l , 2nd e d i t i o n , Warsaw, 1937.
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
63
THE CONTROLLED CONVEKGENCE THEOREM FUK THE APPROXIMATELY CONTINUUUS INTEGKAL OF W K K I L L Soeparna Darinawi jaya U n i v e r s i t d s h j a h Mada, I n d o n e s i a Lee Peng Yee N d t i o r i a l U n i v e r s i t y o f Singapore, Singapore We prove t h e c o n t r o l l e d convergence t h e o r a n f o r t h e a p p r o x i m a t e l y c o n t i n u o u s i n t e g r d l o f E u r k i 1 I and deduce o t h e r convergence theorems as c o r o l l a r i e s . 1. P r e l i m i n a r i e s
We know t h a t t h e Henstock i n t e g r a l i s i n c l u d e d i n t h e a p p r o x i m a t e l y c o n t i n -
For t h e former, t h e r e i s a theorem which i s
uous i n t e g r a l o f B u r k i l l [l,21.
c a l l e d " t h e c o n t r o l l e d convergence theorem" [4,5]
and which i m p l i e s t h e o t h e r
convergerice theorems, f o r example, t h e domindted convergence theorem, t h e monot o n e convergence theorem and t h e u n i forto convergence theorem. The aim o f t h i s paper i s t o show t h a t t h e c o n t r o l l e d convergence theorem can be extended t o t h e a p p r o x i m a t e l y c o n t i n u o u s i n t e g r a l o f B u r k i l l .
Consequently,
i t a l s o i m p l i e s t h e o t h e r c o r r e s p o n d i n y convergerice theorems.
We r e c a l l t h e f o l l o w i n g n o t i o n s [I]. L e t F : [a,b]
+ K,
t h e r e a l l i n e , and E be a c l o s e d subset o f [a,b].
t h a t F i s HC*ap on E or, s i m p l y , F E HC&,(E) every
x
C
(U,1)
a s e t E ~ C[a,,
where
e
bn] w i t h a,
bn
e E$
sequence of c l o s e d s e t s {xi]
such t h a t [d,b]
every i, we say t h a t F i s ACGSp t i i v e n x B [a,b],
-
an) such t h a t
= u
I f there exists
d
~ = and ~ xF B~ H C ; ; ~ ( X ~t)o r
on Ca,bl o r , s i m p l y , F
g
HCG$a,bl.
we a s s o c i a t e a s e t Dx o f d e n s i t y 1 a t x and x
c o l l e c t i o n A o f a l l c l o s e d s u b - i n t e r v a l s [u,v]
n tor
N t h e c o ~ l e c t i o no f d l l
i s a c l o s e d c o n t i g u o u s i n t e r v a l o f E.
b,,]
bn] o f E t h e r e e x i s t s
and I J ( E ~ ) > ( 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
<
aiml))
or, simply,
[A The v a l u e
E
or, simply,
- 1 f(x)(v-u)(
A i s c a l l e d an K & - i n t e y r a l
<
E
E
.
o f f and we w r i t e
* b (Rap) f ( t ) dt a
I
=
A
.
Then t h e r e e x i s t s a f u n c t i o n THEOKEM 1.1 [I]. L e t f E H2p[a,b]. F : [a,b] + K such t h a t ( i ) F e Cap[a,b], ( i i ) F E ACti;3p[a,bl and ( i i i ) F & ( x )
on Ca,bl.
= f ( x ) a.e.
Note t h a t [a,b],
c aP [a,b]
i s t h e s e t o f a l l a p p r o x i m a t e l y continuous f u n c t i o n s on
and F i s t h e R;,,-primitive
o f f on [a,bl.
2. The controlled convergence theorem F i r s t o f a l l , we need t o g e n e r a l i z e t h e u n i f o r m l y convergent sequence o f Given f u n c t i o n s Fn, F : [a,b]
functions.
+
R, a sequence o f f u n c t i o n s {Fn} i s
s a i d t o be l o c a l l y a p p r o x i m a t e l y convergent t o F on [a,b] x a [a,b] at
X
With
i f t o r every
and E > 0 t h e r e e x i s t a p o s i t i v e i n t e g e r nu and a s e t Ux o f d e n s i t y 1 X E
Ox Such t h d t
IFn(U) f o r every u
€ Ox
F(u)( <
E
and n > no.
I t i s easy t o prove t h e f o l l o w i n y theorem.
I t a sequence o f f u n c t i o n s I F n } i s l o c a l l y a p p r o x i m a t e l y convergent t o F on [a,b] and Fn e Cap[a,b] f o r every n , then F E C,,Ca,bl.
THEOKEM 2.2.
I f f o r every n we have f n E Rip[a,b]
w i t h Fn b e i n y i t s R $ p - p r i m i t i v e .
then
t h e sequence of f u n c t i o n s i f n } i s s a i d t o be c o n t r o l - c o n v e r g e n t t o f on [a,b]
i f it s a t i s f i e s the f o l l o w i n g conditions : (i) (ii
{ f n } converges t o f a.e. {Fn} i s ACG$p on [a,b]
on [a,b]. u n i f o r i n l y i n n, i e.,
c l o s e d s e t s { X i } such t h a t [a,bl
=
uy=IX
t h e r e e x i s t s a sequence o f
and Fn
8
AC;3p(X,) u n i f o r i r l y i n
n. ( i i ) {Fn} i s l o c a l l y a p p r o x i m a t e l y convergent t o some F on [a,b].
The Approximately Continuous Integral of Burkill THEOKEH 2.3.
I f {fn} i s control-
(The c o n t r o l l e d convergence theorem)
convergent t o f on [a,b],
t h e n f E KZp[a,b]
b l i m (R* ) tll(t)dt n+m ap a
and *
1
65
b
)
= (R
t(t)dt ,
a
To prove t h e above theorem, we need t h e f o l l o w i n g lemmas, F o r convenience, we w r i t e i n whdt f o l l o w s Fn(u,v)
LEMMA n.
2.4
L e t f,, be Lebesgue i n t e y r d b l e on [a,bl
= Fn(v)
-
F,(u).
w i t h p r i m i t i v e Fn t o r e v e r y
If { f n } s a t i s f i e s c o n d i t i o n s ( i ) dnd
(iV)
on [a,bl u n i f o r m l y i n n , t h e n f o r e v e r y
{Fn} i s a b s o l u t e l y continuous
> 0 t h e r e e x i s t s a p o s i t i v e i n t e g e r nu such t h a t f o r every
E
d i v i s i o n o t [a,bl
a < we have
dl
< b l < a2 < b2
n
1 1J= 1 f o r every m,n >
L E W 2.5
partial
:
bi)
9
-
... < bp < b
F,bi
s
bi
)}I <
E
no.
L e t F,,
be t h e K$,,-priinitive
o f f, 6 K*aP [ a , b l
f o r every n.
I f {tn}
s d t i s f i e s c o n d i t i o n s ( i ) , ( i i i ) and (v)
IF,)
i s K i p on a c l o s e d s e t X u n i t o r m l y i n
Lemma 2.4
Proof
h o l d s w i t h a i , b,
n , t h e n t h e consequence o f
E X f o r i = 1, 2,
..., p.
We d e f i n e f u n c t i o n s Hn such t h a t Hn(x) = F n ( x ) f o r every x 6 X,
a r e p i e c e w i s e l i n e a r dnd c o n t i n u o u s on Xc = [a,bl
-
X and {Hn)
and Hn
converges, say
t o H, on [a,b].
I n view o f c o n d i t i o n s ( i i i ) and ( v ) , t h e f u n c t i o n s Hn a r e Again, we d e f i n e indeed a b s o l u t e l y continuous u n i f o r m l y i n n on [a,b]. f u n c t i o n s h n ( x ) = H,!,(x) a.e. on [a,bl, h ( x ) = f ( x ) when x E X and h ( x ) = H ' ( x ) When x 6 Xc. By Lemma 2.4, f o r every Thus, {h,} converges t o h a.e. on [a,b]. E > U t h e r e e x i s t s a p o s i t i v e i n t e y e r no such t h d t f o r every n > n u and a < al < bl < a2 < < bP < b we have
...
Expecially, f o r
di,
b i E X w i t h i = 1, 2,
..., p,
and t h e p r o o f i s complete.
Proof of Theorem 2.3. [a,b].
AFC, A,.,,
HISO
We may assume t h a t I f n } converges t o t everywhere on
we have fn E K * [a,b],
o f [a,b]
w
t h a t i s , f o r every
such t h a t f o r any A , - p a r t i t i o n
E
> U t h e r e e x i s t s an
Dn = {[UyVl;
XI
we have
S. Darrnawijaya and
66
Let
e
be t h e s e t o f d e n s i t y 1 a t x and x
e
i s , t o r e v e r y u,v
ttiere e x i s t s a p o s i t i v e integer (f,(X) The c o l l e c t i o n A = {[u,v]; Ca,bI.
By Theorem 2.2.
P=
A-partition
e
-
-&.
e ,D:
u,v
<
f(x)(
U;
f o r every x
A.,
e [a,b]
and (2)
u < x < v and x e [a,b]} dP
i s a n HFC o t [a,b].
For every
x) we have
By ( 2 ) , t h e t h i r d t e r m i s l e s s t h a n E ( b - a ) .
By (1) and H e n s t o c k ’ s lemma on
51, t h e second t e r i n o f t h e above i n e q u a l i t y i s l e s s
i a r t i a l sums [j. Theorem
.
e
sucti t h a t x
in
and c o n d i t i o n ( i i i ) , we have F E C
{[u,v];
t h a n 2 1;=1~.2-n-1
f r o m w h i c h we c o n s t r u c t A ~ ,t t l d t
U!
x < v we have [ u , v j
with u
U!
P Y. Lee
Therefore,
i t r e m a i n s t o show t h a t t h e f i r s t t e r m o f t h e
above i n e q u a l i t y i s sinal I . By c o n d i t i o n (ii), t h e r e e x i s t s a sequence o f c l o s e d s e t s { X i } [a,bJ
= U X i and Fn
e AC&(Xi)
subsequence {Fn(,,j)}
For a f i x e d X i , there e x i s t s
u n i f o r m l y i n n.
..., s,
... < bs
i s a subsequence o f {Fn(i-l,J)}.
Now
we h a v e
We may assume t h d t f o r each i, {Fn(i,J)} we c o n s i d e r F n ( J ) = F n ( J , J )
i n p l a c e o f Fn.
I t Ux i s t h e s e t o f d e n s i t y 1 a t x
f r o m w h i c h we c o n s t r u c t t h e HFC, A, above, we m o d i f y U, D,
= U;i
such t h a t rn = n ( J ) > n ( i ) whenever x e Y i =
-
and )f,,,(x)
d
< bl c a2
sucti t h a t f o r dny p a r t i a l d e v i s i o n al
w i t n ak, bk e X i , k = 1, 2,
such t h a t
xi
as f o l l o w s .
-
(XI
u
Xz U
We p u t
...
U Xi-1)
f ( x ) ( < E/(b-a).
I f i n o u r new AFC, A, t h e e n d p o i n t s o f a t y p i c a l i n t e r v a l [u,v]
A - p a r t i t i o n always l i e i n Xi (F(a,b)
in a
f o r some i, t h e n we h a v e
- 1 F m ( ~ , v ) I = 11 {F(U,v) m
<
1 1
-
F,,,(U,V)}(
. .
m
E.2-l-J
=
E
i = l j=1 and t h e p r o o f i s complete. same X i ,
But, however i n g e n e r a l u, v may n o t l i e i n t h e
To overcome t h i s we p r o c e e d d s f o l lows.
since
F,
E AC*
“kY 6k e
xi
and 1;=1(6k
(Xi) u n i t o r r n l y i n n, t h e n f o r e v e r y J t h e r e e x i s t s 6 i j > U ap such t h a t f o r e v e r y sequence o f n o n - o v e r l a p p i n g i n t e r v a l s Lak, e k ] w i t t i
-
a k ) < tiij
and f o r e v e r y A e ( 0 ~ 1 ) t h e r e e x i s t s s e t s
The Approximately Continuous Integral of Burkill EkA
C
-
[a,, B k l w i t h a k , B k e E:
-
Note t h a t (a,b)
and u ( t i ) > ( l - x ) ( ~ ~a k ) such t h a t
X i i s open and i s t h e r e f o r e t h e union o f a f i n i t e o r
L e t EiJ
c o u n t a b l e number o f open i n t e r v a l s .
-
number of i n t e r v a l s i n cd,bl
be t h e u n i o n o f a l l b u t f i n i t e
X i such t h a t t h e t o t a l l e n y t h o f tiJ i s l e s s
So, i f E i j = u k [ u k ,
thdn 6 i j .
67
vk] t h e n f o r e v e r y
x
B (0,1) there exists
> (1-x)(vk
H k c [ u k , v k ] such t h a t u k , vk € H i and p(H;)
-
uk).
Let
.
u
E?. = HA and E * . = U E l l n I f x e Yi n (Xi U Eij)O and m = n ( j ) we r e p l a c e 1J k k 1J* n 1J Ux by O x fl (Xi E . - ) . I f x i s a boundary p o i n t o f X i U E i j , t h e number o f
u
... .
1J
such p o i n t s i s countable, say, xl,
xz,
Since Fn(J) and F a r e a p p r o x i -
mdtely c o n t i n u o u s a t edCh Xky t h e n t h e r e i s a p o s i t i v e i n t e g e r nk and a s e t ok of d e n s i t y 1 a t Xk and Xk e uk such t h a t f o r any u, u ( ~ ~ ( ~ ) ( u ,
and
F ~
e uk we have
J F ( u , ~ ) \<
Hyain, i n t h i s case when x = Xk we r e p l a c e D,
E . z - ~ .
OF n Dk
by
A f t e r some m o d i f i c a t i o n s above, we have a new AFC, A,
P
= {[u,v];
Note t h a t x
x) be a A - p a r t i t i o n .
t h e coorespondiny sum, say IF(a,b)
w i t h m > nk. Let
o f [a,b].
'i,f o r some i. I f x = Xk, t h e n
6
11, i n
- 1 F,,,(u,v)(
=
11 F(u,v) - 1 F,,,(u,v)(
i s small and i n f a c t
/I1 F(u,v) - C1
F,,,(usV)(
< C1(F(u,V)( m
1
<
(E.2
-k
+
C1JF,,,(usV)(
+
dk) = 2.E
k= 1
Now suppose x # Xk f o r every k .
I t u e Xi we a p p l y ( 3 ) . [u,x]
= [u,w]
t o [w,x]
U [w,x]
i f w # x.
I n t h i s case we w r i t e [u,v]
Otherwise u
where w
8
e E * . and 1.I
X i and [u,w]
C
u
e E?
Eij.
1J
.
= [u,x]
U [x,v].
f o r some A ; t h e n w r i t e
Apply t h e i n e q u a l i t y ( 4 )
Consequently, we hdve 1F(a,b)
- 1 F,(u,v)I
< 5~
and t h e p r o o f i s complete.
3. Same c o r o l l a r i e s o f the controlled converyence theorem The f o l l o w i n y theorems may be c o n s i d e r e d as c o r o l l a r i e s o f t h e c o n t r o l l e d converyence theorem.
The p r o o f s a r e s i m i l a r t o t h o s e i n [4] and t h e r e f o r e
omitted. THEOREM 3.1.
The converse o f Theorem 1.1 holds.
THEOREM 3.2.
The c o n t r o l l e d converyence theorem h o l d s t r u e i f c o n d i t i o n s ( i i )
S. Darmawijaya and I? Y. Lee
68
and ( i i i ) a r e replaced by c o n d i t i o n ( i v ) .
THEOKEM 3.3.
The c o n t r o l 1 ed convergence theorem h o l d s t r u e i f c o n d i t i o n s ( i i )
and ( i i i ) a r e replaced by c o n d i t i o n : (vi) H
There e x i s t f u n c t i o n s ti and H such t h a t
e ALti&,[a,bj,
and ti(u,v)
(i,
H E C,,,[a,bl,
< Fn(u,v) < H(u,v) f o r every u, v
ti, 6
Ca,bl,
u < v and
f o r every n.
THEUHEH 3.4.
(The dominated converyence theorem)
The c o n t r o l l e d converyence
theorein h o l d s t r u e if c o n d i t i o n s ( i i ) and ( i i i ) are r e p l a c e d by c o n d i t i o n : (vii)
y ( x ) < f n ( x ) < h ( x ) a.e.
THEOREM 3.5.
on [a,bJ,
where g, h E H$,[a,b].
(The monotone converyence theorein)
The c o n t r o l l e d converyence
theorem h0Ids t r u e i f c o n d i t i o n s (ii)dnd ( i i i ) d r e r e p l d c e d by c o n t i t i o n s : (viii)
f l ( x ) < f2(x) < f3(x)
(ix)
The sequence I F n ( a , b ) )
THEOREM 3.6.
... a.e.
on [a,bJ,
converyes as n +
and m.
(The l o c a l l y approximate converyence theorem)
I f fn B R$,Ca,bl,
f o r every n, and { f n } i s l o c a l l y approxiinate converyent t o f on Ca,b],
then the
consequence o f t h e c o n t r o l l e d converyence theorem h o l d s .
References : 1. B u l l e n , P. S., "The B u r k i l l a p p r o x i m a t e l y continuous i n t e g r a ' I , J. H u s t r d l . Mdth. soc. ( S e r i e s A ) 35(1983), 2 3 6 - 2 5 3 . 2 . B u r k i l l , J. C., "Tne a p p r o x i m a t e l y continuous Perron i n t e y r a 'I, Mdth. Z. 34( 1931) , 271)-278. 3. Henstock, R., "A ttiernann-type i n t e y r d l o f Lebesyue power", Canad. J. Math. 20(1Y68), 79-87. 4. Lee Peny Yee and Chew Tuan Seng, "A b e t t e r converyence theorem f o r Henstock i n t e y r a l " , B u l l . London Math. SOL. 17(1985), 557-564. 5. Lee Peny Yee and Chew Tuan Seny, "A s h o r t p r o o f o f t h e c o n t r o l l e d converyence theorem f o r Henstock i n t e g r a l s " , B u l l . London Math. SOL. l Y ( 1 9 8 7 ) .
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
69
HARMONIC ANALYSIS ON CLASSICAL GROUPS Gony Sheny The C h i n e s e hiv e r s i t y o f S c i e n c e and Techno1 o y y L i S h i X i o n g and Zheny Xue An D e p a r t m e n t o f M a t h e m a t i c s , Anhui U n i v e r s i t y , H e f e i , P e o p l e ' s R e p u b l i c o f China The p u r p o s e o f t h i s a r t i c l e i s t o i n t r o d u c e b r i e f l y t h e p r i n c i p a l r e s u l t s i n 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 g r o u p s and i t s e x t e n s i o n on compact L i e g r o u p s i n China, and a l s o t o i n t r o d u c e b r i e f l y some i m p o r t a n t r e s u l t s i n t h i s d i r e c t i o n abroad.
P r o f . Hua Luo Geny h a v i ny accompl ished h i s famous work "Harmonic A n a l y s i s on C l a s s i c a l Domains i n t h e Theory o f F u n c t i o n s o f S e v e r a l Complex V a r i a b l e s " , a p p l i e d h i s t h e o r y t o 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,
deepened t h e
w e l l - k n o w n Peter-Weyl T h e o r m and i n i t i a t e d t h e r e s e a r c h on 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 i n C h i n a [l]. I n t h e l a t e 1 9 5 0 ' s , based on H u a ' s work a s y s t e m a t i c r e s e a r c h on 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 was c a r r i e d on w e l l [2]-[6].
A s e r i e s o f concepts,
d e f i n i t i o n s and methods were s e t up, w h i c h s u b s e q u e n t l y i n f l u e n c e d t h e r e s e a r c h f o r 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 g r o u p s and compact L i e yroups. t h a t t h e r e s e a r c h was suspended f o r a l o n y t i m e .
It i s a p i t y
The r e a s o n i s now w e l l - k n o w n .
I n t h e l a t e 1970's, t h e research regained i t s s t r e n g t h i n China.
The b a s i c
i d e a i s t h a t u n i t a r y y r o u p s a r e 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 f o r t h e complex c l a s s i c a l domains o f t h e f i r s t c l a s s and b o t h r o t a t i o n g r o u p s and u n i t a r y s y m p l e c t i c y r o u p s a r e c h a r a c t e r i s t i c m a n i f o l d s f o r r e a l c l a s s i c a l domains o f t h e f i r s t c l a s s and f o r t h e c l a s s i c a l domains o f t h e q u a t e r n i o n s r e s p e c t i v e l y . From t h i s p o i n t o f view, Wany Shi Kun, Dong Dao Zhen, He Zu Qi, Chen Guany X i a o and o t h e r s s y s t e m a t i c a l l y s t u d i e d t h e h a r m o n i c a n a l y s i 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 yroups.
L a t e r L i S h i Xiony,
Zheny Xue An, Fan Da Shan, Chen
Shun Fu c o n t i n u e d t h i s r e s e a r c h and e x t e n d e d i t t o compact L i e g r o u p s .
A t p r e s e n t t h e r e s e a r c h i n t h i s d i r e c t i o n i s c a r r i e d on w e l l . S i n c e t h e l a t e 1 9 6 0 ' s , on t h e o t h e r hand, many r e s e a r c h e r s a b r o a d have s t u d i e d t h e h a r m o n i c a n a l y s i s on compact L i e y r o u p s such as E. M. S t e i n , R. Coifman and G. Weiss, J. L. C l e r c , R. J . S t a n t o n and P. A. Tomas, R.
S.
N. J . Weiss, D. L. R a y o z i n , 6. D r e s e l e r , R. A. Mayer, M. E. T a y l o r , M. S u y i u r a , S. G i u l i n , P. M. Soard, G. T r a v a y l i n , 6. George, H. Johuen and Strichartz, others.
S. Gong et al.
I0
The p u r p o s e o f t h i s a r t i c l e i s t o i n t r o d u c e b r i e f l y t h e p r i n c i p a l r e s u l t s i n harmonic 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 i t s e x t e n s i o n on compact L i e g r o u p s i n China, and a l s o t o i n t r o d u c e b r i e f l y some i m p o r t a n t r e s u l t s i n t h i s d i r e c t i o n abroad.
The r e l a t e d p r o o f s a r e o m i t t e d .
1. Poisson Kernels and Abel Sumnation The r e s e a r c h i n 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 was i n i t i a t e d f r o m t h e P o i s s o n k e r n e l s on t h e c l a s s i c a l domains o f s e v e r a l complex v a r i a b l e s d e f i n e d by Hua Luo Geny. L e t RI b e t h e c l a s s i c a l domain c o n s i s t i n y o f a l l complex m a t r i c e s o f o r d e r n such t h a t
- Zll
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
.....
$1 P+1 )
P+l
+
( q,t
r( q-1,t
I f p < 0 , t h e n i t i s easy t o see from (1.13)
that (1.18)
Harmonic Analysis on Classical Groups
13
r e p e a t e d l y , we can o b t a i n
By u s i n y (1.18)
I f p < 0, t h e n t h e r e e x i s t s
( Similarly,
q;t
(
of
(
P.4
1.
oyq )
q-'
r-p
1
k=U
(
i t can be deduced f r o m (1.16)
I n v i r t u e o f (1.1Y) of
1=
and (1.2U),
(
P$ 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
Harmonic Analysis on Classical Groups
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 <
class L i p P
P < I),
lu(u)
-
~ : ( u ) I < A ~ N - ~ ,i f
2)
lu(u)
-
IF;(U)I
<
3)
lu(u)
-
I;(IJ)I
< A ~ N - ~ " + ~ - ~i f,
PROOF.
Fourier
satisfies
1)
where A1,
> (n-l)/n,
suinmable t o i t s e l f , and when u ( U ) b e l o n y s t o
I ~ ( u )( s e e ( 2 . 1 ) )
A
a
~
N
-
a n - n + l > P;
~ ri,~
Oi
f~ a n - n + l =
P;
an-n+l < P ;
A2 and A3 a r e a b s o l u t e c o n s t a n t s . Take
n
= max{s-(atl)(n-l),
Cesaro k e r n e l s K i ( B ) ,
O}.
1 k;(e) 1 ('-') I e I Take s = 2 ( n - 1 )
From an e s t i m a t e o f o n e - d i m e n s i o n a l
we can o b t a i n
i n the definition of
0,
< BNn-l-S
I NO 1'.
and
then
The d i r e c t e s t i m a t e o f ( 2 . 7 )
leads t o t h e conclusion.
Amony Cesaro k e r n e l s , t h e k e r n e l o f F e j e r means has t h e s i m p l e s t e x p r e s s i o n
BN( N+1) I'
where BN i s a number such t h a t t h e i n e y r a l o f (2.8) Bf(N)
=
1
Bf(N) =
on U,,
i s e q u a l t o 1 and
1
-7
N ( f ) c BN(N+l)
F e j e r means o f F o u r i e r s e r i e s (1.9
nN>fl>
1 ... >fn>-nN
o f u(U) reads
Bf(N)N(f)tr(CfAf(U))
The f o l l o w i n y t h e o r e m y i v e s t h e F e j e r means c o e f f i c i e n t s and t h e i n t e y r a l c o n s t a n t s 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 .
I n t h e p r o o f o f t h e theorem, a
c o m p l i c a t e d and i n g e n i o u s s k i l l f o r m a t r i x i n t e g r a t i o n i s used.
The same
S. Gong e t al,
80
lnethod can a l s o be a p p l i e d t o t h e c a l c u l a t i o n o f t h e c o e f f i c i e n t s and t h e i n t e y r a l c o n s t a n t s f o r g e n e r a l (C.a) THEOREM 2.3.
means.
[Sl The F e j e r means c o e f f i c i e n t s o f F o u r i e r s e r i e s on u n i t a r y
groups a r e Bf(N) =
2n
1
x
sl=o kl>O
(-l)n(n-l)/2(lto(l/N))
2n n+k ntk, ...sn=o 1 Cgn Cn ... C f n Cn 1 n
N((N+l)sl-fl
,...,( N + l ) s n - f n ) ,
(2.10)
kn>O
where k . = f . + n - j t n N - ( N + l ) s j , J J equal t o
j = 1,2,
...,n,
and t h e i n t e y r a l c o n s t a n t s RN a r e
(2.11) From t h e d e f i n i t i o n o f N ( f ) and Theorem 2 . 3 , Bf = 1 Here we s k e t c h t h e p r o o f .
+
i t can be seen t h a t
0(1/N).
F o r d e t a i l s , see [3].
Let (2.12) Then i f p c 2n, we have
k+(N+l)s=q
-
-
(2n)!
(p+q-( N + l ) s - l ) ! (q-(N+l)s)!s!(Zn-s)!
2n s=o
c
(2.13)
q- ( N+1) s>O and tltnN
Bf(N)N(f) = (N+l)
-n2
(-1)
n(n-1)/2
-1 BN
a2"
t2+nN aZn
tn+nN
tl+nN '2n-1
t2+nN a2n-l ).***
,..., a2n
tn+nN a2n-l
.............. tltnN a
n+l
t2tnN a n+l
tn+nN
,..., an + l
(2.14)
Harmonic Analysis on Classical Groups
81
( 2n+kl-1 ) !
(2n+kn-1)!
""'
kl!s1!(2n-sl)!
n kl >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 <
lm a
P
eipe.
e < 2n
and i t s F o u r i e r
Suppose t h a t T i s a summation and t h e k e r n e l s o f t h e
S. Gong et a1
82
are (2.18) N a t u r a l l y , t h e r e i s an a s s u m p t i o n o f t h e e x i s t e n c e o f t h e k e r n e l k m ( e ) , 7.e.
If
t h e converyence o f (2.18).
I;=a, eipe P
s f o r m tending t o a l i m i t , then
T,,, +
i s c a l l e d T-summable t o s.
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 \In arid i t s F o u r i e r s e r i e s be
1 N(f)tr(CtAf(U)).
(2.1Y)
f
Ayain l e t -1
T,,,(V)
= 6,
detn
(
umkVk)
-m
(2.20) Then t h e T-means o f (2.19)
is
F
(2.21)
Bf(m)N(f)tr(CfAf(U)),
where
(2.22)
B
m
= c-l
j'
detn (
(2.23)
pmkVk)v,
-m
"n and e
iel
,...,e i o n
.
i n (2.2U)
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.
Generally, replaciny km(81)
...kI,,(en)
i n (2.20)
by k e r n e l s kln(el,e2,,..,en)
o f m u l t i p l e F o u r i e r s e r i e s , we can y i v e a summation on u n i t a r y y r o u p s .
This i s
t h e method from k e r n e l t o sum.
THEOREM 2.4.
c11
L e t k e r n e l s k m ( e ) i n (2.20)
f o r any y i v e n 6 > 0 , where 6 c / e l c
1)
k N ( B ) = O(N-n)
2)
k N ( e ) = O(Nc) f o r any
3)
j'
(TN(V)(v c
satisfy
H, (m
e where 1 > 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]
Harmonic Analysis on Classical Groups
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
Harmonic Analysis on Classical Groups
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
j = 1,2,..
1 . = m.+k-j,
J
J
-!j- ... ( k -
1
,...,( n - 1 - s k ) ( N + l ) - m k ) ,
) - l N ( ( n-l-sl)(N+l)-ml
-
1'
((s +
II
.,k
and t h e i n t e y r a l
( j +;
)2)
U<j < s< k - 1
=
... ( 4 k - 2 ) ! ( N + l ) k ( 2 k - 1 )
(2k)! (2k+2)! 4k
1
x
s 1=u el>l-k
4k
... s I:=o (-1)
Sl+.
..+s k C4k
4k
sl"'csk
3k-l+el '2k
k
k ek>l-k
k x
3k-l+e
...Cilk
II ( ( n-1-s. ) ( N + l ) - k + j j =1
)-l N( ( n - l - s l ) (
J
where e j = P k N - ( N + l ) s j - k + j , The p r o o f o f Theorem 2.7
j = 1,2
N+1),
...,( n - l - s k ) (
N+l))
,
,...,k.
needs t h e method used i n Theorem 2.3
c o m p l i c a t e d and s k i l l f u l c a l c u l a t i o n .
For d e t a i l s ,
see
and needs a
El].
He Zhu Qi and Chen Guang X i a o d e f i n e d Ceasro k e r n e l s and Cesaro summation on u n i t a r y s y m p l e c t i c groups USP(2n), n = 1,2,
... .
L e t u ( U ) be i n t e g r a b l e on USP(2n) and i t s F o u r i e r s e r i e s i s u(U) where f = ( f l , f2
,...,f n ) ,
fl
- 1f
> f2 >
N(f)tr(CfAf(U
... > f n
u
1
(2.34)
a r e i n t e y e r s , Af(U) a r e t h e
u n i t a r y s i n y l e - v a l u e d 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 USP(2n) w h i c h t a k e f as i t s l a b e l s , N ( f ) are t h e orders o f Af(U), x f ( U ) = t r ( A f ( U ) ) a r e t h e characters o f Af(U),
and
cf
= c-1
J
where c i s t h e volume o f USP(2n) and So, Cesaro means o f (2.34)
u ( U)Af
(g')I?,
USP(2n)
is
6
i s t h e volume element.
Harmonic Analysis on Classical Groups
87
(2.35)
where
(2.36)
i s t h e Ceasro (C,a)
kernel.
B E a r e t h o s e numbers such t h a t
I n (2.37),
I
Ki(V);
= 1
USP( 2 n )
f o r N = 1,2,3,
...
THEOREM 2.8.
(He Zu
C il and Chen Guany Xiao, see [ l ] )
on USP(2n), U 6 USP(Zn), t h e n when a > ( 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 ( ~ )
1) 2) 3)
L e t u ( U ) be c o n t i n u o u s t h e F o u r i e r s e r i e s (2.34)
E
L i p p, t h e f o l l o w i n y h o l d
A ~ N - P , i f ( ~ n + l ) a - ~ n +>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 <, p.
I f a = 1, i t i s j u s t t h e F e j e r summation and i t s F e j e r k e r n e l s a r e 1
BN( N+1)n(2n+1) (
-
det(1 VN+l) )2n+l d e t ( 1 - V)
S i m i l a r t o t h e p r o o f o f Theorem 2.3 we can o b t a i n t h e F e j e r summation c o e f f i c i e n t s and i t s i n t e y r a l c o n s t a n t s .
THEOREM 2.9.
(He Z u Qi and Chen Guany Xiao, see 1 1 1 )
The F e j e r summation
c o e f f i c i e n t s o f F o u r i e r s e r i e s on t h e u n i t a r y s y m p l e c t i c y r o u p s a r e t h e followiny
x
4n+2
1
sl=o
... 4n+2 1 sn=o
kl>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)
Harmonic Analysis on Classical Groups
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,
Harmonic Analysis on Classical Groups
-
( - 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
<
+ ,)e1
qS
j ,s
'
< k , and e
=
tiel
/ sin a2k+l
0, e N ( e ) = s i n ( N + l ) e = Zmkik(2k-1)!
2 cos qe, s(ql, . . . , q k )
,...,e
fiek
...3 ! 1 ! , = det(S
,
/ s i n 1 8, C(q,
qS
,...,q k )
(ej)),
S
4
(el
are the characteristic roots o f
= = 2 i s i n qe,
r.
F o r t h 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 on S 0 ( n ) , t h e f o l l o w i n g t h e o r e m i s valid.
THEOREM 3.4.
(Wang Shi Kun and Dong Dao Zheng,
d e f i n e d on SO(n) and
u(r)
€
SO(n), moreover,
i f n = 2 k , where 0 < p
Ck(k-l)+P
F o u r i e r s e r i e s (2.29)
1 sN ( u
r e
-
of
u(r)
converge t o
L e t u ( r ) be a f u n c t i o n see [l])
u(r)
let
6
Ck2+p i f n = 2 k + l and
1, t h e n t h e p a r t i a l sums SN(U'r)
u(r)
u ( r ) l < A max{(N - l / ( k + l ) ( l o g
Of
and
N) k 2 / ( k+l)
,
N-P( l o g N) k-l)
1.
9 r )
The 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 (2.34)
of integrable function
u ( U ) on USP(2n) d e f i n e d by He Zu Qi and Chen Guang X i a o a r e
where ak = f k + n - k + l . I n t h e same way, we can o b t a i n SN(U,U) = c - l
U(V'U)UN( v
J
)i,
(3.16)
USP ( 2 n ) (3.17)
where a r e fli r i c h l e t k e r n e l s
THEOREM 3.5.
.
(He Zu Qi and Chen Guang X i a o , see [ l ] )
L e t u ( U ) be an i n t e g r a b l e
f u n c t i o n on USP(2n), U E USP(2n), t h e n t h e p a r t i a l sums (3.15) s e r i e s (2.34)
can b e e x p r e s s e d as (3.16)
and
of i t s Fourier
S. Gong et al.
94
(3.18) (2n-2k+l) ! d e t ( s i n ( n-p+l)e. ) k=l J l
II
tie1 where e
,...,e
ti en 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.
I t can be o b t a i n e d t h a t
(He Zu Qi and Chen Ggang Xiao, see [ l ] )
THEOREM 3.6.
f u n c t i o n on USP(2n) and u(U)
L
8
Cn 'p,
L e t u(U) be an i n t e g r a b l e
0 < p < 1, t h e n SN(u,U
c o n v e r g e s t o u( U)
u n i f o r m l y and
ISN(uyU)
-
u ( U ) ( < A max{(N-'loy n2 N) l / ( n + l ) ,
N-p(
oy N I n - ' } .
L i Shi X i o n g and Zheng Xue An s t u d i e d t h e c u b i c a l p a r t i a l sums and 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 g r o u p s . The c o o r d i n a t e r e p r e s e n t a t i o n s o f t h e C a r t a n s u b a l g e b r a H I o f any s i m p l e compact L i e a l g e b r a a r e known.
1; H =
Aj
As t o (An-l)u,
we have H I = {hAl.,.An(
U}, t h u s t h e c o o r d i n a t e r e p r e s e n t a t i o n f o r C a r t a n s u b a l g e b r a o f un a r e
=
{ hA1.. .An}.
The c o o r d i n a t e r e p r e s e n t a t i o n f o r C a r t a n s u b a l g e b r a s o f o t h e r
b a s i c compact L i e a l g e b r a s m e n t i o n e d i n s e c t i o n 2 c a n a l s o be d e c i d e d s i m i larly.
I f L i s a compact L i e a l g e b r a , t h e n we t a k e t h e d i r e c t sums o f t h e
above-mentioned c o o r d i n a t e r e p r e s e n t a t i o n s f o r C a r t a n s u b a l g e b r a s o f t h o s e b a s i c compact L i e a l g e b r a s w h i c h a r e i n c l u d e d i n 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 L ( s e e s e c t i o n 2) as t h e c o o r d i n a t e r e p r e s e n t a t i o n s o f t h e C a r t a n s u b a l g e b r a o f L.
These r e p r e s e n t a t i o n s a r e c a l l e d t h e s t a n d a r d c o o r d i n a t e r e p r e s e n t a t i o n s
f o r C a r t a n s u b a l y e b r a o f compact L i e a l g e b r a s . L e t H be a C a r t a n s u b a l g e b r a o f a compact L i e a l g e b r a L, and t h e S t a n d a r d c o o r d i n a t e r e p r e s e n t a t i o n f o r t h e p o i n t s i n H be hA1...A
xl,
..., P a r e n o t A
P n e c e s s a r i l y i n d e p e n d e n t o f each o t h e r .
r a l i t y , assume t h a t XlY...,A
4
.
G e n e r a l l y speaking,
W i t h o u t l o s s o f gene-
i s t h e maximal l i n e a r l y i n d e p e n d e n t system and
t h a t t h e system o f a f f i n e c o o r d i n a t e s composed by t h e maximal system i s c a l l e d t h e r e g u l a r system on H.
where f = (fl,
...,f 4)
Thus any w e i g h t A on H can be u n i q u e l y e x p r e s s e d as
are c a l l e d the regular coordinates f o r A.
L e t L be t h e d i r e c t sum o f a s e m i - s i m p l e compact L i e a l g e b r a L ' and t h e c e n t r e Hk.
N a t u r a l l y , as a system o f v e c t o r s on t h e C a r t a n s u b - a l g e b r a H I o f
L o , t h e system o f r o o t s o f L o i s j u s t t h e system o f v e c t o r s on t h e c o r r e s p o n d i n y C a r t a n s u b a l y e b r a H = HI tB Hk o f L w h i c h i s c a l l e d t h e system o f r o o t s
Harmonic Analysis on Classical Groups
95
f o r L and t h e g r o u p g e n e r a t e d by t h e r e f l e c t i o n s w i t h r e g a r d t o t h e r o o t s i n H i s c a l l e d t h e Weyl y r o u p f o r L. p o s i t i v e roots.
B e s i d e s , @ d e n o t e s t h e h a l f o f t h e sum o f a l l
Thus we v e r i f i e d t h a t t h e e q u i v a l e n t c l a s s o f a l l s i n y l e -
v a l u e d 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 f o r a compact L i e g r o u p i s u n i q u e l y d e t e r m i n e d by t h e h i g h e s t w e i g h t s .
M o r e o v e r , we e x p l i c i t l y e s t a b l i s h t h e method o f
c a l c u l a t i n g t h e regular coordinates f o r t h e highest weights. L e t G be a c o n n e c t e d compact L i e g r o u p and ?ibe t h e s e t o f t h e h i g h e s t w e i g h t s o f a l l n o n e q u i v a l e n t s i n g l e - v a l u e d 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 f o r G. Let AX(g), g f(y)
L(G).
E
E
x
which takes
G, be 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 f o r G
as i t s h i g h e s t w e i y h t , and d X be t h e o r d e r o f A X ( g ) .
Let
The F o u r i e r s e r i e s o f f ( g ) i s u s u a l l y e x p r e s s e d as (3.19)
or
where
(3.20)
c,
If X
I f(g)AX(g-’)dg
=
IG dg
and
G E
6
= 1, and X,(g)
and t h e r e g u l a r c o o r d i n a t e f o r
c u b i c a l p a r t i a l sums o f (3.19) SN(f;Y)
o r (3.20)
x
+ fl i s ( t 1 ,
N>kl,.
N>E1,.
...,E 9 )
then the
are
c..,E >-N c 4 ..,E >-N
=
i s t h e character o f AX(g).
d X t r ( CXAX(Y) ) dx f*XX ( Y )
(1
= fr)N(Y)
9
where D N ( g ) i s D i r i c h l e t k e r n e l s and DN(9) = N>El,.
Assume t h a t D1(q)
c..,a
dXXX(S). 4
>-N
i s a p o l y h e d r o n i n E u c l i d e a n space o f d i m e n s i o n q 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 t h e c o e f f i c i e n t s o f t h e e q u a t i o n s o f a l l faces being integers.
x
+ 6
E
S e t D N ( q ) = { x E E ~ , x = t y , y E D1(q),
0 < t < N).
DN(q) means t h a t t h e R e g u l a r c o o r d i n a t e s f o r A + fl b e l o n g t o UN(q).
Then 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 o f f ( g ) a r e
and D i r i c h l e t k e r n e l s a r e
I n t h i s c o n n e x i o n . we have t h e f o l l o w i n g r e s u l t s .
S. Gong et al.
96
THEOREM 3.7. ( L i Shi X i o n g and Zheng Xue An) L e t G be a compact L i e group, T be a maximal t o r u s o f G, dim G = n, d i m T = q, m = ( n - q ) / 2 and Lebesyue c o n s t a n t s f o r i t s U i r i c h l e t k e r n e l s be PN(G) =
j’
(’N(Y)Idy
G then
1)
AGN[n’31(10g
pN(G)
N)’,
where n I s mod 3, s = U,1,2,
[XI denotes
t h e g r e a t e s t i n t e g e r o f a l l i n t e g e r s t h a t a r e n o t g r e a t e r t h a n x, i f G t a k e s one of
2)
u k , (Bk),,, PN(G)
’
as i t s L i e a l g e b r a ;
(Ck)u, AGN “nt1)’31(10y
Nls,where n
# 3, n + l
5
s mod 3, s = U,1,2;
if G
t a k e s ( A k ) u as i t s L i e a l g e b r a . 3)
pN(G)
5 HGN l o g N,
4)
pN(G)
AGN, i f G t a k e s one of (Al),,,
5)
pN(G) < AGNm(log N)’,
t a k e s one o f 92, (F4),,,
i f G t a k e s ( G Z ) ~as i t s L i e a l g e b r a ;
where s = 2 f o r
( E 6 l u , ( E 7 ) u , (E81u,
(Bl)u,
e6
(Cl),,
as i t s L i e a l g e b r a ;
and s = 1 f o r t h e o t h e r s , i f G
e6, e6, e7 as i t s L i e a l g e b r a ;
...
@ L i s the 6 ) PN(G) = A G P N ( G ~ ) P N ( G ~ ) pN(Gp), if L = L 1 @ L2 @ P r e g u l a r d e c o m p o s i t i o n f o r L i e a l g e b r a L o f G and Gk i s t h e b a s i c compact L i e
group o f w h i c h L i e a l g e b r a i s L k , k = 1,2 7)
,...,p;
Lebesyue c o n s t a n t o f t h e k e r n e l D;(Y)
o f t h e p o l y h e d r a l p a r t i a l sums f o r
F o u r i e r s e r i e s on G s a t i s f i e s
where s < q and s = 1 f o r t h o s e L i e a l g e b r a s i n 1 )
-
5 ) and s = 2 o n l y f o r
e6.
Moreover, t h e c o n c l u s i o n i n 6 ) i s a l s o v a l i d f o r g e n e r a l compact L i e g r o u p s .
8 ) The c o n d i t i o n f o r 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 and 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 and t h e e s t i m a t i o n f o r t h e a p p r o x i m a t i o n o f t h e p a r t i a l sums t o t h e f u n c t i o n s can be deduced f r o m J a c k s o n t h e o r e m ( s e e Theorem 4.14,
3)).
I n Theorem 3.7, 1)
-
d e n o t e s t h e p r i n c i p a l p a r t o f pN(G).
F o r Theorem 3.7,
4 ) , t h e e x a c t v a l u e s f o r c o n s t a n t s AG a r e a l r e a d y o b t a i n e d by us.
U s u a l l y , t h e a b s o l u t e convergence o f F o u r i e r s e r i e s on compact L i e g r o u p s i s e x p r e s s e d by (3.21)
THEOREM 3.8.
( L i Shi X i o n g and Zheng Xue An)
i n Theorem 3.7,
L e t G, T, n, q, m be d e f i n e d as
Harmonic Analysis on Classical Groups
I f f ( g ) E L!'p(G),
1)
and p > 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 < p / ( p - 1 ) ,
f
1 < p c 2, U < s < 1, t h e n F o u r i e r s e r i e s o f f ( y ) s a t i s f i e s
p r o v i d e d k+s > ( ( 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 <
Z,, U <
s < 1, t h e n F o u r i e r s e r i e s o f f ( y ) s a t i s f i e s
p r o v i d e d k+s > ( ( 3 / r ) + ( 3 / p ) - 3 ) m + q / p .
4)
I f f ( g ) 6 L b y s ( G ) , 1 < p < 2, 0 < r < p / ( p - l ) ,
0 < s < 1, k i s non-
negative integer, then
p r o v i d e d k+s > (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
Harmonic Analysis on Classical Groups
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 <
N(f)tr(CfAf(U)) n
.+aE=m,
where
g i v e s us t h e means o f
4(6/R
4.1)
-1
< -, c o n t i n u o u s a t t = 0 and a ( 0 ) = 1..
t
d e f i n e d as f o l l o w s :
1 4(JG/R)
m
I:
fl>" . > 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)(
<
+
(4.3)
m.
f
If t h e l i m i t o f ( 4 . 2 ) e x i s t s f o r R + 4-summable t o a l i m i t .
I n (4.2),
m
then F o u r i e r s e r i e s f o r u(U) i s c a l l e d
b = n(n-1)(2n-l
/6.
Taking
where J s ( u ) i s t h e Bessel f u n c t i o n o f o r d e r s o f t h e f i r s t k i n d .
THEOREM 4.1. e x p r e s s e d as
(see
C61).
I f u ( U ) i s i n t e g r a b l e on Un,
t h e n (4.2)
can be
Harmonic Analysis on Classical Groups
a
D(
x
101
a
acl
(4.4)
acn
I , . . . , -
Here 6 ( t ) must s a t i s f y t h e f o l l o w i n g c o n d i t i o n s : 1)
b ( t ) i s a b s o l u t e l y c o n t i n u o u s on any d e f i n i t e i n t e r v a l ,
2)
jmI $ ( t ) I t ( n - 1 ) i 2 d t
<
(4.5)
-9
0
where 0 < j, (4.4)
,...,j n <
... + j n =
jl +
n-1,
n(n-1)/2.
can be r e w r i t t e n as
=
xf
(A)CfN( f)-l,
where A i s a d i a g o n a l m a t r i x and i t s d i a g o n a l e l e m e n t s a r e e
i5 1
,...,e i s n .
From t h i s , we o b t a i n
1
J
s ~ ( ~ ; u ) A ~ ( u =~ )c o + ( ~ ~ / R ) ~ ( J w R ) - ' . f
'n Thus t h e F o u r i e r s e r i e s of (4.4) all
w h i c h a r e i n t e g r a b l e f u n c t i o n s on Un f o r
R > 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 <
for
1 < t,
< 1,
t
t h e R i e s z summation o f o r d e r 6.
Then, i n t h e A b e l - ,
t h e Gauss-,
and t h e Riesz-summation o f o r d e r 6 o f
F o u r i e r s e r i e s f o r u ( U ) we c o n s i d e r (4.9) el+. 2 G s~(u;u) =
1 e- in/ R
tb
/R
m
..+eE=m
1 ... 'f,
fl'
N(f)tr(CfAf(U)),
( 4.10 )
.
el+. 2 .+e:=m and
Si(u;U)
=
fl'".'fn m = respectively.
2 el+
1
( 1 - b / R 2 ) - 6 ( 1-m/R2)& N( f ) t r ( C f A f ( U ) )
(4.11)
2 2 ...fen
I t i s o b v i o u s t h a t t h e s e t h r e e summations s a t i s f y t h e c o n d i t i o n s
i n Theorem 4.1.
THEOREM 4.2.
(see [S]).
means S i ( u , U )
o f t h e F o u r i e r s e r i e s f o r u(U) converges t o u ( U ) u n i f o r m l y .
2)
L e t u ( U ) E Lp(Un).
for R +
3)
m
1)
L e t u ( U ) be c o n t i n u o u s on Un.
Then SR(u;U) A 6 Lp(Un) and S i ( u ; U )
i n t h e norm o f Lp(Un), where p > 1; and I l S i ( u ; U ) l l
L e t u ( U ) be i n t e g r a b l e .
Then S;(u;U)
L e t u(U) E L i p a .
Then
A (SR(u;U)
-
u ( U ) I < AIK-'
i f 0 < a < 1,and
ISR(u;U) A i f a = 1.
-
P
converges t o u ( U )
< AoIlu(U)IIp.
converges t o u(U) f o r R +
everywhere.
4)
Then t h e Abel
u ( U ) ( < A2R-'log
R,
-
almost
Huniionic Anu1,vsis O I I Classical Groups THEOREM 4.3. Si(u;U)
(see [S]).
1 ) I f u ( U ) i s c o n t i n u o u s on U,
103
t h e n t h e Gauss means
o f i t s F o u r i e r s e r i e s u n i f o r m l y converyes t o u(U) f o r R +
modulus o f c o n t i n u i t y o f u ( U ) i s w ( t ) ,
-
IS;(u;U)
2) I f u(U)
B
-,
and i f t h e
then
u ( U ) ( < A3w(K
-1
),
G t h e n SR(u;U) 6 Lp(Un) and
Lp(U,),
G IIS~(U;U)II < A 4 ~ l u ( U ) i ~ P P’ and SG(u;U) c o n v e r y e s t o u ( U ) f o r K + i n t h e norm o f Lp(U,), where p > 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 <
for R +
be v a l i d f o r a ( t ) .
-,
Si(u;U)
m.
u n i f o r m l y converges t o u(U) p r o v i d e d u(U) i s
continuous. As i n t h e c a s e o f r o t a t i o n y r o u p s , Wang S h i k u n and Dong Daozheng 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 o n r o t a t i o n g r o u p s by s p h e r i c a l means. proved:
They
S. Gong e l a1
104
THEOREM 4.6.
u(r)
(Wang S h i k u n and Dong Daozheng see [l]). Let
on SO(n) and
1
m
where b = 1' + 2'
+
le(JTiIR)N(m)
... + ( k - 1 ) '
I
<
+
be i n t e g r a b l e
-,
(4.12)
i f n = 2k, and b = ( 1 / 2 ) 2 +
... + ( k - 1 / 2 ) 2
if
n = 2 k + l , and t h e n 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 o f t h e s p h e r i c a l means o f Fourier series for
u(r)
is
(4.14)
on any f i n i t e i n t e r v a l ;
= 0 , where 0
THEOREM 4.7.
(Wang S h i k u n and Dong Daozheng, see [ l ] ) .
< j,
,...,j k <
n-2.
By t a k i n g t h e above-
m e n t i o n e d t h r e e f u n c t i o n s as + ( t ) , d e f i n e t h e s o - c a l l e d t h e Abel, t h e Gauss and t h e R i e s z summations o f o r d e r 6 o f F o u r i e r s e r i e s on r o t a t i o n y r o u p s r e s p e c tively.
F o r t h e s e t h r e e summations 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 where
6 > n(n-1)/4-1/2
1)
u(r) 2)
These t h r e e summations u n i f o r m l y c o n v e r g e t o
u(r)
u(r)
for R +
m,
provided
i s c o n t i n u o u s on s O ( n ) .
Si(u;r),
provided 3)
i s needed:
u(r)
Si(U;r),
S i ( u ; r ) , i n t h e of L P ( S O ( n ) ) , p > 1, c o n v e r g e t o
u(r),
E LP(SO(n)).
G SAR ( u ; r ) , SR(u;T),
Si(u;r) a l m o s t e v e r y w h e r e c o n v e r g e t o u ( r ) , p r o v i d e d
i s integrable.
THEOREM 4.8. on SO(n),
(Wang S h i k u n and Dong Daozheng, see [ l ] ) .
Let
u(r)
be c o n t i n u o u s
and $ ( t ) be a b s o l u t e l y c o n t i n u o u s i n any f i n i t e i n t e r v a l and s a t i s f y
Harmonic Analysis on Classical Groups
4)
ti,
a ( --
a
a%
i f 151 > 1 / R ,
,..., -
Ht/2-1(1c1R) = O(R-p-llcl-p-kn+k 2 );
--k-l
Id
ack
t h e n S:(u;r)
105
u n i f o r m l y converges t o
u(r).
He Zuqi and Chen Guangxiao d i s c u s s e d t h e summation by s p h e r i c a l means on u n i t a r y s y m p l e c t i c g r o u p s and o b t a i n e d t h e f o l l o w i n y r e s u l t s .
THEOREM 4.9.
(He Zuqi and Cheny Guangxiao, see E l ] ) .
L e t u U) be i n t e g r a b l e on
USP(2n) and
2)
+ ( t ) i s a b s o l u t e l y c o n t i n u o u s i n any f i n i t e i n t e r v a l
Then t h e s p h e r i c a l means o f F o u r i e r s e r i e s f o r u ( U ) (4.15) al+. 2 whose i n t e g r a l r e p r e s e n t a t i o n i s ( - i I n R ( 2 n ) - n / 2 2-n Si(u;U)
=
n!D(nL,...,lL)n!+(JR)
where Q ( x l
,...,xn)
= xlx 2 . . . ~ n D ( ~ :
J
..+e:=m
-... J
-m
is,
is1
m
$(,e
,...,e
)
-m
,...,x n2 ) ,
tk = fk+n+l-k,
k = 1,2
,...,n,
and
b = n(n+1)(2n+1)/6. By t a k i n g t h e aboveA G m e n t i o n e d t h r e e f u n c t i o n s as + ( t ) , s p h e r i c a l means SR(u;U), SR(u;U) and
THEOREM 4.10. Si(u;U)
(He Zuqi and Chen Guangxiao,
see [ l ] ) .
o f F o u r i e r s e r i e s f o r u(U) a r e d e f i n e d r e s p e c t i v e l y , and 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 ( f o r t h e R i e s z means, t h e c o n d i t i o n 6 > n2 + ( n - 1 ) / 2 needed) :
is
S. Gong et al.
106
1 cont
The t h r e e s p h e r i c a l means c o n v e r g e t o u(U) f o r R + flUOUS
m
i f u(U) i s
on USP(2n).
F o r p > 1, t h e t h r e e s p h e r i c a l means c o n v e r g e t o u ( U ) i n t h e norm o f
2
L ~ ( u s P ( ~ i~ f) )U ( U ) e L ~ ( u s P ( ~ ~ ) ) . 3)
The t h r e e s p h e r i c a l means a l m o s t e v e r y w h e r e c o n v e r g e t o u(U) f o r R +
m
i f u(U) i s i n t e y r a b l e .
THEOREM 4.11. (He Zuqi and Chen Guanxiao, see [l]). c o n d i t i o n s i n Theorem 4.9.
where p > 0 and u(U) i s c o n t i n u o u s . for R +
Let $ ( t ) satisfy the
Moreover
Then S$(u;U)
u n i f o r m l y converges t o u(U)
m.
L i S h i x i o n g and Zheng Xucan d i s c u s s e d t h e s p h e r i c a l means and t h e more g e n e r a l means o f F o u r i e r s e r i e s on compact L i e groups. L e t G be a compact L i e group o f d i m e n s i o n n, T be a maximal t o r u s o f dimens i o n q o f G, rn = ( n - q ) / 2 , B(
,
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 G ,
) be t h e i n v a r i a n t i n n e r p r o d u c t on t h e L i e a l g e b r a o f G , and (
,
) * be
t h e special i n v a r i a n t i n n e r product which i s c a l l e d quasi K i l l i n g - C a r t a n form on compact L i e a l g e b r a s .
The r e l a t e d d e f i n i t i o n can be found i n " F o u r i e r
a n a l y s i s on compact L i e g r o u p s " t o appear i n "Advances i n Flathematics ( i n C h i n e s e ) [21]. Let f(g)
6
L(G).
We c o n s i d e r t h e f o l l o w i n g means o f F o u r i e r s e r i e s f o r f ( g ) .
1) L e t $ ( t ) , H!(t)
and W;(t)
d e f i n e as b e f o r e and c o n s i d e r t h e s p h e r i c a l
means o f F o u r i e r s e r i e s ( 3 . 2 0 ) f o r f ( g )
(4.17-1) t ;(h) 6
6
L ( H ) , ;(h)
be 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
H,
$ ( h ) = ( 2 ~ ) - ~ / ./' $(y)e-iB(hyY)dy H d e r t h e means
lA o(
AEG
I$( ft. ) - l d X f * X X ( S ) ,
(4.17-2)
(4.17-3)
(4.17-4)
Harmonic AnaI.vsis on Classical Groups Take $ ( h ) = W$2-l(lhl) (4.17-1),
i n (4.17-2).
I07
I t i s o b v i o u s t h a t (4.17-2)
becomes
where
( A + B I = lB(A+B,
A+B)) 1/2 ,
and, as a f u n c t i o n on H, g ( h ) i s i n v a r i a n t u n d e r t h e t r a n s f o r m a t i o n o f t h e Weyl yroup.
Thus g ( h ) can be u n i q u e l y e x t e n d e d as a f u n c t i o n on t h e L i e a l y e b r a f o r
G, t h e v a l u e s o f w h i c h a r e @ ( a d h ) = g ( h ) f o r h E H and y Y L e t a1
,...,am be
a l l p o s i t i v e r o o t s on H; p ( h ) = n j = l
be t h e o r t h o n o r m a l b a s i s f o r B( and h = xlXl+x2X2+
...+ x X
q q
,
) i n H;
a = ax
Y
Q(h) = ?r
B(h,aj);
a + x2
1 ax1
X1,X2
a +,,,+ ax2
,...,X 9 x a q axq’
Y
(exp h) =
I f(gt
exp(h)t-’)dt,
G
G, exp be t h e e x p o n e n t i a l mapping; A ( h ) = n y = 1 ( 2 i
sin
B(h,ai));
iB(ho,oh)
ZoeW
(4.17-4)
and C,
E
G.
be a p o i n t i n H; a g a i n l e t W d e n o t e t h e Weyl y r o u p ,
)111 be t h e o r d e r o f t h e Weyl y r o u p ; J, ( h ) = h E H, y , t
x
E
,
e
ho
e H.
Then t h e i n t e y r a l e x p r e s s i o n s f o r (4.17-1)
are respectively
depends o n l y on t h e L i e a l g e b r a .
For t h e means (4.17-1) e x p r e s s i o n s S!,R(f;g)
THEOREH 4.12.
, ,+
%
(4.17-4)
Si,R(f;g),
and f o r t h e i r c o r r e s p o n d i n g i n t e g r a l the following results are valid.
( L i S h i x i o n g and Zheng Xuean).
v a t i v e s o f up t o m - t i m e s on H.
L e t $ ( h ) h a v e L1 p a r t i a l d e r i -
Then t h e f o l l o w i n g h o l d .
S. Gong et al.
108
1 ) I f f ( g ) E L p ( G ) , p > 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$((A+e)/R)ldA <
+
m,
then,
i n t h e sense o f t h a t
A€G
1
A€G
(4.17-j)
l$((A+E)/R)dAf*xX(Y)
1
<
+
m,
a b s o l u t e l y c o n v e r y e s f o r a l m o s t e v e r y g 6 G and j = 1,Z.
And, i n t h e
meantime, ( 4 . 1 7 - j )
i s e q u a l t o S'? ( f ; g ) f o r a l m o s t e v e r y y 6 G and j = 1,2, J sR G(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 , where f ( g ) 6 L(G) b e s i d e s t h e above
if
mentioned c o n d i t i o n s .
5)
(4.17-j)
j = 1,2,3,4,
i n t h e sense o f 4 ) a b s o l u t e l y c o n v e r y e s f o r a l l R and
i f ( $ ( h ) I < C ( l t (h()-n'z-q/2-E,
E
> 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
Harmonic Analysis on Classical Groups
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+Bl
and i n d e p e n d e n t o f f ( g ) .
From 1 ) and 2 ) o f t h i s
theorem, t h e J a c k s o n Theorem f o l l o w s : ER(f) i f f(g)
6
CksW(G).
<
A(G,k,p)
liflk,wR-kw(l/R),
B e s i d e s , t h e B e r n s t e i n Theorem c a n be d i r e c t l y deduced b y
t h e u n i t a r y r e p r e s e n t a t i o n s f o r compact L i e groups. 4)
L e t 11 > ( n - 1 ) / 2 ,
M be an i n t e g e r , and
k M ' where T k ( y ) = f r o m 1,2,..
1
x. and any o f xl,
y
x2,
..., x
j =1 .,M. Then
M IIVR(f;Y)
P4 where sup llVRU < + R>O
THEOREM 4.15.
i s a sum o f k numbers c h o s e n $4
-
f(y)( <
PI
( S U P IIVRII +
-.
( L i S h i x i o n g , Fan Dashan and Zheng Xuean).
Then t h e k e r n e l s o f R i e s z means o f ( 2 k , 6 ) s a t i s f y 1)
2k,6, nKR
l)ER(f),
R>U
(g)iill
A(G,k)
l o y R,
( ( s e e (4.18)).
Take 6, = ( n - 1 ) / 2 .
S. Gong el al.
110
6
nsup ) S i k ' 6 ( f , y ) 1 i i p R>O
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 < p < 2, 6 > ( 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.
Harmonic Analysis on Classical Groups
[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
This Page Intentionally Left Blank
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
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&) < p < 2d/(2d-1-2&) ( HerzC101). On the otherhand if d = 2 and & > 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 <
in
-.
H1
is said to belong to
Hm
if
and
121
<
llflHm
121<1
Let
f E H2
. Then
we have the following properties:
lim f(reie) = f(eie) r-,1 Ilf(rei*)
(I)
-
(11)
2.2 The space N+ We have the following
holomorphic function
A
to
N
If
(
exists a.e. -,
o
as
r
+
1.
see e.g.171)
ftz)
in
lzl
<
1
is said t o belong
if
d 0 < -, s u p J-: log+lf(reie)l O
A function f in N is said to belong to N+ i f the inequality (1.2) holds for all l z l < 1. Thus N c N+ c H2. For details we refer the Duren's book t71. 1.3. Riesz-Thorin interpolatiom theorem and (N,Y,u) are In the following we assume that ( M , ¶ l , f i )
1
S.Igari
118
u-finite measure spaces. Let T be a linear operator from step functions in ( b l . 8 , ~ ) to measurable functions in ( N . 9 . u ) .
-.
Let 1 i po,pl,qo,ql i Suppose that T is a bounded mapping of LP' ( M ) to L9 0 ( N ) and of L P1 ( M ) to
THEOREM(Riesz-Thorin).
L91 ( N ) simultaneously and with norm I f 0 < 0 < 1 and
then
T
is a bounded mapping of
Co and
L Pe (M)
C1 respectively.
Lqe ( N )
to
with norm
ce L c y c ; . Several proofs of Riesz-Thorin theorem are known. Our proof is a slight variant of that of Caider6n and Zygmund "21. I t is essentialy same to that of Rochberg and Weiss C131 and i t will suggest the method used in Chapter 1 1 . We divide the unit circle 8D C-?t,?t) into two intervals I o , I l of length Zx(l-8) and 2 x 0 respectively. Let a ( z ) be l/po in int(Io) and = a function in H2 such that Re a(eie) p1 in int(I1), and 3111 a ( 0 ) = 0. For non-zero s mple function w in ( M , ! R , N ) and complex number l z l < 1 define WZ(X)
= llwl Po"(z) e i arg w(x), w(x)
pe
w(x) # 0 and = 0 otherwise, where "(2) = l/pe - a ( z ) . For 1 i p L ==,p' denotes the conjugate index of p, that is 1 = l/p + l/p'. Let 8 ( z ) be a function in H2 which is defined as a(&) for and q i . For given non-zero simple function f in ( N , ~ , u ) let q6l(z) 968 (2) e i arg f(x),f(x)l FZ(x) = Ilfl if
q i
f(x)
if
LEMMA.
f
0
and
= 0 otherwise, where
8(2).
ie
j'
it3 = 0,l.
nwu
I
uwe
and
(ii)
-
With the above notations we have
(i)
j
S ( z ) = l/qb
IIFe
k=llfY
qi
Wo(x) = w(x)
for and
pe eie
E
int(I.) J
Fo(x) = f(x).
PROOF. For (ii) remark that a(0) = l/pe and follows from the definitions of a and 8 .
8tO) = i/qb.
(i)
f
Interpolation of Operators in Lebesgue Spaces
PROOF OF THEOREM.
Let
w
and
f
119
be non zero simp e functions
in (M,W,p) and (N,%,v) respectively and define Wz and FZ as above. Put @ ( z ) = fN TWZ FZ dv. Then @ ( z ) E Hm, s nce Be a is bounded. Thus log I @ ( O ) I i logl@(eie)I d8
cX
.
By Holder's inequality and by our assumption we have
for
z = e i t E int(1.). J
IS
log
On the other hand
TW f dvl i iog(iiwii
pe
iifu,,)
e
+
l
@(O) =
co
(i-e)iog
Tw f dv. Thus +
eiog
cl.
Thus
Taking supremum over f
such that
l/fllq, = 1 , we get the theorem.
CHAPTER I 1
INTERPOLATION OF OPERATORS IN LEBESGUE SPACES WITH MIXED NORM 2.1. Lebesgue spaces with mixed norm Let d be a positive integer and (M.,W.,p.)(j = O,l, d-1) J J J d d d be 0-finite measure spaces. Let (M ,0l , p ) be the product measure space dil(M., W . , p . ) . For a subset p = (po,pl,. ,pm-l) j=O J J J of (O,l,...,d-1) let (M(p),fl(p),p(p)) = (M.,fR.,p.). Thus JgP J 1 J dp(p)(x ,...,x ) = dp (X )...dp (X ) . If pc= (0.1 d-1) PO 'm- 1 Po Po 'm-1 'm-1
...,
..
,...,
ons
f
in
Md such that
denotes the space of measurable funcllfll, = ( f If!' dpd )'Is < if s Md
;L~(M(P)) Lt(LS)
denotes the space of measurable functions
-
f
in
It <
Md such
-.
The definition for the cases s = or/and t = w 1 1 be obvious. In the following we assume that d = m + n , where m and n are positive integer, 1 i s,t i and u is defined by l/u = C m / s + n/t I/d. For 1 i s i , s ' denotes the conjugate exponent defined by 1 =
-
120
S. Igari
l/s + l/s'. 2 . 2 Auxiliary functions ; the case m L n Let P be the family of index sets p of ( 0 , ...,d-1 ) such that card(p) = rn. Let Q = ( p E Q : 0 E p ) and R = P - Q. For q E Q put Rq ( r E R ; card(rnq) m-n ) . Divide the unit circle 8D into card(P) congruent arcs I P' p E P. Assume 1 i s S t i Let a o ( z ) and a (2) be func2 9 tions in the Hardy space H whose real parts are defined by Table 1 and which satisfy Jm a o ( 0 ) Jm a q ( 0 ) = 0 .
-.
Definition of a o ( z ) and a
q
(2)
The case m 2 n Re
ao(e
I
l/s
%
l/s
q'
i e)
Table 1 Re a (eie) 9
0
6 Q
I C I"
-
R
-
l/t
l/s
0
l/s
6
l/t
(l/s-l/t )/card(Rq)
IP
9
(l/s-l/t )/card(Rq) R
0
R - R
9
l/t
By the mean value theorem we have
0
1
Interpolation of Operators in Lebesgue Spaces
For simple function W z and FZ as follows.
w
and
f
on
Md and
lzl
121
<
1
define
and
FZtx>
where and
LEMMA 1. Let d = m + n, in L n 2 1 and and f be non-zero simple functions fn followings. (i) Wo(x) = w(x) and Fo(x) = f(x). (ii) For p E P If
.
t ,s :P ) i II~U,, z = e i O E in (Ip). f is of the form fo(xo)fl(xl) fd- ( x ~ )-, ~then , z = e i O E int(1 1 . nFZll( t , s' :p) = Ilfllu, P
awzll
(iii)
1 i s i t i -. Let w Then we have the
M
(
...
For a proof see 1 1 1 1 . Auxiliary functions ; the case m < n d = mk + r , where k L 2 and m L r > 0, so that n = m(k 1) + r. We define a family P of m integers pa = a a a { pl,p2,. . . , p , ), a 1,2, d , by four cases : Let a = m j + b. Case A : 0 < j < k and 0 i b < m-r. Case B : 0 L j < k-1 and m-r S b < m. Case C : j = k and m-r i b < m. Case D : mk i a < b. Let a = m j + b correspond t o the point (mj+b,b) in Fig.2. Then the set pa is defined as a successive sequence indicated in Fig.2. Let a o ( z ) , aa(z) ( m i a < m(k+l)) and y ( z ) be functions 2 in H whose real parts are defined by Table 3 and which satisfy 2.3 Let
-
...,
Jm ao(0) = Jm aa(o) = Re y ( 0 ) 0. d F o r non-zero simple functions w and f in M and define Wz and FZ as follows.
lzl
< 1
122
+
P .r)
O
G3
E
M
N I * .r-
cr,
E
+k
Y
: E
I
rl
x
E
v
I
N
h
\
I: E
v
n E
.‘3
E
0
S. lgari
/
k
f
L
3 E 1
a
h
Interpolation of Operators in Lebesgue Spaces Definition of a o ( z ) and a
cl
( 2 )
Table 3
The c a s e m < n
I"....\ ! to
lea(e
ie
Re v ( e
l/s
i e)
0
l/u-l/t
I :
li
123
l/S
0
0
l/u-l/t
l/t
0
0
l/u-l/t
l/t
0
0
l/u-l/t
l/t
0
0
l/u-l/t
l/t-l/s l/t
a
0
l/U-l/t
l/t
0
0
l/u-l/t
l/s-l/t
I
.
1
.
l/t
a
0
l/u-l/t
l/t
0
0
l/u-l/t
I :
l/t-l/s
if
w(x)
it
0
lit
0
l/t
0
l/t
0
and
= 0
otherwise, w h e r e
U
l/u-l/t
Aw(z) = I I W I I ~ ~ ( ~ )and ,
S. Igari
124
if
f(x)
#
= 0 otherwise, where
and
0
Bf(z) = Ilfll-U'Y(z! U'
Let P = ( pa 1 and divide the unit circle 8D into card(P) congruent arcs I .Then we get a similar lemma as in 52.2. 2.4. a' 2.4. Interpolation theorems Let d = m + n. Let P be the family of m integers defined in 52.2 or 52.3 according to m 2 n or m < n and Ip' P E p, be arcs of the preceeding sections. Let ( M,I,p ) and ( N,Y.v ) d d d be 0-finite measure spaces. ( M(p),R(p),p(p)), ( N ,n , v ) , etc. will denote the spaces defined in 52.2. THEOREM 1. Let T be a linear operator of simple functions in M d to measurable functions in N . Let v(eie) be a measurable Define v by function in 8D such that 1 i v(eie) i
-.
1= v Sao (i)
1 i u o i u1 i
Let
2n
*
and
m
l/u Suppse that
1 4 8
v(eie)
(
m/uo+ n/ul)/d.
I I T ~ I Iie)i c(~~')II~II (ul,u
(2.1)
v(e 0 for simple functions w and e ie E int(Ip), p E P , where is a measurable function in 8D. Then
C(eie) (2.2)
IITwll, i C llwllu, where
c
exp JeD1og C(eie) 48 2n.
Let 1 A u 1 i uo i m . Then under the assumption (2.1) we have (2.2) for all functions w of the product form wo(x 0 )wl(xl) (ii)
...Wd-l(xd-l).
THEOREM 2 . Let T be a linear operator of simple functions in M d to measurable functions in Nd. (i) Let 1 i uo i u 1 i and 1 5 v1 < vo i m . Suppose that
-
IITWll(v 1 ,vo:p)i cpiiwn (ul,uo:p) for all If then
w
and
l/u =
(
p
E
P.
m/uo+ n/ul)/d
and
l/v =
(
m/vo+ n/vl)/d,
(2.3)
Interpolation of Operators in Lebesgue Spaces
125
IITwllv A C Ilwllu,
(2.4)
where
C = ( n C ) 1 /card (P) P PEP (ii) Let 1 i u1 i uo i 0 and 1 S v1 < v 0 I( m. Then under the assumption of (2.3) we have (2.4) for all functions w of product form. For a proof we refer t o Igari C 1 1 1 . APPLICATIONS OF INTERPOLATION THEOREMS TO SOME PROBLEMS ARISING IN FOURIER ANALYSIS
CHAPTER 1 1 1
In this chapter we shall show three examples of applications of the interpolation theorems given in Chapter 1 1 . They are closely related to the spherical summation problem of multiple Fourier transform. We omit the proofs since the proofs of Theorems 3 and 6 in 553.1 and 3.2 are given in 1111, and that of Theorem 8 will appear in Igari C121. 3.1. Estimates of Riesz-Bochner means Let P be the family of all subsets p c ( 0.1, ...,d-1 ) such that card(p) = d-2. Using the notations in Chapter I 1 we have THEOREM 3 .
for all
If
&
>
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) < P < 2, then IlsA(f)llp i c UflIp. (3.2) holds for 1 i p < 2d/(d+l) for radial functions. Tomas 1141 pproved that (3.2) is valid for general f if 1 i p < 2(d+l)/(d+3), but i t fails for p > 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 < p i (d+1)/2,
IIM9fIlp i C(log N) Nd/p- 1 II f II
.
for some constant 8 > 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.
<
I n general, t h e matrix
h a s a t l e a s t one r e c u r r e n t s t a t e .
D e f i n i t i o n 2.8.
m
%(s x s)
=
M3(s
X
S)
= M(s
Mr(s
X
s) =
A state
i
11
i s r e c u r r e n t i f a l l of i t s s t a t e s are r e c u r r e n t .
P
Define
M1(s x s ) = {P E M(s x s ) : {P E M ( s S)
X
MZ(s x s)
x
is irreducible},
P
s): P
- M1(s
U M3(s
X
is reducible recurrent}, S)
- M2(s x s ) ,
x s).
M3(s x s )
t r a n s i e n t s t a t e s and X
0.
i s called recurrent if
i
and t r a n s i e n t i f C p!n) n = l 11 We s a y t h a t t h e m a t r i x P
M(s
for all
For each p o s i t i v e 11 i s denoted by Pn = ( p(n) ,. ) =
m
A state
m
Note t h a t
lirn p . ( n ) = p . n 1
is c a l l e d p o s i t i v e if p . . > 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 < i 2 <
=
Then we have
2
2 5 n 5 s-1. Define t h e sequence s ) , m 2 1, a s fo1lows:
x
11
If
where
where
= (p. .(m))F M2(s
'm
F (Xn,a)
= pi,
p
S , and
(p,P ) Markov measures. We s e e r e a d i l y t h a t m 1 s-1 1, and P,,, * C c L k 1C %IR,T: A S ) . k=O
s
Mr(R,T)
=
%(R,T).
P r o o f . Let
P = ( p . .) C % ( s x s ) . Then t h e r e i s a unique p a r t i t i o n 11 2 5 n 5 s , o f S such t h a t e a c h P = ( p . . ) . . i s an
{Ck}:=l,
irreducible stochastic matrix. p r o b a b i l i t y v e c t o r such t h a t pk
.
Let
k pk = ( p i ( k ) ) i
pkPk = pk
and l e t
1 1 1.1
ck
C Ck
b e t h e unique
nk C Il be t h e e x t e n s i o n of
W e s e e e a s i l y t h a t Il(P) i s t h e convex h u l l o f {nkI1 n n n(P) = { 1 Cknk: 0 5 $ c 1, c Ck = 11 k= 1 k =1
L;
n, i . e . ,
.
Let
pk
be t h e
(nk,P)
denote t h e s e t of a l l Markov measures i n matrix
P
, i.e.,
convex h u l l of
M2(R,T: P)
{pkl1 i-
. Let M2(R,T: P) 'kP t h a t a r e g e n e r a t e d by t h e
Markov measure, i . e . , pk = p
=
,
M(R,T)
{ppp: P 6 Il(P)}.
Then
M2(Q,T: P)
i s the
Markov Measures
139
T h e r e f o r e , we o b t a i n M2(R,T) = U{M2(R,T: P ) : P
c
P = (p. .) 11
Suppose {C,
,..., Cn-l,Dl,
2
Mj(s
c
M2(s
s).
X
5)).
X
S
Then
, such t h a t
5 n 5 s
is uniquely p a r t i t i o n e d i n t o
i s t h e nonempty s e t o f t r a n s i e n t
D
Pk = ( p . . ) . . 1J 1 t J ck 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 . Let b e t h e 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 Pk and l e t nk c ll he t h e
s t a t e s and each pk
extension of
.
pk
If
M3(R,T: P)
d e n o t e s t h e s e t of a l l Markov measures
t h a t a r e induced by t h e m a t r i x P , t h e n n- 1 n- 1 M3(R,T: P) = { C \ p k : 0 5 ck 5 1, C \ = 11 k=1 k=l
in
M(R,T)
uk
where p'
=
denotes the
( p : .)
by
13
each
c
(i,j)
13
D
S
X
Markov measure.
,P)
('k
p:. = p..
f o r each
Define t h e m a t r i x
( i , j ) 6 (S-D) x S
and
P ' C M2(s x s )
and
13
.
I t follows t h a t
M3(R,T: P) c M2(Q,T: P ' )
so t h a t
Mr(R,T)
=
M2(R,T).
We now prove t h e f o l l o w i n g analogue o f Theorem 2 . 1 0 . __ Proof.
+
u
I t f o l l o w s from Lemma 4 . 5 , t o g e t h e r w i t h Lemma 3 . 2 , t h a t
where
(pn,P,)
c
M2(R,T)
u
To prove t h e compactness o f
=
Mr(R,T)
c
Pn
and
c
p
M(R,T).
Mr(s x s ) , pn
c
t h e r e i s a unique p a i r
( p , P ) 6 II
X
M1(s
Let
Suppose
1-1,
be t h e he t h e
p
p C M1(R,T).
such t h a t
s)
X
M2(R,T)
M2(R,T), suppose
II(Pn), and l e t
P 6 M(s x s ) , p 6 n ( P ) .
Markov measure w i t h
Lemma 4 . 4 , and
pn
Markov measure w i t h
(p,P)
See a l s o Remark 2 . 1 1
i s a compact, connected nonconvex s e t .
M2(R,T)
i s a connected nonconvex s e t .
un
for
c M2(R,T).
Consequently, Mj(Q,T) c M2(R,T) Theorem 4 . 7 .
p:. = 6 . . 11 11
pP
By p
=
I t f o l l o w s from p a r t ( i i ) o f Lemma 2 . 6 , t o g e t h e r w i t h 1~ PP ' Theorem 2 . 1 0 , t h a t P + P c Mr(s x s ) , a c o n t r a d i c t i o n . T h e r e f o r e we must have
=
c
p
M2(R,T).
The n e x t r e s u l t i s an analogue o f P r o p o s i t i o n 2 . 1 2 . Proposition 4.8.
Proof. sequence
{un} i n
M1(R,T)
Markov measure i n t h a t each
M1(R,T)
i s dense i n
M(O,T).
I t i s enough t o show t h a t , f o r each
such t h a t
M2(R,T).
Pk = ( p . . ) . '1 ' 9
. 3
'
1-1,
Suppose t h a t
+
u .
p 6 M2(R,T), t h e r e i s a
Let
{Ck}F=l
(i) choose
pk
Assume t h a t
ik,j,
such t h a t
c
Ck
.
ck
0 <
Then
be t h e
(n.P)
is a p a r t i t i o n of
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 .
be 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 of t h e m a t r i x
the extension of
u
p =
m k = l CkunkP where
'
Pk
, and l e t ?rk c
0 5 c
m
pk I[ be
C C k = l .
' k=l
$ < 1 f o r a l l k c { 1 , 2 ,..., m}. F o r each k , p . . > 0 . Let no he a p o s i t i v e i n t e g e r
such t h a t
pikjk> 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
The e n t r o p y map
[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,<) = l o g s n-1 _ . n p f o r a l l n 2 1, s o H ( v T ’ 5 ) = log s
From t h e p r o o f o f Lemma 5 . 2 , we g e t
p
t h a t , by C o r o l l a r y 4 . 2 . 1 o f W a l t e r s 171. p(x. = i . J I’ f o r a l l n Z 1.
o
o
0 5 j 5 n-1) I j’ By t h e uniqueness theorem of f i n i t e measures, we o b t a i n 5 j 5 n-1) = sn =
~ ( x .= i
p = x .
i s e v i d e n t , and t h e p r o o f i s complete.
The i m p l i c a t i o n ( i i ) = ( i ) Lemma 5 . 4 . (i) (ii)
Let
h (T) = 0 !J
11
B(n,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 :
( 6 . . ) . B e r n o u l l i measure f o r some
is the
l~
c
. j
11 1
Proof. -
Clearly ( i i )
=)
(i).
To prove ( i )
c
( i i ) , let
S p
. be the
B e r n o u l l i measure.
Suppose
h 1-1 (T) = 0 , i . e . , - C.p. log p. = 0 1 1
have
for all
i
p.log p. = 0
Z.p. = 1 1 1
,
, t h e r e i s a unique j E
so that S
pi = 0
such t h a t
or
.
1 for all
pi = 6 . . 11
for all
p Then we
i .
Since
i .
)-I is t h e ( 6 . . ) . B e r n o u l l i measure. 11 1 R e n e x t lemma i s a g e n e r a l i z a t i o n o f Lemma 5 . 4 .
Therefore
Lemma 5 . 5 . (i)
Let
h (T) = 0
lJ
p
.
M(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 :
C-W. Kim
146 (ii) p
'There e x i s t a p e r m u t a t i o n m a t r i x
of
which induce t h e measure
P
Proof. -
(i)
(ii):
measure where p'
=
(pi)i
P'
Note t h a t
and a s t a t i o n a r y d i s t r i b u t i o n
P
.
(i) holds,
Let
be t h e
p
Let E
s ) , p = (pi) t I I ( P ) .
M(S x
11
F = S - E.
and
Suppose
P = ( p . .) C
ii
= (p. . ) . .
c
{i
S : p . > 01
i s s t o c h a s t i c , and
11 I,] E is a p o s i t i v e p r o b a b i l i t y v e c t o r with
I:
(p,P) Markov =
. An
p'P' = p'
elementary c a l c u l a t i o n y i e I d s h (T) = X i
v
pij log p. .)pi 11
E(- Z j
0 ,
=
so t h a t -
cj c
f o r each
p i j log p . .
~
c
i
11
1
for a l l
i,j C E
.
i C E , t h e r e i s , f o r each
f o r each
pij = 1
unique q ( i ) C E
c
i
E
, a
such t h a t
p.. = 6
for all di)j Then t h e mapping rp: E -* E
j C E
11
that
0
, equivalently,
E
or
p.. = 0 11 Since Z j
=
j # rp(i)
must be s u r j e c t i v e .
for a l l i C 1
p . . = 0 for all 11 We have 0 < p . = Ci
, a contradiction, Since
p.p.. = 0 1
cp
I f not, there is
j
such
E
, equivalently,
i c E .
1
s e t , t h e mapping
.
11
is b i j e c t i v e , so t h a t
P'
E
is a f i n i t e
i s a permutation matrix.
Note t h a t f o r e a c h i C E and each j c S cp(i)j 4 , t h e n P = P' , s o t h a t we a r e done.
p.. = 6 11
If
F =
# @
Suppose
F
q..
p.. 11
11
=
.
Define t h e p e r m u t a t i o n m a t r i x
f o r each
(i,j) 6 E
X
.
remains t o show t h a t , f o r each
io,...,xn = in ) equivalently, p(xo
=
=
Q = ( q 11 . . )1. 9 .1 6
(p,Q)
0 5 j 5 n) = 0
=
io,..., xn = i n ) ,
n E 1.
, then p(x.
Let = i .
. . ,inti
io,.
n = 0,l
11
191
s
.
be any s t a t e s .
0 C j C ntl) = 0 =
I 1' 1 I' V ( X . = i . 0 5 j 5 n t l ) . I f ~ ( x =. i . 0 C j 5 n) > 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 .
The e n t r o p y map
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.
The e n t r o p y map
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.
The e n t r o p y map
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.
The e n t r o p y map
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.
The e n t r o p y map
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 ) .
This Page Intentionally Left Blank
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
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 <
S u b j e c t C l a s s i f i c a t i o n (AMS 1 9 8 0 ) : 20B05, 4 3 8 2 2 , 46G10. Key Words a n d P h r a s e s : B a n a c h m o d u l e , homomorphism, i n v a r i a n t operator, m u l t i p l i e r , Bochner i n t e g r a l , vect o r m e a s u r e , Radon Nikodym p r o p e r t y ,
H - C Laiand T.-K. Chang
152
For t w o B a n a c h s p a c e s X a n d Y , a b o u n d e d l i n e a r o p e r a t o r T from a B a n a c h f u n c t i o n s p a c e E ( G , Y ) t o a n o t h e r B a n a c h f u n c t i o n s p a c e F ( G , X ) i s s a i d t o b e a n invariant operator i f T commutes with t r a n s l a t i o n operator
-ra(a
G).
E
Through o u t t h i s p a p e r , t h e
s p a c e of a l l i n v a r i a n t o p e r a t o r s from
to
E(G,Y)
is de-
F(G,X)
n o t e d by (E(G,Y),
F(G,X)).
Our p u r p o s e o f t h i s p a p e r i s t o c h a r a c t e r i z e t h e s p a c e
of
inva-
r i a n t o p e r a t o r 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 . If V a n d W a r e A-module, i t i s known ( s e e R i e f f e l [ l l ] c f . also Lai [ 7 ] , [ 8 ] ) t h a t (1.1) HomA(V,W*) 2 (V
W)
i n which a l i n e a r o p e r a t o r
T
E
IJJ
tinuous l i n e a r functional
Here V
to
T
T(av) = aT(v) where
for
v
correspondingtoaconi s g i v e n by
V 8 W A
V, w
E
E
HomA(V,W*)
a
for all
satisfies
A, v
E
is a c o n t i n u o u s l i n e a r o p e r a t o r from
T
6yW/K,
V,
E
where
K
tensor product space form :
V
to
W*;
V QA W
is is t h e closed linear subspaceof t h e p r o j e c t i v e
d e n o t e s t h e A-module t e n s o r p r o d u c t s p a c e of V
W.
E
is t h e s p a c e of a l l A-module homomorphism from
t h a t i s , each
W*,
*
HomA(A,W )
on
( T v ) ( w ) = $ ( v 8 w)
* HomA(V,W )
*
av 8 w
-
V
8Y
W
v 8 aw
g e n e r a t e d by
a
f o r any
E
and
V
the
A, v
W, t h a t
element E
V, w
F
of W.
If X is a B a n a c h A-module, t h e n L p ( G , X ) , 1 < p < m 1 nach L (G,A)-module. I n [7] and [ 8 ] , Lai c h a r a c t e r i z e d s p a c e s of m o d u l e homomorphisms a s w e l l a s The module
isaBavarious
Ranach-valued f u n c t i o n
s p a c e s d e f i n e d on a l o c a l l y compact a b e l i a n g r o u p tain appropriate conditions,
the
G
homomorphism
under space
ceris
g e n e r a l l y c a l l e d t h e muZtipZier space.
I t i s w e l l known t h a t , i n
t h e scalar-valued function spaces over
G , a bounded l i n e a r o p e r a -
t o r is a m u l t i p l i e r i f and o n l y i f it is a n
invariant
operator.
For e x a m p l e , (1.2)
(L1 ( G ) , L 1 ( G ) )
Hom Id
where
M(G)
t h a t is, i f
G
1
1
( L ( G ) , L (G)) ;M(G)
(GI
d e n o t e s t h e s p a c e of b o u n d e d r e g u l a r n e a s u r e s o n
T
i s a b o u n d e d l i n e a r o p e r a t o r on
f o l l o w i n g s t a t e m e n t s are e q u i v a l e n t :
L1(G)
then
G,
the
Banach-valued Function Spaces
*
*
(a) (b)
T ( f * g ) = Tf .rsT = T T for
(c)
t h e r e is a u n i q u e m e a s u r e
g = f s E G
*
Tf = p
153
Tg f o r a l l f , g E L'(G) w h e r e ~ ~ f ( =t f) ( t s -1 ) = f ( t - s )
f
p
M(G)
E
such t h a t
f o r any
M o r e o v e r i t i s a l s o known t h a t
Horn
(1.3)
(L1(G
L (GI 1
p
and t h e r e l a t i o n s h i p between b o t h s i d e s of
m
by t h e f o l l o w i n g e q u i v a l e n t s t a t e m e n t : n e a r o p e r a t o r of T ~ =T T-rs
to
L1(G)
Lp(G).
Let
T
is given
="
b e a bounded
li-
Then
if and o n l y i f t h e r e e x i s t s a f u n c t i o n
g
E
Lp(G)
such t h a t Tf = f
*
for a l l
g
f
E
LL(G).
However, i n t h e B an ach - v al u ed f u n c t i o n s p a c e s , an i n v a r i a n t
ope-
I n [12], T e w a r i , D u t t a a n d V a i d -
rator need not be a m u l t i p l i e r . ya proved t h e following theorem. THEOREM A
([12]: T h e o r e m 3).
If
the
d i m A > 1,
dimension
t h e n t h e r e is a bounded l i n e a r i n v a r i a n t o p e r a t o r
T
of
L1(G,A)
s u c h th;ct T
t
1 HomA(L ( G , A ) , L 1 ( G , A ) ) .
Using t h i s r e s u l t , t h e y d i s p r o v e t h e Akinyele's r e s u l t s
operator on
t h e e q u i v a l e n c e between t h e m u l t i p l i e r and i n v a r i a n t L'(G,A).
I n [12], t h e y p r o v e d t h a t HomA(L1(G,A), L'(G,A))
(1.4)
provided
about
= M(G,A)
h a s a n i d e n t i t y o f norm 1.
A
This result
is
extended
b y L a i 181 a s i n t h e f o l l o w i n g t h e o r e m s . ( [ 8 , T h e o r e m 91 ) .
THEOREM B
Let
A
a l g e b r a w i t h a n i d e n t i t y o f norm 1 a n d X
b e a commutative
a B a n a c h A-module.
t h e f o l l o w i n g statements are e q u i v a l e n t : (a) (b)
Horn
(L'(G,A), L'(G,x)) L (G,A) t h e r e e x i s t s a u n i q u e p E M(G,X)
T
E
Tf = f Moreover
*
p
for a l l
such t h a t
f E. L1(G,A).
Banach Then
H.-C Laiand T.-K. Chang
154
X = A , (1.5)
Evidently i f
is r e d u c e d t o ( 1 . 4 ) .
THEOREM C ( 8 , T h e o r e m 6 ) .
g e b r a w i t h i d e n t i t y o f norm l a n d g i c a l d u a l and b i d u a l s p a c e s
b e a commutative Banach a l -
A
Let X
X*,
a n A-module,
X**
of
X
have
dym p r o p e r t y i n t h e w i d e sense w i t h r e s p e c t o t
I f t h e topolot h e Radon Niko-
G, then the
fol-
lowing s t a t e m e n t s are e q u i v a l e n t :
(b)
t h e r e e x i s t s a unique
Tf
= f
*
g
Lp(G,X)
E
for all
g
f
such t h a t t
L'(G,A).
Moreover, (1.6)
( L ~ ( G , A ) LP(G,x)) , z LP(G,x), 1 < p <
Hom
U?.
L (G,A) R e c e n t l y , Quek [ l o , T h e o r e m 91 p r o v e d t h a t i f
X
h a s t h e wide
Radon Nikodym p r o p e r t y t h e n t h e i s o m e t r i c i s o m o r p h i s m of ( 1 . 6 ) i n Q u e k ' s r e s u l t i m p o r v e s Theorem C s i n c e i f
Theorem C h o l d s .
embedded as a c l o s e d s u b s p a c e of Theorem 2
X**
i n t h e norm t o p o l o g y ,
D i e s t e l a n d Uhl[l; p . 8 1 1 i m p l i e s t h a t
01
X**
w i d e Radon Nikodym p r o p e r t y w h e n e v e r A s t h e r e m a r k i n [12: p.2291
indicates,
X
X
is the
has the
has.
it wouldbe i n t e r e s t i n g
t o c h a r a c t e r i z e t h e set o f a l l b o u n d e d l i n e a r i n v a r i a n t o p e r s t o r s on v a r i o u s 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 o v e r
G.
In t h i s paper,
we s h a l l characterize the spaces
(Ll(G,Y), Lp(G,X)), for 1 5 p a n d e s t a b l i s h t h e r e l a t i o n s h i p s i n t h e case Y = A between 1 I:om ( L ( G , A ) , Lp(G,X)) a n d i n v a r i a n t o p e r a t o r s . M o r e o v e r , L~ ( G , A ) s i n c e T h e o r e m A s h o w s t h a t n o t e v e r y i n v a r i a n t o p e r a t o r is a mul<
m
t i p l i e r , t h e question
arises
naturally
under
t hat
conditions
( L ' ( G , A ) , L P ( G , x ) ) is a m u l t i p l i e r . W e w i l l prove t h a t a necessary and s u f f i c i e n t condition is t h a t
every invariant operator i n A =
a,
2.
INVARIANT OPERATORS
t h e complex f i e l d ,
We w i l l p r o v e f i r s t t h a t i f e v e r y i n v a r i a n t o p e r a t o r of
A =
in
to
L'(G)
(1.5)
and
(1.6) then
LP(G,X), 1 5 p <
will
be a m u l t i p l i e r . TEEOREM 1. rator
T:
Let
L'(G)
+
X
A bounded linear opei s an i n v a r i a n t o p e r a t o r
b e a Banach s p a c e .
LP(G,x),
15p
i f and o n l y i f it is a m u l t i p l i e r .
<
m
155
Banach-valued Function Spaces
Proof.
is an i n v a r i a n t o p e r a t o r , t h e n f o r any
T
If
x*
E
X*,
t h e t o p o l o g i c a l d u a l s p a c e , d e f i n e a mapping
i t is a b o u n d e d l i n e a r i n v a r i a n t o p e r a t o r .
*
Indeed, T
is c l e a r l y l i n e a r w h e n e v e r
T
i s , and
X
5 IIx*ll IlTll llflll shows t h e c o n t i n u i t y o f
Tx*
(see Lai
T
[8]).
Now l e t
11 /Ipx
where
~
s,
Lp(G,X)
b e a t r a n s l a t i o n o p e r a t o r , we
G
E
i s t h e norm o f
then have T ~ ( Tf ) ( t ) =
T ,F(ts-')
X*
X
=
x*
= X*
0
Tf(ts-l)
0
~ ~ ( T f ) ( t )
t h a t is, TsTx* T h i s shows t h a t
= Tx*Ts.
is i n v a r i a n t whenever
Tx*
I t i s known t h a t ( s e e ( 1 . 2 ) f o r i m )
and
t h e i n v a r i a n t o p e r a t o r s and m u l t i p l i e r s
t h e case o f s c a l a r - v a l u e d any
p = 1
T
x*
E
function spaces.
is. (1.3)
for
1 < p
are e q u i v a l e n t
It follows
that
in for
x*, x*
*
T(f
*
9 ) = Tx*(f =
* *
f
= f =
x*
g)
Tx*g (x* 0
(f
0
Tg)
*
Tg)
1
for a l l
f, g
E
L (G).
for a l l
f, g
E
1 L (G).
Thus (a)
x*
0
Now i f w e t a k e
T(f
f,g
*
g ) = x* E
0
*
(f
Tg)
Cc(G), t h e c o n t i n u o u s f u n c t i o n s w i t h compa-
c t s u p p o r t , t h e n t h e s u p p o r t of
f
*
Tg
is c o n t a i n e d i n a c o m p a c t
H.-C. Lai and T.-K. Chang
156
K = s u p p f C G , a n d s o from C o r o l l a r y 7 o f D i e s t e l a n d U h l [l, p . 4 8 1 , w e see t h a t t h e i d e n t i t y ( a ) i m p l i e s
subset
(b)
T(f
*
*
g) = f
Tg
almost e v e r y w h e r e a s an e l e m e n t i n Since f,g
is d e n s e i n
Cc(G) L1(G).
E
The
L1(G),
for all
Lp(G,X)
(b) holds i n
f,g
Lp(G,X)
E
Cc(c;).
for
all
Hence
"if part"
complete.
o f t h e o r e m i s known.
Therefore t h e
proof
is
Q.E.D.
A p p l y i n g T h e o r e m 1, w e c a n e s t a b l i s h t h e
following
theorem
for invariant operators. THEOREM 2 .
Let
X
Y
and
two
be
Banach
space.
Then
the
f o l l o w i n g two s t a t e m e n t s a r e e q u i v a l e n t (i) (ii)
T E (L'(G,Y), L'(G,x)) t h e r e e x i s t s a unique
L
linear operator o f
to
T(f 8 y )
= f
*
Y
Ly
E
E(Y, M ( G , X ) ) ,
the
bounded
M(G,X), s u c h t h a t
for a l l
f
E
L'(G),
y
L1(G),
y
Y.
Moreover 1 1 ( L ( G , Y ) , L (G,X)
(2.1)
Proof.
( i ) ==a
we can w r i t e
E(Y, M(G,X)
(ii).
T(f Q y ) = T(fy)
Y, we d e f i n e
for a l l
f
E
E
Y.
For
T : L1(G) -+ L1(G,X) by Y T f = T ( f y ) f o r a l l f E L1(G). Y Evidently, T i s t r a n s l a t i o n i n v a r i a n t w h e n e v e r T i s . So t h a t Y 1 is a Ty E (L1(G), L ( G , X ) ) . A p p l y i n g T h e o r e m 1, w e see t h a t T Y mult iplier , That is, T E Hom ( L 1 ( G ) , L1(G,X)). Y L each
y
E
Banach-valued Function Spaces
I t f o l l o w s f r o m Theorem B, by t a k i n g M(G,X)
E
VY
Y
*
M(G,X)
for all
Y
Note t h a t
/ I T y / I = llpyll.
-f
that there exists
(c,
a
such t h a t T f = f Y
and
A =
157
I/TyII
f
L'(G)
E
5 IIylMlTII.
Thus
the
mapping
d e f i n e d by
L: Y
_ j
pY
is bounded l i n e a r s u c h t h a t T(fy) = (ii)
4
*
f
L(Y)
L
Conversely i f
M(G,X)), we d e f i n e d a mapping
:(Y,
E
1
rA
T:
(G)
Y
x
1 TL(f,y) = f* L(y)
by Then
Ti
I I L I / 5 IlTIl.
with
(i).
+
L 1( G , x )
for all
f
1 L (G), y
E
Y.
E
is a b i l i n e a r c o n t i n u o u s o p e r a t o r , and by t h e u n i v e r s a l
p r o p e r t y o f t e n s o r p r o d u c t , t h e r e e x i s t s a l i n e a r map TL:
8
L 1( G )
Y
Y = L 1( G , Y )
>
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 <
can be charac-
H - C Lai and T.-K. Chang
158
t e r i z e d a s g o o d a s t h e p r o o f o f T h e o r e m 1.
W e s t a t e i t as i n t h e
following theorem. Let
THEOREM 3.
X
and
Y
b e Banach s p a c e s .
X
If
has
the
G, then t h e following
w i d e Radon Nikodyrn p r o p e r t y w i t h r e s p e c t t o two s t a t e m e n t s are e q u i v a l e n t :
T E (L'(G,Y), L ~ ( G , x ) ) t h e r e e x i s t s L E E(Y, L p ( G , X ) ) , 1
(i) (ii)
*
T(f 0 y ) = T(fy) = f
i
p <
€or a l l
L(y)
m
f
E
such
that
L'(G),
y
t
Y.
Moreover
1 (L ( G , Y ) , L ~ ( G , x ) )2 REMARK 1.
Note t h a t i f
(i)
E(Y, LP(G,x)),
Y =
T h e n T h e r o e m 2 a n d 3 re-
(c.
d u c e t o T h e o r e m 1. (ii)
If
Y =
t h e n Theorelrs 2 and 3 are c o i n c i d e
X
(c =
with
t h e usual m u l t i p l i e r s , t h a t is,
1
1
(L ( G ) , L ( G ) ) Z Hom
(L1(G),
L1(G))
Z M(G)
L (GI
1
1
(L ( G ) , L ~ ( G ) z ) IIom
(L ( G I , L P ( G ) )
r
L~(G),
L (GI MULTIPLIERS OF VECTOR-VALUED FUNCTION SPACE
3.
bra.
i n Theorems 2 and 3 , b e a commutative Banach a l g e -
Y = A,
Let
Then w e h a v e t h e f o l l o w i n g c h a r a c t e r i z a t i o n s , Let
A
b e a commutative Banach a l g e b r a
have i d e n t i t y ) and
X
a B a n a c h A-module,
THEOREM 4 .
Hom
(3.1)
(L1(G,A),
L1(G,X))
(need not
Then
2 HomA(A, M ( G , X ) ) .
L (G,A) Proof. Horn
We
have
known
(L1(G,A),
L (G,A) (L1(G,A), L1(G,X). e x i s t s a unique
Here
L(a)
E
*
f
=
I ( A , M(G,X))
E
M(G,X) f
*
L(a) and
grable function over v o l u t i o n , and
G
L(a)
Hence i t is a n e l e m e n t o f T
and
L
a
multiplier
T c
Y = A,
with
T
E
there
such t h a t
for all
f
E
L1(G),
a
E
A.
*
L ( a ) is a n X-valued Bochner i n t e 1 s i n c e L (G) a c t s o n M(G,X) u n d e r c o n f
v a n i s h e r s o n t h e s i n g u l a r p a r t o f M(G,X).
L 1( G , X )
i n ( a ) is w e l l posed. 1 E L (G), a, b
Moreover, f o r f , g
operator
is an i n v a r i a n t o p e r a t o r , t h u s
A c c o r d i n g t o Theorem 2 L
T(fa)
(a)
that
L1(G,X))
and t h e r e l a t i o n s h i p
E
A,
between
159
Banach-valued Function Spaces
T(fa
*
gb)
=
T((f
T(fa
*
gb)
=
fa
*
g)ab)
=
*
(f
g)
*
L(ab)
and
*
T(gb)
(f
*
g)
a, b
E
A.
=
*
aL(b)
we then have L(ab) This shows that L
=
L
is an A-module homomorphism, that is
HomA(A, M(G,X)).
F_
Conversely, for (b)
for a l l
aL(b)
TL(fa)
=
L c HomA(A, bI(G,X)), we define 1 f * L(a) f o r all f E L (G), a
E
A.
1
Then fa E L (G,A), TL is a bounded li.near mapping from L1(G,A) 1 to L1(G,X) since L1(G) * M(G,X) is laid in the space of L (G,X). 1 We want to show that T is an L (G,A)-module homomorphism. In 1 1 1 fact, f o r any f E L (G) and a E A , fa E L (G,A) = L (G) 6, A , thus each h E L1(G) 8, A can be written by h = gb for some g E L1(G) and b E A. Then TL(h
*
fa)
=
TL(gb
=
TL((f g * f
=
= (g =
=
and
TL
is an
*
* * *
fa)
g)ba) L(ab) f ) * bL(a)
(bg) * (f h * TL(fa)
*
L(a))
1 L (G,A)-module homomorphism.
It is easy to show IITII = /ILII for T and L in the relations Therefore the isometric isomorphism of (3.1) is proved. Q.E.D. of (a) and (b).
Under the same argument as in Theorem 4, we have the following theorem. THEOREM 5. Let A be a commutative Banach algebra and X an A-module. Suppose that X h a s the wide Radon Nikodym property with respect to G. Then (3.2)
Hom
(Ll(G,A), Lp(G,X)) :HomA(A, Lp(G,X)) L (G,A)
for
1 < p <
m.
REMARK 2. If A has an identity of norm 1 in Theorems 4 5, then (3.1) is isometrically isomorphic to M ( G , X ) , and
aiid
H - C Lai and T.-K. Chang
160
is i s o m e t r i c a l l y isomorphic t o
(3.2)
Lp(G,X).
NECESSARY CONDITION FOR AN INVARIANT OPERATOR TO BE A MULTIP-
4.
LIER
an
invariant
o p e r a t o r t o b e a m u l t i p l i e r i n Banach f u n c t i o n s a p c e s .
T h i s s e c t i o n g i v e s a main c h a r a c t e r i z a t i o n
Although
for
a m u l t i p l i e r is an i n v a r i a n t o p e r a t o r , t h e c o n v e r s e is n o t f o r e x a m p l e o n e c a n c o n s u l t Theorem A .
The
following
true, theorem
g i v e s a s l i g h t e x t e n s i o n o f Theorem A. THEOREM 6 . L e t A b e a c o m m u t a t i v e B a n a c h a l g e b r a o f dimens i o n l a r g e r t h a n 1 t h a t h a s a n i d e n t i t y o f norm 1, X b e a n A-mod u l e . Then t h e r e e x i s t s a bounded linear i n v a r i a n t o p e r a t o r T of
to
L1(G,A)
Proof.
Lp(G,X)
such t h a t
We o n l y p r o v e t h e case when
p = 1.
If
1 < p < m,the
p r o o f o f t h e o r e m is t h e same, mutatis mutandis, a s t h e
case
x
Let
b e a n y nonzero c h a r a c t e r o n
t i v e l i n e a r f u n c i t o n a l on
e
where
t o r on
is t h e u n i t o f
Since
A.
an element
b
E
A
{ea}
a
for all
is a bounded l i n e a r o p e r a -
$
Then
A.
G. A
+
by
A
+
A,
E
E
A}
i; A .
Thus
? b
For each M(G,X)
c1
and
(Lp(G,X)
L
L1(G)
having
1< p <
f o r any
a
E
L'(G)
x
A
-
p,(a)
A.
E
*
f
E
is b o u n d e d b i l i n e a r , i t f o l l o w s f r o m t h e u n i v e r s a l
1
,
L (G)
ay
A
-
L'(G,A)
@
are
1 L (G,x)
t e n s o r p r o d u c t t h a t t h e r e e x i s t s a b o u n d e d l i n e a r map T ~ :
ma-
by
m)
is a bounded l i n e a r o p e r a t o r s i n c e L and I t is e a s y t o see t h a t t h e m a p p i n g bounded l i n e a r . (f,a)
compact
HomA(A,X), w e d e f i n e a
E
for
pa
g i v e n by
is
= b$(e).
L p ) = L(+(a)ea Then
there
such t h a t
b e an approximate i d e n t i t y of
support i n pa:
I$: A
d i m A > 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
<
is a m u l t i p l i e r i f a n d o n l y
m
Ic.
Proof.
T h e s u f f i c i e n t c o n d i t i o n f o l l o w s f r o m T h e o r e m 1.
w e h a v e o n l y t o show t h e c o n d i t i o n
A z
(c
is n e c e s s a r y .
Thus
Suppose
that
A p p l y i n g Theorem 6 , i f
A
(c
and so
dim A > 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
<
bi
w i t h ai,
1
li ( b i -
A f u n c t i o n F i s s a i d t o be
> 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
<
w(F;Cai,bil)
E
1
where w denotes t h e o s c i l l a t i o n o f F over [ai,bil. ACG* i f [a,b]
A f u n c t i o n F i s s a i d t o be
i s t h e u n i o n of a sequence o f c l o s e d s e t s X i such t h a t on each Xi
t h e f u n c t i o n F i s AC*(Xi).
A f u n c t i o n f i s s a i d t o be Denjoy i n t e g r a b l e on
Ca,bl i f t h e r e e x i s t s a f u n c t i o n F which i s c o n t i n u o u s on [a,b] t h a t i t s d e r i v a t i v e F ’ ( x ) = f ( x ) almost everywhere i n [a,b]. s u l t s c o n c e r n i n g t h e Denjoy i n t e g r a l , see [ 9 l and [ S l .
and ACG* such For f u r t h e r re-
I t i s well-known t h a t
t h e Denjoy i n t e g r a l i s e q u i v a l e n t t o t h e P e r r o n i n t e g r a l , and a l s o t h e Henstock-Kurzweil i n t e g r a l [41. A f u n c t i o n H i s s a i d t o be a major f u n c t i o n o f f i n [a,bl
f(x) < where
0
0
H(x)
denotes t h e lower d e r i v a t i v e .
#
-
m
if
f o r every x
A f u n c t i o n G i s s a i d t o be a m i n o r
f u n c t i o n o f f i n [a,b] i f - G i s a major f u n c t i o n o f - f i n [a,b]. The f o l l o w i n g convergence theorem was proved i n [8] as a consequence o f t h e c o n t r o l l e d convergence theorem.
Here we g i v e an independent p r o o f .
GENERALIZED DOMINATED CONVERGENCE THEOREM. [a,b]
L e t fn be Denjoy i n t e g r a b l e on
such t h a t f n ( x ) + f ( x ) almost everywhere i n [a,b]
p r i m i t i v e s Fn o f fn converge u n i f o r m l y on [a,b].
as n +
m,
and t h e
I f fn have a t l e a s t one
common major f u n c t i o n and a t l e a s t one common m i n o r f u n c t i o n i n [a,b],
i s Denjoy i n t e g r a b l e on Ca,bl and we have
1,” f n ( x ) d x
+
1,b
f(x)dx
as n +
-.
then f
P.Y.Lee
164
We say t h a t a p o i n t x i s r e g u l a r i f t h e consequence o f t h e t h e o r e m
Proof.
Then t h e s e t Q o f a l l p o i n t s x
h o l d s f o r some open s u b i n t e r v a l c o n t a i n i n g x.
L e t H and G b e r e s p e c t i v e l y t h e common m a j o r and m i n o r
n o t r e g u l a r i s closed. f u n c t i o n s o f f.
Then b o t h a r e VRG*
( s e e [9,
page 2341).
Hence, i n v i e w of
B a i r e ' s c a t e g o r y t h e o r e m , t h e s e t o f r e g u l a r p o i n t s i s non-errpty. L e t (ai,bi),
i = 1,2,
..., be t h e s u b i n t e r v a l s
i n [a,b]
Then f i s D e n j o y i n t e g r a b l e o n any i n t e r v a l [ u , v ]
t o Q.
F n c o n v e r y e s u n i f o r m l y on [a,b],
f i s D e n j o y i n t e g r a b l e on [ai,bi]
and
lai fn(x)dx
which a r e contiyuous
+
bif ( x ) d x
as
n +
Since
c (ai,bi).
f o r each i
-.
i Again, by B a i r e ' s c a t e g o r y theorem, t h e r e i s a p o r t i o n Qo o f Q such t h a t H and G a r e VB* on Qo.
Take J o t o b e t h e s m a l l e s t i n t e r v a l t h a t c o n t a i n s Qo.
Then i t f o l l o w s f r o m L e b e s g u e ' s d o m i n a t e d convergence theorem t h a t f i s Lebesgue i n t e g r a b l e on t h e c l o s u r e o f Qo.
Also, i n view o f t h e f a c t t h a t
< w ( H ; I ) + w(G;I) f o r any i n t e r v a l I b e l o n g i n g t o Jo - Qo, t h e s e r i e s o f o s c i l l a t i o n s o f Fn o v e r t h e i n t e r v a l s i n J o c o n t i g u o u s t o Qo c o n v e r g e s u n i f o r m l y i n n. By t h e f a c t w(Fn;I)
t h a t t h e Oenjoy i n t e g r a l i s c l o s e d u n d e r Harnack e x t e n s i o n (see [9,
f i s Oenjoy i n t e g r a b l e on J o w h i c h i s a c o n t r a d i c t i o n .
page 24Y]),
Hence t h e p r o o f i s
complete.
I f t h e common m a j o r and m i n o r f u n c t i o n s a r e b o t h c o n t i n u o u s , t h e n by M a r c i n k i e w i c z ' s t h e o r e m [9, page 2531 t h e f u n c t i o n s s u p { f n ; n > 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)(
<
E(V-U(
6 ( 5 ) < u < 5 < v < 5 + 6 ( 5 ) , t h e n t h e consequence o f
t h e theorem holds. We remark t h a t t h e c o n d i t i o n i n C o r o l l a r y 2 may h o l d o n l y f o r c, E [a,b]
-
D
where D i s c o u n t a b l e and f o r each 5 E D t h e sequence Fn converges u n i f o r m l y i n an open n e i g h b o u r h o o d o f 5.
The Denjoji Integral COROLLARY 3.
165
L e t fn be D e n j o y i n t e y r a b l e on Ca,bl w i t h p r i m i t i v e Fn such
t h a t f n ( x ) + f ( x ) almost everywhere i n [a,bl d i f f e r e n t i a b l e i n [a,b],
i.e.,
as n
-t
m.
I f Fn a r e u n i f o r m l y
and E > 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
<
E(v-u(
6(6) < u < 5 < v < 5 + 6 ( 6 ) and f o r a l l n, t h e n t h e consequence o f
t h e theorem holds. We remark t h a t o t h e r c o n v e r y e n c e theorems by D j v a r s h e i s h v i l i [ 2 ] ,
Lee and
and Grimshaw [3] w i l l a l s o f o l l o w f r o m t h e above theorem.
Chew [ 6 , 8 ] ,
Further-
more, t h e t h e o r e m can be e x t e n d e d t o more g e n e r a l i n t e y r a l s , f o r example, t h e R u r k i l l approximately continuous i n t e g r a l [l].
References
c11
P. S. B u l l e n , The R u r k i l l a p p r o x i m a t e l y c o n t i n u o u s i n t e g r a l , J. A u s t r a l . Math. SOC.
c21
A.
( s e r i e s A ) 3 5 ( 1 9 8 3 ) , 236-253.
G. D j v a r s h e i s h v i l i , On a sequence o f i n t e y r a l s i n t h e sense o f D e n j o y ,
Akad. Nauk G r u z i n . SSR T r u d y Mat.
I n s t . Rajmadze 1 8 ( 1 9 5 1 ) , 221-236.
MR
14-628. E. Grimshaw, A c o n v e r g e n c e t h e o r e m f o r n o n - a b s o l u t e l y c o n v e r g e n t
C31
M.
C41
R.
H e n s t o c k , T h e o r y o f i n t e g r a t i o n , London 1963.
C51
R.
L. J e f f e r y , The t h e o r y o f f u n c t i o n s o f a r e a l v a r i a b l e , U n i v . o f
i n t e g r a l s , J. London Math. SOC. 4 ( 1 9 2 9 ) , 439-444.
T o r o n t o P r e s s 1951.
C6l
P.
Y.
c71
P. Y.
Lee and T. S.
integrals,
Chew, A b e t t e r c o n v e r g e n c e t h e o r e m f o r H e n s t o c k
B u l l . London Math. SOC. 1 7 ( 1 9 8 5 ) , 557-564.
Lee and T. S. Chew, A R i e s z - t y p e d e f i n i t i o n o f t h e D e n j o y i n t e g r a l ,
Real A n a l y s i s Exchange 11( 1 9 8 5 / 8 6 ) , 221-227.
C8l
P. Y.
Lee and T. S. Chew, On Convergence theorems f o r t h e n o n a b s o l u t e
i n t e g r a l s , B u l l . A u s t r a l i a n Math. SOC. 3 4 ( 1 9 8 6 ) , 133-140.
C91
S. Saks, T h e o r y o f t h e i n t e g r a l , Warsaw 1937.
National U n i v e r s i t y o f Singapore Republic o f Singapore
This Page Intentionally Left Blank
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
I67
Factorization Theorem for the Real Hardy Spaces
A
Akihiko MIYACHI Department of Mathematics, Hitotsubashi University Kunitachi, Tokyo, 186 Japan * ) The purpose of this article is to give a generalization of the factorization theorem for the real Hardy spaces and its application to the majorant property of the real Hardy spaces.
1.
INTRODUCTION First we shall recall the factorization theorem for the classical Hardy
spaces. Let
gp,
0 < p <
m,
denotes the classical Hardy space over the upper half
plane, i.e., this is the set of those functions F
which are holomorphic in
the upper half plane and for which
0 < p , q, r
The factorization theorem for these spaces reads as follows: Let < a
l / p = l/q
and
plane belongs to that
F = GH
gp
+ l/r
; then, a holomorphic function
if and o n l y if there exist
(pointwise product of
H ).
and
G
G
E
F
on the upper half
gq and H
E
Er
such
As for this theorem, see e.g.
Zygmund [ Z O ; Chapt. VII, 171, Duren [8; esp. Chapters 2 and 111 , or Koosis [ 1 2 ; esp. Chapters IV and VIJ. We can restate this theorem in terms of the Hilbert transform and the boundary values of the holomorphic functions. We define Re Hp, 0 C p < follows:
f
belongs to
Re Hp
if
real line) and if there exists an
f
F
Ep
(the
such that
=
function F
-f
is uniquely determined by
relation, then we define
E
as
is a tempered distribution on in
lim Re F( + iy), Y + O where the limit is taken in the sense of tempered distribution. f
m,
f.
If
f
and
The above
F have the above
by
lim Im F ( * + iy), Y + O where the limit is taken, again, in the sense of tempered distribution. This =
Y
f
is called the Hilbert transform of
f.
Now, taking the boundary values of
* ) Partly supported by the Japan Society for the Promotion of Science and by
the Grant-in-Aid for Scientific Research (C 61540088), the Ministry of Education, Japan.
A. Miyachi
168
the real parts of
F = GH, we can restate the factorization theorem as follows:
Let
be the same as before; then, a tempered distribution
p, q, and
R
r
belongs to Re He
if and only if there exist
g
Re Hq
E
and
h
E
f on
Re Hr
such that N
N
g h - gh.
f
(1)
(This is n o t a precise statement unless we determine the meaning of the right N N
hand side of (1); the terms
gh
and
meanings as tempered distributions if F
=
GH, we use the formula F =
g h , each by itself, may n o t have the p < 1.)
-6 GH,
If, instead of the formula
then we obtain another restatement
of the factorization theorem, i.e., we see that the factorization theorem for the spaces Re He
holds if we replace (1) by
f = g x +:ha Coifman-Rochberg-Weiss [ 61 , Uchiyama [ 181 , [ 191 , Chanillo [ 51, Komori [ 111, and the author [14], [15] obtained generalizations, in a weak form, of the above factorization theorem for Re Hp Remark (iv) in the next section).
to the case of the Hp
spaces over
En
(see
The purpose of the present article is to give
a further generalization of these results and, as an application, to give a proof of the majorant property of the Hp 2.
spaces.
PRELIMINARIES Hereafter, we fix a Euclidean space
En ; the letter
n
always denotes the
dimension of this space.
5
We denote by functions on
En
and
5'
the Schwartz class of rapidly decreasing smooth
and the space of tempered distributions on
En
-
respectively.
The Fourier transform and the inverse Fourier transform are denoted by
and
F - ~respectively. For p > 0, we denote by
He
the HP-space given by C . Fefferman and E. M.
Stein [ 9 ; 1111, i.e., this is the set of those f
in
2'
for which the func-
tions f*(X)
=
sup
(t-n
d.) * f)(x) I
t > 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
<
depending only on
l/p
T, p, q, and
-
l/q
n
for which the inequality HP holds for all
f
5
in
n Hp.
A s for a proof of this proposition, see e.g. [ 4 ; 141.
2
If p
1, then we ( A s for
can also easily prove it by using the atomic decomposition for Hp. the atomic decomposition, see [ 1 3 ] . )
where
m
j
in K ( A ) .
is the multiplier corresponding to If
J
is the empty set, t h e n m j
T
This product is an operator
j'
~
Tj
shall be understood as the
identity operator.
MAIN RESULT
3.
The following is the main result of this article. Theorem 1. Let
..., N, j = 1,
N
A=xjfl,
be a positive integer and let
*.*,
N.
g
For
and
h
respect to and
11, . a * ,
N}
+
-
l/p = l/q
l/r
.
J , and
Jc
5
E
n Hq
of
{ 1,
*..,
m,
m
and all h
j
corresponding to
j
T. J
N 1,
depending only on
5
c
T
j
is a homogeneous function of degree t'O, m.(tt;) = t j m j ( c ) , J j = 1, .*., N; ( b ) For every 5 E
Rn \ { O }
such that
K(A.), J
with
l/r > 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 <
A/n < 1
and
2, we set
in
where the summation is taken over all subsets J denotes the cardinality of
A
A j < n.
be nonnegative numbers satisfying
5 # 0 ,
En \
{O},
there exists an
n
E
Factorization Theorem for the Real Hardy Spaces
1 and Then, every
f
# 0 for j with A.
m.(n) J
J
in He
171
> 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
<
A
< n, and
p
>
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 <
-,
nonnegative i n t e g e r s a t i s f y i n g
If
tion.
0
2
l / p - l / q = p / n , and i f
p < n,
is a
k
k > 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 < p <
m,
0 < q <
m,
and
l/p - l/q
P/n,
then
T h i s lemma i s due t o C h a n i l l o [ 5 ; Lemma 21. Now w e s h a l l prove Theorem 1 ( i ) .
For s u b s e t s
J
{ 1,
of
9 * * ,
1
N
we use
t h e following notations:
By s l i g h t l y modifying t h e arguments i n [ 1 4 ] , w e can prove t h e f o l l o w i n g : There exist
C , k , k ' , u , and
holds f o r a l l
g, h c
5
large positive integers, i n g only on
T1,
* * * ,
v
such t h a t t h e i n e q u a l i t y
and a l l
x
E
0 < u < q,
En;
here
k
and
0 < v < r , and
TN, p. q , r , k , k', u , and
v.
k' C
are s u f f i c i e n t l y
i s a c o n s t a n t depend-
From t h i s i n e q u a l i t y , w e
can e a s i l y deduce Theorem 1 (i) w i t h t h e a i d of P r o p o s i t i o n 2 , Lemmas 1 and 2 , and H B l d e r ' s i n e q u a l i t y . Next we s h a l l s k e t c h t h e proof of Theorem 1 ( i i ) . For a p o s i t i v e i n t e g e r M and f o r 2 L - f u n c t i o n s f on
t h e set of t h o s e
p
En
with
0 < p
2
-
1, w e d e n o t e by
whose F o u r i e r t r a n s f o r m s
f
A
P J
satisfy
Factorization Theorem for the Real Hardy Spaces
173
and
5
t > 0.
f o r some
0 < p 5 - 1 and decomposed as f o l l o w s : Lemma 3 .
ak
where
Here
If
a r e complex numbers,
M > 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.
This Page Intentionally Left Blank
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 <
m,
m
2
-n(1-p)Il/p-1/21,
and
if
161
2
k
and
la1
is bounded in hp
(if
for
2 p
k'
with
2 1)
or
k Lp
and k' sufficiently large, then (if
a(X,D)
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
<
or
K
<, , ! ,K
K'
then t h e r e a r e
f o r which t h e a s s o c i a t e d p s e u d o - d i f f e r e n t i a l
(K,K')
Lp; w e s h a l l a l s o c o n s i d e r t h e s p a c e s
o p e r a t o r s a r e n o t bounded i n
he
and
Lp.
The c o n d i t i o n such as (1.1) w a s i n t r o d u c e d by HBrmander [ 9 ] .
Remark.
n e g a t i v e r e s u l t s mentioned below Theorem A was a l s o p o i n t e d o u t by him. 0 5 - 6 = p < 1.
d i f f i c u l t y of t h e proof o f Theorem A l i e s i n t h e c a s e
C . Fefferman [61 ( t h e c a s e 1 < p <
c f . a l s o Wang-Li
m;
2 1).
0 < p
I n some c a s e s , t h e c r i t i c a l
[131 ( t h e c a s e
orders
mentioned above have a l r e a d y been o b t a i n e d .
K;)
found i n Cordes [ 5 ; Theorem D] ( t h e c a s e
0 < p <
[11] (the case
0 - 6 _5 p _5 1, and
p = 2
and
They are
6 = p = O ) , Miyachi [ l o ] ,
-
0 5 6 = p < l), Muramatu [12] ( t h e c a s e
and
6 < l), and Sugimoto
Theorem
[16; p . 1 9 4 ] ) , and
Paivarinta-Somersalo and
The The
p = 2),
A f o r t h i s case i s d u e t o Calder6n-Vaillancourt [ 2 ] , [3] (the case
K~
K
0 t h e n t h e p s e u d o - d i f f e r e n t i a l o p e r a t o r s w i t h symbols i n t h e c l a s s
K ,;
(K,K')
bmo
2 1)
p
T h i s c l a s s can b e c o n s i d e r e d , f i g u r a t i v e l y , a s t h e c l a s s
K'.
of t h o s e symbols which s a t i s f y (1.1) f o r
Sm P,6
(if
W e s h a l l do t h i s i n a g e n e r a l i z e d s e t t i n g , t h a t i s , w e s h a l l
Theorem A h o l d s .
K'
he
k
The purpose of t h i s a r t i c l e i s t o f i n d t h e smallest
and
That i s ,
m >
p > 1).
(if
LP
R
and
cannot b e r e l a x e d .
a r e a s mentioned i n t h e above theorem and i f
151 ( t h e c a s e
0 < p <
and
p = 2,
6 = P
= 0).
2.
NOTATIONS AND FUNCTION SPACES
The f o l l o w i n g n o t a t i o n s a r e used throughout t h i s a r t i c l e .
R d e n o t e s t h e s e t of r e a l numbers. We f i x a Euclidean s p a c e
En;
the l e t t e r
n
= (El,
5 )
always d e n o t e s t h e dimension o f
t h i s space. If
..*, xn)
x = (xl,
I;=,
xS =
xjSj,
1x1 =
If
a
= (a1, * * *
la/ = a
+
m.0
+
,
-..
5
<x)
=
* * a ,
( l + l x l 2"1/2 1
a r e elements of
.
En,
then
(a , an) i s an n - t u p l e of n o n n e g a t i v e i n t e g e r s . 1' an) i s a m u l t i - i n d e x , t h e n t h e l e n g t h la1 i s d e f i n e d by
a
A multi-index
and
G, and
a
=
and t h e d i f f e r e n t i a l o p e r a t o r
a z f w = (a/aXl)'1
---
(a/axn)anf(x),
a'
i s d e f i n e d by
179
Estimates for Pseudo-differential Operators where
x = (xl,
En
u e
If
* - *
and
f
i s a f u n c t i o n on
En,
+
f (x)
-
A'(u)f (x) = f (x+Zu)
2f (x+u)
then
.
The F o u r i e r t r a n s f o r m i s d e f i n e d by
?(c)
=
I
-ixE f(x)e dx.
Rn -
W e s h a l l explain the function spaces considered i n t h i s a r t i c l e .
S(gn) on
En
and
S(&"xKn)
a r e t h e s p a c e s of r a p i d l y d e c r e a s i n g smooth f u n c t i o n s
En .En r e s p e c t i v e l y . p 2 m, denotes t h e set
and
Lp, 0 <
f
of t h o s e measurable f u n c t i o n s
on
En
f o r which t h e f o l l o w i n g s are f i n i t e :
[."I
f
Ilfll
Lp = (
(x) I d ' .1
l"
If(x) I
e s s e n t i a l supremum of
0 < p 5 1, i s d e f i n e d a s f o l l o w s .
1(t-"d.)*f)
Then
if
p =
a.
f
on
in
s(En)
d,
En,
Lp.
belong t o
f*
llfll
lIf*II
Hp =
I t i s known t h a t
For
f
I.
in
f
on
En
for
H p , we d e f i n e
LP'
does n o t depend on t h e c h o i c e of
He
such
we define the
0 < p 5 - 1, i s t h e s e t of t h o s e tempered d i s t r i b u t i o n s
He,
which
(x)
p < m
Fix a function
d,(x)dx # 0. For a tempered d i s t r i b u t i o n I * function f on En by f*(X) = s u p t>O
if
d,.
hp, 0 < p 5 - 1, i s d e f i n e d a s f o l l o w s . L e t @ b e t h e same as above. For a f on En, w e d e f i n e t h e f u n c t i o n f*" on En by
tempered d i s t r i b u t i o n f*J(X)
I(t-"d.)*f)
sup O
=
(x) 1 .
h P , 0 < p 5 1, i s t h e s e t of t h o s e tempered d i s t r i b u t i o n s -
Then which
f*'l
belong t o
llfll
/If*'l/l
=
I t i s known t h a t
BMO
Lp.
For
f
=
sup h J I f ( x ) - f
' Q, and
f
Q
for
@.
Q Idx
<
f
on
En
f o r which
m,
Q
where t h e supremum i s t a k e n o v e r a l l cubes measure of
En
LP
i s t h e s e t of t h o s e l o c a l l y i n t e g r a b l e f u n c t i o n s
IlfII BMO
on
hp, we d e f i n e
in
does n o t depend on t h e c h o i c e of
hp
f
=
1Ql-'l
f(x)dx.
Q
Q
in
En,
IQI
d e n o t e s t h e Lebesgue
180
is the set of those locally integrable functions f
bmo
on
En
for which
where the notations are the same as above. As for He
BMO, see Fefferman-Stein [ 7 ] .
and
As f o r
hp
and bmo, see
Goldberg [81. If
is in S(gnxEn),
a
(a(X,D)*g) (y) where the function K K(x,z) Note that if
I
=
we define the operator a(X,D)* g(x)K(x,x-y)dx,
is defined by
(2il)-"leizra(x,C)dC.
=
a belongs to S-(znxEn),
are well defined on
?(En)
3.
f and
g
then the operators a(X,D)
and
a(X,D)*
and the equality
,/g(x) (a(x,D)f) (x)dx holds for all
by
I(a(X,D)*g)
=
(y)f(y)dy
g(En).
in
CLASSES OF SYMBOLS We shall introduce the following class.
R, 0 2
2 1, K
>
be the nonnegative integers satisfying k <
K
Definition. Let m c Then
Sm
P,6
6, p
0, and 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,<)I (ii) if
161
= k,
la1
5
A(~)m+~IfiI-pIal; k', u
I 2 A mt6K-P
la:(u)a:aia(x,c) (iii) if
(iv) if
Here A
101 6 k, la1 In:(n)a!a;a(x,c)
En, and
E
k ' , ri
=
I
E
1111
151 = k , la1 = k', u , ~A:(u)n~(rl)a~a~a(x,~) I
En, and
rl E
(EY6, then
IUIK-k;
A m+6 15 I - p K ' j
2
5
:<E)'/4,
I K'-k'
En, 1u1 2 ( 5 ) - & ,
then
;
and
In1 ~ ( < ) ~ / 4then ,
2 A < s > ~ ~ ~ -~P~ ~~ K' - k l ~ l K l - k '
is a constant which does not depend on a, 5, x, 5 , u , and
smallest such constant A It is easy to see that Banach spaces.
is denoted by Sm
P,6
(K,K')
rl.
The
m,p,6,K,Kl.
with the norms
11 11 m , P , G , K , K ~
These are generalizations of the Lipschitz spaces :B
are over
to the funrtion Rn, which are the special ones of the Besov spaces Ba P99'
Es tiniates for Pseudo-differential Operators
En x En.
spaces over
181
Many p r o p e r t i e s of t h e L i p s c h i t z s p a c e s :B Sm ( K , K ' ) . P16
generalized t o our spaces
can b e
Here w e s h a l l mention onl;
one such
property. Proposition. m(8) =
Let
+
(l-e)m(o)
m(O),
em(l),
R,
m(1) c =
K~
0 5 - P , 6 5 1,
+
(1-8)Ko
O K ~ , and
KO,
(1-8)~;
=
K'
e
Ki,
K1,
K;
>
+
eK;.
o <8 < l ,
0,
Then
where t h e l e f t hand s i d e d e n o t e s t h e complex i n t e r m e d i a t e s p a c e .
A s f o r a proof of t h i s p r o p o s i t i o n , t o g e t h e r w i t h o t h e r p r o p e r t i e s of t h e spaces
[ l l ; § 2 ] , and t h e i n t e r p o l a t i o n argument i n
Sm ( K , K ' ) , s e e [ l o ; 521, P,6
[ l l ; 54, P r o o f s of Theorems 3 . 1 and 3.21. The main r e s u l t s of t h i s a r t i c l e , which w i l l b e g i v e n i n t h e n e x t s e c t i o n ,
w i l l t r e a t only those c l a s s e s
4.
Sm
P,6
2
0
with
(K,K')
6 5-
p
5- 1 and
6 < 1.
MAIN RESULTS.
W e s h a l l use the following notation. spaces over d e f i n e d on
sn. Suppose n o n n e g a t i v e and
Y
such a way t h a t
1 1 11 z.
respectively.
Z
llfll
=
m
if
f
Then w e s h a l l w r i t e as
constant
depending o n l y on
C
(Y,Z) b e a c o u p l e of f u n c t i o n
Let
1 1 11
functions
W e s h a l l extend does n o t b e l o n g t o
Ym (K,K')c t ( Y , Z ) P,6
n , m, p , 6 ,
K,
Y
are
11 11
and
in
and s i m i l a r l y f o r
i f there e x i s t s a
Y , and
K',
/I 11
and
1 1 I/
f o r which t h e
2
inequality (4.1)
holds f o r a l l ($,&(K,K'))*
(4.2)
z2 c
Ila(X.D)fll a
E
c
L(Y,Z)
s(gnxEn)
Ila(Xm*fll
holds with
llall m , p , 6 , K , K '
and a l l
f
llfll E
y
z 2 c Ilall m , P , 6 , K , K ' I l f l l
C , a , and
S i m i l a r l y , w e s h a l l w r i t e as
s(Rn).
i f the inequality
f
y
b e i n g t h e same a s above.
#
W e s h a l l u s e t h e symbol
t o i n d i c a t e t h e n e g a t i o n s of t h e above s t a t e m e n t s . Remark.
Although t h e c l a s s
restriction that
a
is i n
i s n o t dense i n
s(EnxRn)
s(gnxEn)
I n f a c t , i f t h e i n e q u a l i t y (4.1) h o l d s f o r a l l and i f
Z
Sm
P,6
the
(K,K'),
mentioned above i s n o t an e s s e n t i a l one. a
t
s(EnXRn)
and a l l
f
E
?(En)
h a s a c e r t a i n good p r o p e r t y , t h e n (4.1) h o l d s , p o s s i b l y w i t h a l a r g e r
A p r o p e r t y of Z c o n s t a n t C , f o r a l l a E Sm ( K , K ' ) and a l l f E ?(En). P,6 which g u a r a n t e e s t h i s i s as f o l l o w s : I f { g . ) i s a sequence of u n i f o r m l y J = A < m and i f g conbounded smooth f u n c t i o n s on En w i t h s u p j IIgjII j t o a f u n c t i o n g u n i f o r m l y on e a c h compact s u b s e t of En, verges a s j +
-
then
llgll
2
CIA.
In f a c t , f o r every
a
E
Sm P,6
2
we can f i n d a sequence
(K,K'),
{ a . } of f u n c t i o n s i n s(ZnxEn) such t h a t a . ( x , C ) 1 J a ( x , S ) uniformly on e a c h compact s u b s e t of R n x R n
-5
converges as and
j
-f
(Iaj(I m,P,6,K,Kl
m
to
2
A. Miyachi
182
'la''
with
m,p,6,K,K'
depending o n l y on
C"
n , m, P , 6 ,
and
K,
{ a . } i s s u c h a s e q u e n c e , f i s i n S-(gn), and g = a . ( X , D ) f , J j J a sequence o f u n i f o r m l y bounded smooth f u n c t i o n s , Ilgj
and
gj
II z 2
g = a(X,D)f
converges t o
then
{g.) J
is
y'
Sn.
u n i f o r m l y on e a c h compact s u b s e t of
h a s t h e p r o p e r t y mentioned above, t h e n , by a l i m i t i n g argument
Z
Hence, i f
IIfll
C"C llall m , P , 6 , K , K '
If
K'.
{ a . } , we s e e t h a t t h e i n e q u a l i t y ( 4 . 1 ) w i t h C J h o l d s f o r a l l a c Sm ( K , K ' ) a n d a l l f E s ( Z n ) . The P,6 (0 < p 2 m ) , Hp (0 < p 2 l ) , he (0 < p 2 l ) , BMO, and bmo have
i n v o l v i n g t h e a b o v e sequence r e p l a c e d by spaces
C'C"C
Lp
a
general
Sm
E
P,6
*
I f one f i n d s a n i c e r e p r e s e n t a t i o n f o r
Z.
that property f o r
a(X,D) f
for
( K , K ' ) , t h e n t h e similar argument may show t h a t i n e q u a l i t y a
( 4 . 2 ) can a l s o b e a u t o m a t i c a l l y e x t e n d e d t o g e n e r a l
E
Now we s h a l l g i v e t h e main theorems o f t h i s a r t i c l e .
Sm
(K,K').
P,6
In t h e following
theorems, w e assume
Theorem 1. n(1-p)/p
(1) I f
(2) I f
0 < p
and
K'
> n/p,
(if
p = I)*)
2
(3) If
2 < p <
and
> n/2,
K'
m
(4) I f
=
or
Ym P,6
2
2
Ym
then
K
> n/2,
K'
2 < p <
> 0, and
2
-n(l-p)
(4) I f (Y;,*(K,K'))*
c
2m 5
Theorem 3 .
z
(Y;,~(K,K'))
m
2
K'
2 < p <
c
+ m,
1 1 L(h ,h )
2).
> n/2,
( 1 - 6 ) ~> 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 < p 5 - 2 , -n(l-p)/p
+
(2) I f
or
( Y J ; , ~ ( K , K ' ) ) *c G(Lp,Lp). t m , and
K'
> 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 < p 2 2 )
then
0 < p < 1)
-n(l-p)(I/Z-l/p),
K'
-n(l-p) ( 1 - l / p )
t m, and
n(l-p)(l-l/p)
(1-6)~>
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< p
-n(l-p)(l/p-1/2),
Ym
0 < p 2 1, m = - n ( l - p ) ( l / p - 1 / 2 ) ,
(1) I f
then
2
2
c L(Lp,Lp).
(K,K')
P,6
-n(l-p)(l/p-
(if m
m
> n/p, then
L(hp,Lp)
c
2
1 < p 5 2, -n(l-p)/2
(3) I f
m
(K,K')
-n(l-p)/2
> 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
< p
1 / 2 ) , and
5 1) o r
> 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.
which s a t i s f i e s
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 < p
2
1< p
(if
(5) I f
m
t
&
and e v e r y
0 < p
(if
2
1)
# L(Lp,Lp)
or
0.
K
n/2, then f o r every m c g and every K > 0 i t holds t h a t for 2 p < m and Ym~ , & ( K , KPOL ( L ~ , B M O ) .
<
K'
# L(Hp,Lp)
(K,n/p)
P-6
# L(Lp,Lp)
(K,K')
yP,6
Theorem 4 . L(Hp,Lp)
0 < p < 1 and
(1) I f
m
f o r every
and e v e r y
t
a(x,S)
( 2 ) T h e r e e x i s t s a symbol
1
2
< p
2, -n(l-p)/2
( 4 ) If
2 < p <
n(1-p)(l-
(5) I f
<
+ L(LP,LP)
(y,m ,6(K,K'))*
then
+ m,
l/p)
-n(l-p)
a(X,D)"
(K,K'))*#
i s n o t bounded from
m 2 -n(l-p)(l/p-1/2), f o r every
-n(l-p)/2
P,6
> 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
which s a t i s f i e s
(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 ) ~<
0.
>
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
<
2
p
&
t
K'
and e v e r y
1< p 5 - 2
and
and ever)
K
2 < p <
then
m,
> 0.
K'
1 and
K
K'
n(l/p-1/2),
then
# L(Hp,Lp)
(~;,&(K,K'))*
for
> 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 < p < 1, 0 < p
p >
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
<
2- 1; t h i s can b e
shown by arguments s i m i l a r t o t h o s e i n [ l o ; 141 o r [ l l ; 141. Problems s i m i l a r t o t h e above one arises i n c o n n e c t i o n w i t h t h e c o n d i t i o n
with
m = -n(l-p)/p
i n Theorem 1 (1) w i t h
> 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
=
<
he
or
K~
K~
or
(K,K')
and
c K'
K'
K~
Lp
=
or
L(Y,Z) <
= K;.
~ ~ ( n , m , p , G , Y , z ) and
holds i f
K
>
K~
The problem i s :
K .;
Theorems
where 0 bmo, 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 t h e and
K'
=
K'
K;(n,m,p,G,Y,Z),
>
Try t o show
K;
b u t does n o t Y;,&(K,K')
c
L(Y,Z)
I n f a c t , t h i s w i l l n o t b r i n g many r e s u l t s s o f a r
Sm (K,K')''). But, i f w e i n t r o d u c e some P,6 c l a s s e s of symbols which are g e n e r a l i z a t i o n s of t h e Besov s p a c e s 'B over PY9 Rn t o t h e f u n c t i o n s p a c e s o v e r En.&", then we can e x p e c t much. Muramatu
a s we c o n s i d e r o n l y t h e c l a s s e s
1121 and Sugimoto [15] have a l r e a d y o b t a i n e d some r e s u l t s i n t h i s d i r e c t i o n .
6.
SKETCHES OF THE PROOFS Theorems 1
- 4 are g e n e r a l i z a t i o n s of
W e can prove Theorems 1 papers.
-
t h e theorems g i v e n i n [ l o ] and [ I l l .
4 by o n l y s l i g h t l y modifying t h e arguments i n t h e s e
So we s h a l l omit t h e d e t a i l s and g i v e o n l y t h e s k e t c h e s o f t h e P r o o f s .
F i r s t , we s h a l l g i v e a s k e t c h of t h e proof of Theorem 1 (2) f o r t h e c a s e p
# 1 and m
=
-n(l-p)(l/p-l/2),
which i s t y p i c a l of o u r argument.
The proof
goes by t h r e e s t e p s . The f i r s t s t e p i s t o prove t h e r e s u l t f o r p = 2 , i . e . , t h e 2 r e s u l t f o r t h e L 4 o u n d e d n e s s . The f a c t is t h a t Theorem 1 (2) f o r p = 2 and m = 0
[lo;
h a s been proved by Cordes [5; Theorem D] ( t h e c a s e
p = 6 = 0; c f . a l s o
141) and Muramatu [12; Theorems 4.5 and 4.61 ( t h e g e n e r a l c a s e ) .
(In fact,
Muramatu's theorems are s h a r p e r than Cordes's theorem and o u r theorem; Muramatu
I) by
W e might be a b l e t o r e p l a c e K = n / p - n.
K
> n/p
-n
i n Theorem 2 (1) i n t h e c a s e
p < 1
Estimates for Pseudo-differential Operators treats t h e case the case
n(l-p)/2(1-6)
=
K
> n(l-p)/2(1-6)
K
and
and
K'
(1-6)~> n ( l - p ) / 2 ,
and
+
> n/p
K'
n / 2 , w h e r e a s Cordes and w e t r e a t
=
The s e c o n d s t e p i s t o p r o v e t h e
> n/2.)
K'
f o l l o w i n g weak v e r s i o n o f t h e a s s e r t i o n : i f
1, t h e n
185
0 < p < 1, m = - n ( l - P ) ( l / p - l / Z ) , m Y (K,K') c L(hp,Lp). We c a n P9
p r o v e t h i s as f o l l o w s :
2
6
We u s e C o r d e s ' s o r Muramatu's L -boundedness theorem 2 mentioned above t o show t h e w e i g h t e d L -estimate, and t h e n w e u s e H a l d e r ' s inequality t o obtain the desired for
LP-estimate.
The a t o m i c d e c o m p o s i t i o n theorem
s p a c e s ( s e e Goldberg [ B ; Lemma 51) i s of much h e l p i n t e c h n i c a l c a l c u -
hp
lations i n t h i s step.
As f o r d e t a i l s i n t h i s s t e p , c f .
o r [ll; P r o p o s i t i o n 4.11.
The t h i r d s t e p i s t o use t h e complex i n t e r p o l a t i o n .
Suppose, f o r s i m p l i c i t y , 1 < p < 2 .
0 < p < 1 i f we replace
the case
0 < q < 1, and l e t
such t h a t
[lo; P r o p o s i t i o n 4.11
(The f o l l o w i n g argument h o l d s t r u e a l s o i n L(Lp,Lp)
by
Take a number
t(hP,LP).)
b e t h e number s u c h t h a t
0
+
l / p = (l-B)/q
q 6/2.
The r e s u l t s i n t h e f i r s t and t h e s e c o n d s t e p s r e a d a s
€1
O [*.-"+€,-+Y p , 6 1-6 2 2
L(L 2 , L 2 )
c
2
and
where
m(q) = - n ( l - p ) ( l / q - l / Z )
and
E
i s a n a r b i t r a r y p o s i t i v e number.
Hence,
by i n t e r p o l a t i o n ( s e e t h e p r o p o s i t i o n i n S e c t i o n 2 o f t h i s a r t i c l e and [ 1 4 ] a n d
[l; § 3 ] ) , w e o b t a i n
where
q
tends t o
proves t h e desired r e s u l t s i n c e
-n(l-p)(l/p-l/Z).
+ m,
a l i t y , (l-p)n/Z
0 --5 m
S-
p
> 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 <
(1-6)K
>
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.
This Page Intentionally Left Blank
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 .
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
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 < p < 00 , q = 00. if 0
(I,"(f'(t)t'/p)qt-ldt)l/(
> 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 < p 5 q < f .
to L(/31,m;I') and that u : G where 1/Bl = l/q - l/f
< 00.
+
Then there exists a C > 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 < p 5 q < p' and let 1/p = l/p
+ l/q - 1. If u : r
4
R+ belongs to
&(PI00;I-) then there exists a C > 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 < p 5 q
(a) If u :I' -+
(b) If u : G
f
< p' and l/p = l/p + l/q - 1.
R+ belonga to L(P,0o;r)then there exists a C > 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 < p 5 q < p', let 0 q =a
+ l / p + l/q - 1. Then there exists a C > 0
Proof. Let r = (a + l/p)-I. Then 1
60
Ia <
l/p' and let
that for all f E L1(G)we have
q
< t'.
According to'Lemma 2, if
~ ( 7=) 117Il-Q then u E L(q-',co;I') with q = l/q - l/#. Also, if u(z) = u-l
11~11"
then
E L(a-l, c0;G) with a = I/r - l/p. Thus an application of Theorem 1 completes the
proof of this corollary.
3. NECESSARY CONDITIONS
For the one weight norm inequalities for the Fourier transform that were proved in Corollary 1 we have the following complementary result.
l/p = l/p + l/q - 1. holds then u E L(p,00;l').
Theorem 2. Let 1 < p 5 q < p' and let (a) If u :r
+ R+ is
radial and if (2.2)
(b) If u : G -+ R+ is radial and if (2.3) holds then u-l E L(P,m;G).
Proof. (a) Since u is a radial function u can be represented as
47)=
5 abtrbtl\rb(7)
k-w
Let f(z)= b b + , ( z )so that i ( 7 ) = (mb+1)-'trktl(7),cf. (1.2). The order-boundedness of G implies that
199
Therefore,
Ilill~2 2 Also
Thus we see that a, 2 C(rnn)-'/fl for each n E Z, that is, u ( t ) 2 Cllzll'/fl. Thus
L e m m a 2(a) implies that u-l E L(P,ao;G).
AB a partial converse to Theorem 1 we present the following necessary conditions for the two weights inequality (2.1).
< p 5 2 and 1 < r < p 5 q < f. Let l/Bl = 119- 1 / f and Assume u : I' --* R+ and u : G + R+ are radial functions for which
Theorem S. Let 1 l/@, = I/r (2.4) holds.
- I/p.
C W. Onneweer
200
(a) If (u(z))-l 2 C ~ ~ ~ then~ u~E ;C(&,co;I'). - ~ ~ f l ~ (b) If ~ ( 72)C ~ ~ ~ then ~ ~u-'- lE~L(P~,co;G). " Proof. Let u and u have the same representations as in the proof of Theorem 2. If we
take f(z)= &.\a.+,(z) then we have
I 113~11, ICII~VII, I Co,(m,)--"P . Thus for all n E Z we have a, I Con(mn)'-l/~l/*. If u(z) I C ~ ~ Z I Ithen ' / ~ *for each n E 2, 0, 5 C(n~,,)-~/fl#. Therefore, in this case an I C(mn)-'/pl, that is, ~ ( 7 5) C ~ ~ ~ ~ an(mn+l)-'(mn+l-
which implies that u E &(@I,
mnI1"
0o;r).
On the other hand, if u(7) 2 Cl1711-1/fl~then we easily see that U-'
(an)-1
an 2 C(mn)-l/bl for each n E
Z and
5 C(mn)-'/fl*, that is, (U(Z))-' 5 C ~ ~ Z which ~ ~ -implies ~ / ~that ,
E t(B2,m;G).
Remark. After this paper had been written the author learned of a paper on the name subject by T. S. Quek in which he gives conditions for radial weight functions u and u that are equivalent to inequality (2.1), see [13].
REFERENCES
[I] Benedetto, J. J. and Heinig, H. P., Weighted Hardy spaces and the Laplace transform (Springer Lecture Notes in Math. 992,240-277,Springer Verlag, 1982). [2] Benedetto, J. J., Heinig, H. P. and Johnson, R., Boundary values of functions in weighted Hardy spaces (preprint, 1984).
[3]Bergh, J. and Efstr6m, J., Interpolation Spaces (Springer Verlag, 1976). [4]Edwards, R. E. and Gaudry, G. I., Littlewood-Paley and Multiplier Theory (Ergeb. Math. 90,Springer Verlag, 1977). [5] Heinig, H. P., Weighted norm inequalitiea for classes of operators, Indiana Univ. Math. J. 33 (1984),573-582.
161 Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis, Vol. II (Grundl. der Math 152, Springer Verlag, 1970). [7]Jurkat, W. B. and Sampson, G., On rearrangement and weight inequalities for the Fourier transform, Indiana Univ. Math. J. 33 (1984),257-270. [8] Muckenhoupt, B., Weighted norm inequalities for classical operators (Amer. Math.
SOC.,Proc. Symp. Pure Math. 35(I) (1979),69-83).
191 Muckenhoupt, B., Weighted norm inequalities for the Fourier transform, h a . Amer. Math. Soc. 276 (1983),729-742.
~ - 1 ~ 8
Weighted Norm Inequalirks for rhe Fourier Transform
201
[lo] Muckenhoupt, B., A note on two weight function conditions for a Fourier transform norm inequality, Proc. Amer. Math. Soe. 88 (1983),97-100.
Ill] 'hibleaon, M. H.,Fourier Analyeis on Local Fields (Math. Notes, Princeton Univ. Press, 1975). [12]Vilenkin, N. Ja., On a class of complete orthonormal systems, Amer. Math. SOC. b s l . (2) 28 (1963), 1-35. [13]Quek, T. S., Weighted Norm Inequalities for the Fourier Transform on Certain Totally Disconnected Groups (Preprint).
This Page Intentionally Left Blank
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
203
M u l t i p l i e r s o f Segal Algebras Ouyany Guanyzhong U e p a r t m e n t o f M a t h e m a t i c s , Fudan U n i v e r s i t y , Shanyhai, C h i n a . The main r e s u l t o f t h i s a r t i c l e i s t h a t : M(L1(G),
S(G))
M(G) and some a p p l i c a t i o n s o f t h i s r e s u l t .
Y
= M(G),
G))
Al,p(G)
when G i s
L e t S1(G) and S 2 ( G ) be t w o Seyal a l y e b r a s on G, where G i s a l o c a l l y compact a b e l i a n g r o u p ; S1(G) and S2(G) may be t h e same.
The bounded l i n e a r o p e r a t o r
T : S1(G) + S2(G) i s c a l l e d a m u l t i p l i e r f r o m S1(G) t o S2(G) i f f T commutes w i t h e v e r y t r a n s l a t i o n o p e r a t o r T ~ ,t h a t i s T o r x = T ~ o T . We d e n o t e t h e w h o l e
o f t h e s e m u l t i p l i e r s by M ( S l ( G ) , S z ( G ) ) .
It i s a Banach a l g e b r a .
p a p e r , we w i l l d i s c u s s t h e c h a r a c t e r o f M(Sl(G),
In this
S2(G)) f o r some S 1 ( G ) and
Sz(G).
1.
M u l t i p l i e r s f r o m L1(G) t o Seyal a l g e b r a S(G).
2.
M u l t i p l i e r s o f Seyal a l g e b r a Ap(G).
3.
Seyal a l y e b r a Al,p(G)
$1.
and i t s m u l t i p l i e r s .
M u l t i p l i e r s from L l ( G ) t o Segal algebra S(G)
We have known t h a t f o r e v e r y T E M ( L l ( G ) , S ( G ) ) t h e r e e x i s t s a u n i q u e measure p
E
M(G) (M(G) i s a Banach a l g e b r a w h i c h c o n s i s t s o f a l l bounded
r e g u l a r Bore1 measures on G), such t h a t T f = t h e w h o l e o f t h e s e measures
u by MS(G).
on t h e y i v e n Seyal a l g e b r a S(G).
u * f f o r every f
E L1(G).
Denote
It i s o b v i o u s t h a t Ms(G) i s d e p e n d e n t
Now t h e p r o b l e m we f a c e i s t h a t MS(G) may n o t
be t h e w h o l e M(G), and i t may o n l y be a p r o p e r s u b a l g e b r a o f M(G).
In this
case, what c h a r a c t e r does M S ( G ) have, a n d how i s i t d e t e r m i n e d b y S ( G ) ?
H. G o l d b e r g and S. S e l t z e r [l] have g i v e n a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n on MS(G). do f u r t h e r r e s e a r c h .
B u t t h e c o n d i t i o n i s so complex t h a t no one c a n u s e i t t o Here we w i l l g i v e a much s i m p l e r n e c e s s a r y and
204
G.Z. Ouyung
s u f f i c i e n t cond t i o n . Suppose t h a t (i)
s an a p p r o x i m a t e i d e n t i t y o f L1(G) and I I U ~ I I =~ 1 f o r
{ua]
e v e r y a , ;a has a compact s u p p o r t . (ii)
Ms(G) = { p
*
M(G) : tiua
E
< C u } , where Cu i s a c o n s t a n t o n l y
piis
I n MS(G), d e f i n e t h e norm by
dependent on u.
~
= l d m IIU,*~II~.
II~II,, S
The f o l 1 owi ny a r e e q u i Val e n t :
Theorem 1 C6l. (1)
T E M(L1 G), S ( G ) ) ,
(2)
There ex s t s a u n i q u e p E MS(G) such t h a t Tf =
,*
f f o r every
f E Ll(G).
Moreover, t h e c o r r e s p o n d e n c e between T and p d e f i n e s an i s o m e t r i c a l y e b r a i c i s o m o r p h i s m f r o m M(L1(G), S ( G ) ) o n t o Ms(G).
Proof.
Suppose p
MS(G), t h e n ua*p
E
S(G).
E
*
Now we a r e g o i n y t o p r o v e T f = p
f E L1(G).
F o r any y i v e n f
E
L1(G),
Let T f = p
*
f f o r every
f E S(G).
denote g a = p * u a * u a * f ;
ya E S(G) because ua ya
-
S(G) and S(G) i s an i d e a l i n L1(G).
E
y 1 = (p
*
ua
*
ua
*
-
f)
(p '
a
*
f
-
*
u
u 1
*
a
u 1
*
f)
a
( u a * f - u l * f ) .
= ( p * u a + p * u l ) * a
N o t i c e t h a t ua
*
Consider
a
f E L1(G),
so we o b t a i n
*
u lIls'IIua*f
a
llya
-
y I l l s < IIp
ua + l.
*
a
<
*
Z C I I ~ ~f
-
-
u l*flll a
a
*
flll
+ 0,
( a , a'
a
therefore, there exists y
m 1:
E
S(G) such t h a t y
= g
in
(S(G), II 1 1 ~ ) .
O f c o u r s e , t h e above l i m i t i s a l s o t e n a b l e i n L1(G).
Then
+
-1;
Multipliers of Segal Algebras A
14" y, A
But 9,
*
A
= u(u,)
2-
f and [ua}
205
A
= g.
i s an approximate i d e n t i t y , t h e r e f o r e A,.
lim y a
,.A
a
uf.
=
A
T f = u*f
Thus we have proved t h a t uf = g i.e.
E
S(G).
Besides,
.
we o b t a i n IITII < I I ~ I I ~ S
Conversely, suppose T c e r t a i n l y e x i s t s a unique
Now we w i l l prove t h a t t h e r e
N(Ll(G),S(G)).
E
u
MS(G) such t h a t T f = u * f f o r e v e r y f
E
L1(G).
E
We have known, f o r every above-mentioned T, t h e r e e x i s t s a unique measure E M(G) such t h a t
Tf = u
*
f f o r every f
E
We w i l l prove
L1(G).
u
E
MS(G).
Since T i s bounded, iiTti = sup 1 I f1 I = P'*fllsl
Take f = ua, n o t i c e t h a t iiuaiil
*
= 1, t h u s IIU * ~ I
So we have proved u
E
MS(G) and 11~11
=
IIU,*~II~
< IITII.
MS
The p r o o f o f Theorem 1 has been accomplished. Note 1.
I t i s n o t d i f f i c u l t t o p r o v e t h a t FI,(G)
approximate i d e n t i t y {u,] i d e n t i t y o f L1(G) and p Note 2.
o f L1(G), i.e. E
i s independent o f t h e
i f { v a } i s a l s o an approximate
M(G), t h e n I I U ~ * ~ I
Theorem 1 can be denoted by M(L1(G),
S(G))
MS(G).
Now, we w i l l use Theorem 1 t o s t u d y M(L,(G),
S(G)) f o r some c o n c r e t e SeSal
alyebras.
Corollary 1.
Suppose t h e Seyal a l g e b r a Ap(G) i s t h e f o l l o w i n g :
G.Z. Ouyang
206 A (G) = { f P
L1(G) : f
E
Lp(G)},
E
A
"fit*
= "fal
llfll
f
P
P'
and suppose B(G) = { p
E
M(G) :
Then B(G) = MA ( G ) , t h a t i s B(G)
P
LP(i)].
E
M(L1(G), A ( t i ) ) , P
"?' means i s o m e t r i c and a l g e b r a i s o m o r p h i c .
where
R e f e r t o t h e above n o t e
2. Proof.
I t f o l l o w s from Theorem 1 t h a t p
.
I I u ~ * ~ I I A< C
P
MA (G) i f and o n l y i f
E
P
But
U IIUa*pIIA
= IIU *!.Illl
+ II(Ua*p)
= IIua*plll
+ IIUa.llll
IIp
P *
A
.
.
and itua.pt~
P
i s bounded i f f i i j i i P <
C o r o l l a r y 2.
-.
A
P
The p r o o f o f C o r o l l a r y 1 i s c o m p l e t e .
Suppose t h e Segal a l g e b r a F ( H ) i s t h e f o l l o w i n g :
F ( R ) = { f E L ~ ( R ): l i m ? ( n ) I n n = U} n+m
+ sup{ l i ( n )
llfllF = Ilflll
1
I n n},
and suppose C ( R ) = {!.I
E
M(G)
: { I j ( n ) l I n n}
Then C(R) = MF(R), t h a t i s C ( R ) 3_ M(L1(R),
Proof.
I t f o l l o w s f r o m Theorem 1 t h a t p A
iium*piiF = iium*util
(m = 1,2,3 L e t u,
be a F e j g r kernel
E
i s bounded}.
F(R)). MF(R) i f and o n l y i f *
+ S;P{ lu,(n)p(n)(
I n n}
< c
-4
,...) .
(m = 1 , 2 , 3 , ...), t h u s
iium*pul
<
up11
and
Multipliers of Segal Algebras
201
{ l i m ( n ) L ( n ) [ I n n } < Cu i f f { l b ( n ) l I n n } i s bounded.
Then sp;
The p r o o f i s
complete.
Corollary 3.
Suppose t h e Seyal a l y e b r a S f ( K ) i s t h e f o l l o w i n y :
+ IIf*yllm,
= IlYIll
IIyII
sf
where f
L1(R) and f
E
0 , Co(R)
#
i s a l i n e a r space c o n s i s t i n g o f a l l t h e
c o n t i n u o u s f u n c t i o n s on R w h i c h a p p r o a c h t o z e r o a t i n f i n i t y w i t h t h e norm II iim
,
and suppose D(K) =
Then D ( R ) = MS ( R ) , f
Proof. itun*uii
Sf
{u
that i s
M(K) : p
E
D(R)
+ iiun*u*fiim < C IIun*ulll
*
*
f i s the (C,l)
bounded i f f iiu*fii,
$2.
<
E
E
+ IIUn*!J*fllm
<
ll!JIl
Lm(K)}.
Sf(R)). MS ( R ) i f and o n l y i f f
( n = 1,2,3,
!J
L e t un be a F e j G r k e r n e l ( n = 1,2,3, un
f
= M(Ll(R),
I t f o l l o w s f r o m Theorem 1 t h a t 11
= iiun*uiil
*
...). B u t
+ IIUn*u*fllm.
...); n o t i c e t h a t
sum o f t h e F o u r i e r i n t e y r a l of
*
p
,, *
f
E
L1(R),
f. Then IIun*,,*fllm
The p r o o f i s c o m p l e t e .
m.
M u l t i p l i e r s of Segal algebra $(G)
L e t G be a l o c a l l y compact b u t noncompact a b e l i a n yroup, and
A (G) = { f P
E
= llflll
llfllA
P
L1(G) :
+
Ilfll
iE
Lp(t)},
1< p <
m,
P'
It has been known t h a t Ap(G) i s a Segal a l y e b r a .
M(G) [S]. We have known t h a t M(Ap(G), A ( G ) ) P In 1 o f t h i s p a p e r , we have p r o v e d M(L1(G), Ap(G)) Suppose f
s e c t i o n , M(Ap(G), L l ( G ) ) w i l l be c o n s i d e r e d . P(L1(G))
= {f
E
L1(G)
:
7
E
E
Cc(i)},
MA ( G ) . In this P P ( L ~ ( G ) ) , where
is
G.Z. Ouyang
208
c c ( t ) i s a l i n e a r space c o n s i s t i n y o f a l l t h e continuous f u n c t i o n s on t h e compact support. Suppose
, T~ ,
T~
0
i d e n t i t y o f G.
6
with
P ( L ~ ( G ) )i s dense i n every Segal a l g e b r a S(G).
1
...,
,
T
xn
... a r e t r a n s l a t i o n
o p e r a t o r s , xo = e t h e
Uefine f
n
f
X
T X f + T x f+...+T 0 1
=
n
ntl
We have
in E Cc(?i) For any
E
> 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
-
<
YIll
E.
Let
By t h e homogeneous s t r u c t u r e o f L l ( G ) , i t i s easy t o see t h a t
-
'If,
Lemna.
..., xn, ... i n G such t h a t
(a)
iiynn2
+
0,
(n
+
-1.
(b)
iifnii2 +
0,
(n +
-1.
(c)
There i s a subsequence xn
1< p <
Proof. Xo,
XI,
E.
Suppose G i s a l o c a l l y compact b u t noncompact a b e l i a n group.
e x i s t s a sequence x o , xl,
( k + -),
gnu1 <
,x ,
1
-.
..., x 'k , ..., such t h a t
Suppose supp g = K , K i s a compact s e t i n G.
..., x,, ... . i n G such t h a t n 1
(K,,)
=
0,
J
Hence we o b t a i n
n
SUPP T ~ . Y 1
SUPP T ~ . Y=
J
0, i f j,
i
# j.
There
I f n m p + 0, k
There i s a sequence
Multipliers of Segal Algebras
209
But
Therefore
For ( b ) , we have
Then ( a ) has been p r o v e d .
tIfnl12
Notice that f
-
< llfn-gnl12 +
llYnl12.
y i s a bounded f u n c t i o n on ti s i n c e y
E
C,(G)
and f
E
Co(G).
Suppose iif(x)
-
y ( x ) i i - < C,
C i s a constant.
Therefore
-
Ilf,
< c llfn-gnII1
ynll;
+
(n +
0,
m).
( b ) has been proved. A t l a s t we w i l l p r o v e ( c ) .
( n + -).
t h a t I I ? ~ I I+~ 0
in (Y) + k
in (y)
0
(k +
+ 0 (k +
E
P ( L l ( G ) ) and tifnl12 + 0
(n +
m ) ,
Hence t h e r e e x i s t s a subsequence I f n ] of k
a l m o s t e v e r y w h e r e on
m)
m)
f,
f o r e v e r y p o i n t on
*G.
But every
implyiny
Ifn],
i s continuous, in
so
k
t.
k N o t ic e
.
1+(Xl ,Y )+. .+(xn,Y ) fn(Y) =
. f(Y),
n+l ((xi,y)I
< 1,
i = 1, 2,
..., o b v i o u s l y , (in (Y)] < k
But
in E Cc(c)
and
?
E
li(Y1J. Y
E
t.
C c ( 6 ) , Lebesyue Dominated Converyence Theorem i n d i c a t e s
k tifn
n k p
The lemma h a s been p r o v e d .
+ U
(k
+
-).
G.Z. Ouyang
210
Theorem 2.
Suppose G i s a l o c a l l y compact b u t noncompact a b e l i a n yroup,
then M(Ap(G), L1(G)) where
M(G).
i s o f t h e same meaning as i n 51.
Proof.
Suppose T
Hence f o r e v e r y
> 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
+
E
< IITII(II~,II~+
Now we e l e c t a subsequence
(Tf)lll
T
= (n+l)llTfnlil
'n
< IITII.II~
II? II
n P
) +
Ifn}
of
E.
Ifn} m a k i n g
( c ) o f t h e lemma t e n a b l e ,
k Therefore, according t o t h e f o l l o w i n g i n e q u a l i t y :
and as L l ( G ) i s a homogeneous Banach a l g e b r a , we o b t a i n iiTfiil
< IITII
f o r every f
iifiil
E
P(Ll(G)).
I t i n d i c a t e s t h a t T d e f i n e s a bounded l i n e a r o p e r a t o r f r o m P(L1(G)) t o L1(G)
and T commutes w i t h any t r a n s l a t i o n ,
T d e f i n e s a u n i q u e bounded l i n e a r
o p e r a t o r T (we s t i l l u s e t h e same s i g n t o d e n o t e i t ) f r o m L1(G) t o L1(G) T commutes w i t h t r a n s l a t i o n , and t h e norm
because P(L1(G))
i s dense i n L1(G).
d o e s n ' t chanye.
A c c o r d i n y t o a known t h e o r e m a b o u t t h e m u l t i p l i e r s of L1(G),
t h e r e e x i s t s a u n i q u e measure 11
E
*
f
Tf = p
M(G) such t h a t f o r every f
E
Ll(G).
Multipliers of Segal Algebras
u e M(G) , define
for
Conversely,
*
Tf =
*
Obviously,
f
E
f
211
an o p e r a t o r T by
f o r every f
E
Ap(G).
L1(G) because L 1 ( G ) i s an i d e a l i n M(G).
Then, a c c o r d i n g t o
t h e p r o p e r t y o f c o n v o l u t i o n , we can v e r i f y t h a t T i s a bounded l i n e a r o p e r a t o r f r o m Ap(G) t o L1(G) and
T
commutes w i t h any t r a n s l a t i o n ,
i.e.
T
E
M(Ap(G),
Ll(G)).
So much f o r t h e p r o o f o f Theorem 2. I f G i s a d i s c r e t e group, Theorem 2 w i l l become t h e f o l l o w i n y t h e o r e m .
Theorem 3.
Suppose G i s d i s c r e t e a b e l i a n y r o u p , t h e n M(Ap(G),
Proof.
Ll(G))
'
Ap(G).
When G i s d i s c r e t e , we have = L1(G) = M(G),
L l ( G ) - n o r m and M(G)-norm a r e t h e same.
+
llfllA = llflll
Ilfll
P
^G
Because
P'
IIfll,
<
llflll,
< CIIflll, where C i s a c o n s t a n t . P o b t a i n t h a t L1(G)-norm and A (G)-norm a r e e q u i v a l e n t . P and
i s compact, we have Il?ll
Suppose T
E
T h e r e f o r e we
M(Ap(G), L 1 ( G ) ) and e i s an i d e n t i t y o f Ap(G) = L1(G).
Obviously,
Tf
= T(e
and IITII = iiTeiiM = iiTeiil, Conversely,
let y
Tf = g
*
E
f
f ) = Te
Te
E
*
f f o r every f
E
Hp(G),
Ap(G).
Ap(G), d e f i n e f o r every f
I t i s easy t o v e r i f y t h a t T
43.
*
Seyal a l y e b r a A I , ~ ( G )
E
E
Ap(G).
M(Ap(G), L l ( G ) ) .
and i t s m u l t i p l i e r s
Denote t h e space w h i c h c o n s i s t s o f t h e f o l l o w i n y f u n c t i o n s b y m
where 1 < p, q <
1
m,
-P+ Lq
= 1,
$ 1. ( x ) = v i ( x -
1 ) and
P
(G):
212
G.Z. Ouyang OD
I:
IIUiIlP i=l
<
IIViIq
m.
Ti (G) i s d e f i n e d by
The norm on
P
m
= inf{
iifi
AIJ
1 ~ i u ~ i i ~1i i qv}. i.i i=l
Now we w i l l d i s c u s s an i m p o r t a n t s u b a l g e b r a AI,~(G) A1,p
( G ) = K p ( G ) fl
o f Tp(G):
;
i t s norm i s d e f i n e d by nfn
= iifiil
Herz
G.
[Z] has p r o v e d t h a t
multiplication. alyebra.
Then Her2 [ S ]
.
+ nfiiA
A1,P
P
aP (G)
i s a Banach a l y e b r a under t h e p o i n t w i s e
has p r o v e d t h a t
aP (ti) i s
L a i and Chen [4] have p r o v e d t h a t Al,p(G)
a l y e b r a under t h e p o i n t w i s e m u l t i p l i c a t i o n .
a r e y u l a r Tauberian
i s a c o m m u t a t i v e Banach
Now we w i l l d i s c u s s t h e space
A1,P( G) under c o n v o l u t ion. Theorem 4. Proof.
G) i s a c h a r a c t e r S e y a l a l y e b r a w i t h c o n v o l u t i o n p r o d u c t .
Al,p
O n l y t h e f o l l o w i n y f a c t s need t o be proved.
(a)
F o r any f
(b)
Al,&
(c)
For every f
nyfiiA
G) i s dense i n L1(G). E
A 1 ,P (G) and e v e r y y
E
*G,
yf
E
Al,p(G)
and
= iifiiA
1,P
1,P.
Now, we w i l l p r o v e t h e s e f a c t s .
(a)
It i s obvious t h a t f
*
ui,
ti
si
E
L (G), P
vi,
y
E
E
L1(G) f o r e v e r y f, y
Lq(G),
1 + P
=;
4
1.
E
AI,~(G).
Let
Multipliers of Segal Algebras
213
then f * y = L1(G) and s i
E
t i E Lq(G), hence f
*
f
E
C(f*Si)*ti; i=1
*
Lp(G), t h e n f Y
s i E Lp(G) and iif*s.ii
1 P
c iifiil
IIS.II 1
P'
But
AI,~(G).
E
Besides , Nf*91
+ flf*guSip
= IIf*glll A1Y P
m
,I llsillp.llt.Il 1 1 =1
c 111111 llglll + " f a 1
4 m
m
For every
s{
u;,
E
E
> 0, t h e r e e x i s t u i ' , v i ' s i ' , t i ' , such t h a t
L (G), v;, P
ti
E
Lq(G)
,1+
1=
1
and above i n e q u a l i t i e s h o l d , and
m
1
lls;llp
IlVillq c IlgIlK + P
i=1
E l
Thus we o b t a i n
N o t i c e t h a t L1(G) has t h e f a c t o r i z a t i o n p r o p e r t y , so t h e r e e x i s t y,
(b) h
E
L l ( G ) such t h a t f = g
so f o r any
E
*
h for f
> 0 there e x i s t 151
-
p, N
ynl
E
L1(G).
Besides, Cc(G) i s dense i n L1(G),
E Cc(G) such t h a t
<
nh
E,
- Till
<
E.
Hence we o b t a i n
nf But
p*
'Fi
E
Cc(G)
*
- *
Fill
= iig
*
h
-g*
Cc(G), i t i n d i c a t e s t h a t
N
c
C
Fill <
(G)
*C
CE.
C
( G ) i s dense i n L1(G).
G.Z. Ouyang
214
has been proved.
(b)
m
(c)
Let
y
E
f =
G,
1
i=1
* ti,
ui
m
1
i=1
iiuiiiP
yf =
But yui
E
< nfnA
+
E.
E
A
P
,Iyui * vvi.
1 =1
L (G) and I I ~ U ~ =I I I ~I U ~ I I ~ t, h u s y f P
= iifiil,
nyfiil
iiviiiq
1.P
(G).
Besides, by
we have
Repeat t h e above process and n o t i c e y-’
y = 1, so we o b t a i n
a contradiction. Thus Theorem 4 has been proved. L e t G be a l o c a l l y compact but noncompact a b e l i a n yroup.
Theorem 5.
M(Al,p(G),
Proof.
Let T
t h e r e e x i s t xu,
E
L1(G))
M(G)*
For each f
M(AlYp(G), Ll(G)).
..., xn, ... i n G,
XI,
‘
Then
E
Al,p(G)
and any
E
> 0,
xo = e t h e i d e n t i t y o f G. such t h a t
< STII (llfnlll
+ Ilf
II
) +
E
AP Tx
where f n =
f+Tx
f
f+...+T
‘n
O
ntl Let
1
i=1
E
The p r o o f i s s i m i l a r t o t h e p r o o f o f Theorem 2.
m
f =
For
.
ui
* ti,
ui
E
Lp(G),
> 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),
<
E.
S i n c e Cc(G) i s dense i n Lp(G) and i n L q ( G ) ,
i = 1, 2 ,
..., N ,
1IU.-(1.11
1 1 P
such t h a t
<;,
1lV.-6.11 1 1 q
N
N
M = max (
1 1iu.14 i=l 1 P '
Let
< Fl' 2
1
IIV.II 1
4'
N).
N
J, =
,Ic i * 1=1
;j9
N
+ 1 (ui-ai) * i=l N
iigN-J,ll'A;
< i1 p =l
N
nuiiip.iIvi-B
.II
1 9
+
1 Bui-aillp.llVi-B i=l
'.I14
there exists
216
G.Z. Ouyong
Therefore iiTfiil
+
IITII [ilfniil
N o t i c e t h a t p > 1, a l s o
E
48
1 - - 1 + (n+llP
N
1
1.
llaillp.I16.11
i=1
and n a r e a r b i t r a r y , and llfnII1 < IIfal, iiTflll
so we o b t a i n
c iiTn 1 1 f t 1 ~ .
Then i t i s s i m i l a r t o t h e p r o o f o f Theorem 2, t h e d i f f e r e n c e i s t h a t :
we
have used t h e d e n s i t y o f P ( L l ( G ) ) i n L1(G) i n t h e p r o o f o f Theorem 2, b u t here we use t h e d e n s i t y o f t h e Segal a l g e b r a Al,p(G) t h e r e e x i s t s a unique measure Tf = Conversely, l e t p
E
u
We can o b t a i n t h a t
M(G) such t h a t
u * f
f o r every f
E
AI,~(G).
D e f i n e an o p e r a t o r T by
M(G). Tf =
E
i n L1(G).
u *
It i s easy t o v e r i f y t h a t T
f
f o r every f
M(Al,p(G),
E
E
AL,~(G).
Ll(G)).
Theorem 5 has been proved.
Theorem 6 .
L e t G be a l o c a l l y compact but noncompact a b e l i a n group. M(AlYp(G),
Proof.
Al,p(G))
Then
M(G).
Since f o r every f
itfnl c iifii
AlYp(G),
E
A1,P each T
E
M(AlYp(G),
A l Y p ( G ) ) d e f i n e s a unique element i n M(AlYp(G), L l ( G ) ) . E
M(S) such t h a t
f E Al,p
(G) f o r every
Then a c c o r d i n g t o Theorem 5, t h e r e e x i s t s a unique measure p Tf =
u *
f
f o r every f E A1
(G) Y P
and null = nTIi. Conversely, f E AlYp(G).
if p
E
t h e n it i s obvious t h a t p
M(G),
So we can see t h a t Al,p(G)
*
i s an i d e a l i n t h e measure a l g e b r a
M(G) and
Uefine an o p e r a t o r T by Tf = p
I t i s easy t o see t h a t T
E
*
f
f o r every f
M(AlYp(G), AI,~(G)).
E
A
1I P
(G).
Multipliers of Segal Algebras
217
The p r o o f o f Theorem 6 has been accomplished. When G i s a d i s c r e t e group, as i n t h e p r o o f o f Theorem 3, we have: Theorem 7.
I f G i s a d i s c r e t e abe
M(A1 , p ( G l s L1
Some open problems: 1.
F o r each Seyal a l y e b r a S(G), what i s t h e c h a r a c t e r o f m u l t i p l i e r s from
S(G) t o L1(G) ? 2.
Unni
[71
has proved t h a t :
For each T
E
M(S(G), S(G)), t h e r e e x i s t s a
P(G), such t h a t T f = u * f f o r e v e r y f E S(G). Denote t h e whole o f t h e above pseudomeasures u by Ps(G). I t i s p r o b a b l e t h a t Ps(G) i s unique pseudomeasure u
E
o n l y a p r o p e r subset o f P(G). algebras.
For example, Ps(G) = M(G) f o r some Segal
Now t h e problem i s t h a t :
What i s a necessary and s u f f i c i e n t
c o n d i t i o n d e s c r i b i n y Ps(G) ? 3.
More g e n e r a l l y , l e t S1(G) and S2(G) be two Seyal a l y e b r a s .
What i s t h e
c h a r a c t e r o f M(S1(G), S*(G)) ?
References [l]Goldbery, K. K. and S e l t z e r , S. E . , U n i f o r m l y c o n c e n t r a t e d sequences and m u l t i p l i e r s o f Segal a l g e b r a s , J. Math. Anal. and Appl., 59(1977), 488-497. [ 2 ] Herz, C., Theory o f p-space w i t h an a p p l i c a t i o n t o c o n v o l u t i o n o p e r a t o r s , Trans. Amer. Math. SOC. 154( 1971), 69-82. [3] Herz, C., Harmonic s y n t h e s i s f o r subgroups, Ann. I n s t . F o u r i e r (Grenoble), 23( 1973), 91-123. [ 4 ] L a i , H. and Chen, I . , Harmonic a n a l y s i s on t h e F o u r i e r a l g e b r a Al,p(G), J. A u s t r a l . Math. SOC. ( S e r i e s A) 3U(1981), 438-452. Larsen, R., An i n t r o d u c t i o n t o t h e t h e o r y o f m u l t i p l i e r s , S p r i n y e r - V e r l a y [ S] (1971). [6] Ouyang, G., M u l t i p l i e r o p e r a t o r s f r o m L1(G) t o a Segal a l g e b r a , Chinese Annals o f Math. Vo1.5 (Ser. A), 2(1984), 247-252 (Chinese). Unni, L. R., A n o t e on m u l t i p l i e r s o f a Segal a l g e b r a , S t u d i a Math. [7] 99(1974), 125-127.
This Page Intentionally Left Blank
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
219
Differentiation in Banach spaces
Minos Petrakis and J . J . Uhl. J r .
1. P a r t 1.
Norm differentiation.
This section is a commentary on that part of mathematics having its origin with the following question of Tamarkin's: Tamarkin's Question: What are the Banach spaces X Lipschitz function f : [O.l] + X
such that each
is norm differentiable almost everywhere?
To get some familiarity with the question, we'll look at some examples. Example 1.
The function f : [O.l] + L1[O.l]
defined by
=
f(t)
x
[ O . t]'
0 < t < 1 is Lipschitz but is nowhere differentiable. To check this, note that if 0 < s < t < 1. then Ilf(t) - f(s)Il, It -
51:
therefore f
is Lipschitz. Also if 0
f t+h -
lim ( h-10
f(t)
Example 2. The function f 0
<
t
:
1.
Note that evidently for 0
Ilf(t)
<
t
<
1, then
-
h 4
1
a,
h and 0
= lim '[t,t+hl
in measure but not in L -norm because
(lsin(nnx)dx)n=l
<
=
1
t t+h
1
~
1
=, 1
[O.l] + co defined by
f(t)
for small h.
=
is Lipschitz but not differentiable a.e..
<s
1
< lldx =
- f(s)ll
( t - sI;
cO therefore f
is Lipschitz. But if
f'(T)
-
exists at some t
in [O.l].
then f'(t)(
= (sin(nlrt))
E
which is impossible except at certain selected
co
-t
from a null set.
M. Petrakis and J.J. Uhl, Jr.
'220
Example 3.
rf
f
:
[O,l]
+
el
is Lipschitz, then f
is differentiable
almost everywhere.
To see why, write f = ( f 1' f2'....fn....) m
and compute variation of
1
f =
m
variation of fn =
n=1
.
1
1 J I~;I J,
n=1 0
=
m
1 Ifhi.
This shows (fh(t))n=l
m
E 8,
for almost all
t
in [O,l] and it
n=1 is not much more work to show
= fh(t))
f'(t)
for almost all t. Upon seeing this little argument, Dunford and Morse [6] in 1936 came up with the following definition. Definition 4.
A Banach space
X has a boundedly complete basis (xn) if
(a) Each x E X has the unique expansion m
1 anxn
x =
n=1 where the an ' s are real numbers uniquely determined by x. (b) If (p,) is a sequence of real numbers and m
1
c
supll m n=l
PnXnll
then the series m
n=l is convergent in the norm of X.
<
m,
and
Differentiation in Banach Spaces
Now suppose f
:
22 1
[O.l] + X is Lipschitz. and write
n=l Use (b) to show
is convergent in X for almost all t and then directly compute m
This proves Dunford’s and Morse’s fifty year old theorem: Theorem 5. If X has a boundedly complete basis, then every Liuschitz f : [O,l] + X is differentiable almost evervwhere. Most familiar Banach spaces (e.g. 8 1 p < m) have boundedly complete P bases. Putting (1) and (2) together with (4) proves that no basis of co or is boundedly complete. This shows that differentiation can be used to L1 study individual Banach spaces. This is correct and much of the work of the last twenty years is motivated by this little observation. The first response ever given to Tamarkin’s question was by Tamarkin’s student Clarkson. Recall that a Banach space X is uniformlv convex if for each E > 0 there is a 6 > 0 such that
IIx
-
yll
<
e
provided llxll = llyll = 1
I I X
+ yll > 1 -
and 6.
In 1936. Clarkson [3] introduced the notion of uniform convexity to prove the following theorem
222
M.Petrakis and J.J. Uhl, Jr.
Example 5. If X is uniformly convex, then every Lipschitz f : [O.l] + X is differentiable a.e. Clarkson's theorem marked one of the very first investigations into the geometry of Banach spaces. It provides a concrete link between differentiation and geometry and shows, among other things, that neither
co
nor L1 have equivalent unifomrly convex norms. In case it is not obvious, what we have been talking about is a property of Banach spaces now well known as the Radon-Nikodym property. Definition 7. A Banach space has the Radon-Nikodvm property (RNP) if every Lipschitz f : [O,l] -B X is differentiable a.e. To get more information on RNP. we'll set up some machinery, the idea of which is present in the classic 1940 paper of Dunform and Pettis [S]. Take a Lipschitz function f [O,l] into intervals, set
:
[O,l]
-B
X and for each partition
T
of
T
where that
'[a,b) Ilf II, T
Definition 8 .
is the characteristic or indicator function of
[a,b).
Note
is no greater than the Lipschitz constant for f.
Let (wn)
for all n the partition
be a sequence of partitions of T
~
+ refines ~
lim rnax Ib n-ro [a,b]€Tn
T
n'
-
[O,l]
such that
If
a1 = 0 .
(fr ) a difference quotient martingale. It is not hard to prove n the following theorem.
we call
Theorem 9. A Lipschitz function f : [O.l] + X is differentiable everywhere if and only if there is a difference quotient martiwale
Differentiation in Banach Spaces
In which case
223
lim fK = f' a.e. and all other difference quotient n n
martinvales converge to f' as well. Some might recognize Theorem 9 is a primitive version of the martingale convergence theorem but this sets us off our track. Suffice it to say that once Theorem 9 is understood, then the Chatterji-Ionescu-Tuleca (see [3,13]) characterization of RNP via martingales is an immediate corollary. Not so easy but not terribly hard is Pettis's theorem: Theorem 10. Suppose X is separable. A Lipschitz function f
is differentiable almost g :
[O,l]
+
:
[O.l]
+
X
everwhere if and only if there is a function
X such that
-
x*(f(t)
f(0)) =
p*!3 0
*
for all x in a separatinp; subset of the dual space X*. f' = g a.e.
In this case,
Pettis used this theorem to prove that an absolutely continuous f : [O.l] + X is differentiable almost everywhere in norm if and only if f is differentiable weakly at each point of a subset of [O,l] whose complement is Lebesgue null. Here are some related consequences. The first is due to Dunford and Pettis [S]. Corollary 12. Separable dual spaces have RNP. Proof.
Let f
:
[O,l]
+
X
quotient martingale. Since Ilf U constant for f.
fK ) be a difference n Itm is no greater than the Lipschitz
be Lipschitz and let
n
we see that for each t
is bounded. Let g(t)
in [O.l],
*
the sequence (fr (t)) n
be an arbitrary weak -cluster point of
(fr (t)). n
From real variable theory,
(f(t)
-
f(0))x
=
:J
x
for all x in X. Apply Pettis's theorem to learn f' = g a.e. The next corollary is due to Dunford-Pettis-Phillips.
M. Petrakis and J.J. Uhl, Jr.
224
Corollarv 13. Reflexive spaces have RNP. Since Lipschitz functions on [O.l] have separable ranges, it is clear that a Banach space has RNP if and only if each of its separable subspaces has RNP. But each separable subspace of a reflexive space is a separable dual space and hence has the RNP. Proved by a similar argument is the next corollary which has been described by some as the Uhl Tool. Corollare 14. If X is a Banach space such that each separable subspace of X has a separable dual, then the dual X* has RNP. A famous theorem of Stegall's [19] shows that the converge of Corollary 14 is true and hence the statement of Corollary 14 characterizes the RNP
in
dual spaces. It is well known that neither L1 nor
co are dual spaces. The usual
argument proceeds by showing that the unit ball of each contains no extreme points. But all this shows is that in their given norms they are not isometric to dual spaces. The above theorems can be used to give more information, a fact first noted by Gelfand. Corollarv 15. Neither co nor L1 are isomorphic to subspaces of a separable dual space. Now let's take up the question of RNP and geometry. One way to block the RNP for a Banach space X is to find a Lipschitz function f : [O.l] X that has a difference quotient martingale (fr ) such that there is a n 6 > 0 with the property that
Ilf"
(t)
n+1
-
f,, (t)llX n
.
... for all t E [O,l) and all n = 1,2. the dyadic partitions:
f
H2
= f(1/2) - f(0 1/2 - 0 '[0,1/2)
= "2'[0,1/2)
L
Let's see what this means for
f(1) +
+
6
1
"3'[1/2.1)
- f(1/2 - 1/2
'[1/2,1)
+
Differentiation in Banach Spaces
22s
etc. Now note x2 + x3
x1 =
2
and in general X
xn = 2n + X2n+1 2 i.e. the sequence (x,)
partitions
is a bounded tree in
. X. Also for the dyadic
T
n'
llfT
(t)
-
-
11%
n+ 1
X2kll
=
fT (t)ll
n
or 11%
- x2k+lll
depending on whether t is in the left hand side or the right hand side of the interval in K in which t is found. n Definition 16. A bounded tree (x,) there exists a 6
>
in a Banach space is called a 6-tree if
0 such that Ilxn
-
x2nII
>
6
and IIX
n
- x2n+lll
Here are two easy examples of 6-trees: Example 17. A-6-tree
in L1[O.l]
>
6.
M. Petrakis and J.J. Uhl, Jr.
226
Example 18. A 6-tree in co
'\ XI
x2 =
(-1.-1.0.0. ...)
= (O.O,O,O, . . . )
x3 =
(-1.1.0,o. ...)
(1 ,- 1.0.0 , . . . )
Note that if X contains a 6-tree (xn),
(l,l,O,O, ...).
then reading the above backwards
and supplying a small continuity argument produces a Lipschitz f : [ O . l ] such that
llfs (t) n
- fs
(t)llX
2 6
n+ 1
... .
for all t E C0.l) and n = 1.2,
This proves
Theorem 19. If X contains a bounded &tree. then X fails RNP. Stegall [19] proved the converse dual Banach spaces. Attempts to block non-dyadic difference quotient martingales give rise to the idea of a &bush. One way to help a Banach space to have RNP is to eliminate the possibility of 6-trees or &bushes. The idea is due to Rieffel [lS]. Definition 20. A subset D of a Banach space is not dentable if there exists an e > 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
<
E}
A slice of a set D is any set of the form {x E D : x*(x)
> 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.
<
1/2.
M.Petrakis and J. J. Uhl. Jr.
228
The slices are nested and their diameters tend to zero. By completeness, they all contain a common point; this point is an extreme point of D. One outstanding question is
Question 24. Are KMP and RNP equivalent properties? Huff and Morris [12] proved that for dual spaces, the two properties are equivalent. For non-dual spaces this problem has remained highly resistant to a solution. And it is not from lack of attention that this problem is not
*
a yet solved. The implication KMP RNP is vexing because there is no is separably determined (see the proof of Corollary 13 to see why RNP is separably determined). Nevertheless Schachermayer [lS] has obtained the following tantalizing facts. (1) If X is isomorphic to its square. then KMP and RNP a s equivalent for X. priori reason to believe KMP
(2) T h e KMP in dual spaces is separablv determined. (3) If e2(x) KMP, then x h a s RNP! Another way of studying RNP
is via operators on L1[O.l].
T : L1[O,l] + X be a bounded linear operator
Note that if 0 <
<
( t
<
<
1.
IlTll
Let
.ne by
then
IIX(s,tllll = IlTll t -
+
SI.
Therefore is Lipschitz with constant IlTll Similarly, Lebesgue-Steiltjes integration with respect to a Lipschitz function : [O,l] + X defines a bounded linear operator T : L1[O.l] + X for which
for all t
+
in [O,l].
Definition 25. A bounded linear operator T : L1[O,l] + X is called representable if the norm derivative
229
Differentiation in Banach Spaces
d
iE T(XIO, t]) ex sts almos everywhere. In view of the discussion a we. we mve Corollarv 26. A Banach space has RNP
into
operators from L1[O.l]
if and only if all bounded linear X are representable.
Let’s take a short look at the meaning of representability. Take an operator T : L1[O,l] + X and let
If T is representable and f E L1[O.l]
is a step function, then
fd+ =
Tf = J[O.l] The upshot of this is that
f+’
(Bochner integral)
Tf = “0.11 for all f E L1[O.l].
T
:
L1[O.l]
+
In other words, a bounded liner operator
X is representable if and only if there is a bounded
measurable g : [O,l] + X
(all derivatives are measurable) such that
Tf = S[o.llfg for all f E L1[O.l].
Weakly compact operators on L1[O.l]
are
representable for precisely the same reason that reflexive spaces have RNP. Here are some facts. F
a 27. Representable operators are almost compact.
Proof.
Let T : L1[O.l]
X have the action
Tf = J[o.11ff3
M. Petrakis and J.J. Uhl, Jr
230
for a bounded measurable g : [0,1]+ X. Let E > 0 and choose a sequence (g,) of simple functions converging to g a.e. Use Egorov’s theorem to
E [0.1] whose comploement has measure less than e then lim g = g uniformly on E. Note that if f E L1[O,l]. n-ro
find a measurable set E such that
Tf
=
SEfg +
SCo.l,\Efg
and that
is compact because it is the operator limit of finite rank operators. Thus
T is almost compact in this sense. The following fact is not well known, but it has been around for many years. Our proof follows that of Gretsky-Uhl [lo]. Corollary 28.
Proof.
Let W
Weakly compact subsets of
E Lm[O.l]
measurable set E and such that
E [O.l]
L,[O,l]
are almost compact.
be weakly compact. Given
>
E
0 we want to find a
such that [O,l] \ E has measure less than
{fx,
:
f
E
E
W}
is compact in Lm[O.l].
To this end, find a reflexive Banach space R and an operator T
:
R
+
L,
such that
(Here L1[O.l].
%
*
Let S = T
is the closed unit ball of R.)
Since R*
restricted to
has RNP. the operator
*
S:L1+R
is representable. A short computation shows S* = T. By Fact 27. there is a
Differentiation in Banach Spaces
measurable set E
E [O.l] such
23 1
[O.l] \ E has measure less than e
that
and such
f
i s compact from L1[O,l]
R*.
to
(S(* X , )
+
S(f%)
Therefore
*
:
R
+
Loo[O,l]
E W} = XE
*
w
%T(BR)
XET
is compact. Thus
{fs f :
is compact in L,[O,l].
This completes the proof.
Here is a quick corollary. Corollary 29. Weakly convergent sequences in L,[O.l]
are almost everywhere
convergent. The last theorem we’ll look at in this section is an old chesnut due to Dunford and Pettis. Let us agree that a bounded linear operator
T
:
L1[O,l]
+
X
is a Dunford-Pettis operator if T sends weakly compact
sets onto norm compact sets. Corollary 30. RepresFntable operators on L1[O.l]
are Dunford-Pettis
operators. Proof.
Let
W C L1[O.l]
be weakly compact and recall that this ensures that
E > 0 and use Fact 27 and the weak compactness of W to find a measurable subset E of [O.l] such that f + T(f%) is compact on
Let
L1[O.l]
and
J
If1
SUP f€E [O.l]\E
< &.
M. Petrukis arid J. J. Uhl, Jr.
232 Now
T(%W)
radius
T(%W), E .
e/2.
is compact and hence i t can be covered by finitely many balls of
Since every point i n
we see that T(W)
Therefore
T(W)
T(W)
is within
e/2
of a point i n
can be covered by finitely many balls of radius
is totally bounded.
Differentiation in Banach Spaces
P a 11. Dunford-Pettis operators on L1[O.l].
233
This section is devoted to a
study of Dunford-Pettis operators on L [O,l]. The main purpose of this 1 section is to make a case for the position that maybe Dunford-Pettis operators on L1[O,l]
can be profitably studied by a program modeled on the
RNP study of Part I. The main fact about Dunford-Pettis operators is the following elementary observation:
F a 1. A bounded linear operator T
:
L1[O,l]
+
X is a Dunford-Pettis
operator if and only if the restriction T : Lm[O.l]
Proof. If T
:
because B
is a weakly compact subset of L1[O.l].
Lm
L1[O.l]
+
X is Dunford-Pettis. then T(B
T
:
L1[O.l]
Lm[O,l]
+
and recall that W
uniformly in f E W.
:
L1[O.l]
is a bounded linear operator such
is uniformly integrable. Hence
let
E
>
J"
1ijin]=
o
0 and pick no
such that
Now
1
+ T(fX [lfl>n0]) :
is compact
X is compact. Let W be a weakly compact subset of
: 1
Since T
)
Lm
For the converse, suppose T that
is compact.
Lm[O.l]
+
:
X is compact it follows that
M. Petrukis and J. J. UhI, Jr
234
is relatively compact and therefore can be covered by finitely many balls of radius ~ / 2 . From the discussion above, we see that T(W) can be covered by finitely many balls of radius e . This proves T(W) is relatively compact and hence compact. The possibility of a strong analogy between representable operators and hnford-Pettis operators is contained in the next result. Theorem 2. L A T : L1[O,l] -+ X be a bounded linear operator. For O < t < l . set f(t)
and let (f
lr
)
= T(X
Cost])
be a difference quotient martingale. Then
n
(a) the operator T
is representable if and only if
(fr ) is n
L1-Cauchy; i.e. 1im
lr
n
m
and (b) the operator T is hnford-Pettis if and only if
(f, ) n
is Pettis
Cauchy: 1.e.
*
IlX*ll
Proof.
- x*fs I
Ix fs
lim sup J[0,1]
m
= 0.
n
The equivalence (a) is just Theorem 1.9. To prove (b). define the
operators E,
on L,[O.
11 by
n
where ~ ( 1 ) is the Lebesgue measure of the interval I. Recall that a
Differentiation in Banach Spaces
bounded subset K of L1[O.l]
is relatively norm compact if and only if
lim E, (f) = f uniformly in f E k. Now T n n
only if T* : X*
L1[O,l]
235
:
Lm[O.l]
*X
is compact if and
is compact which, in turn, is equivalent to
saying
**
**
lim En T x = T x n n
* <
uniformly in Ilx 1 I
1. This is the same as saying lim HE, T* - E, T*II = m n
o
m.n
for the operator norm. But a quick calculation shows that for fixed m
and
n IIE
R
*
*
T - En T II = SUP m n IlX*ll
Putting it all together proves that T
:
*
* n
Lm[O.l]
+
X is compact if and only
if
A glance at Fact 1 completes the proof.
Definition 3. A Banach space X has the complete continuity property ( E P ) if each T : L1[O,l] + X is a Dunford-Pettis operator. The property CCP is also known in the literature as CRP (compact range property). In terms of differentiation, a Banach space X has CCP if and only if every Lipschitz f : [O.l] + X is scalarly uniformly differentiable in one sense that for all difference quotient martingales
<
uniformly in II~*II 1.
M.Petrakis and J. J. Uhl, Jr.
236
Evidently any Banach space with RNP has CCP, but the converse is false as can be seen from the following theorem, which is an easy consequence of Rosenthal's fundamental el theorem. Theorem 4 . Let X be a separable Banach space. The dual X* and only if X contains no copy of the space e l .
Proof. Suppose X contains no copy of el. Let T
:
L1[O.l]
CCP
+
X be
If
a
bounded linear operator. By a compactness argument on the difference ) in the style the proof of quotient martingales (fn ) (for f(t) = T(X n LO. tl * d Corollary 12 produces a bounded function g : [O.l] + X (satisfying dt f(t)
*
x = x
for all x
g
Define an operator S
:
X
E
X) such that for each 0
+ L1[O.l]
by Sx = g
sequence in X. Since X contains no copy of tells us that
(x,)
E
L1[O,I]
x. Let (x,)
el.
be a bounded
Rosenthal's theorem
has a weakly Cauchy subsequence (x ).
Observe that
nj
since g : [O,l]
+
X*.
xn ) is a pointwise conversent .I sequence (g xn). The bounded conversence
the sequence (g
subsequence of the L,-bounded theorem guarantees that
(g
x ) is conversent in L -norm. Hence the 1 "j * * is also is compact and S : L,[O.I] + X
operator S : X + L [O.l] 1 compact. But a brief computation shows S* is exactly the restriction of T to L,[O.l]. Hence T : Lm[O,l] + X is compact. This proves T : L1[O,l] +
X is Dunford-Pettis. For the rest of the proof, see Riddle-Uhl [15]. Using this theorem, we can see that if X is the James Tree space, then X* fails RNP. but X* has OCP. Next we shall take a look at a tree-structure that plays the same role far CCP as the 6-tree structure plays for RNP. Definition 5. Let 6
>
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
<
E
or
Ilu
- x2II <
Ilu - xII
<
6.
then
e.
X has the approximate Krein-Milman property (AKMP) if every non-empty bounded subset of X has an e-strongly extreme point for A Banach space
E > 0. Rosenthal and Kunen [17] proved that if a Banach space X has the approximate Krein-Milman property, then X contains no 6-trees. Combining
every
this with Bourgain's theorem proves the following observation. Theorem 13. If a Banach space X has the approximate Krein-Milman property.
then X
has CCP.
Since the Bourgain-Rosenthal space BR has AKMP, this space also has
CCP. On the other hand. the space BR fails KMP and hence the AKMP does not imply KMP. In addition the following question is wide-open.
M. Petrakis and J J Uhl, JK
240
Question 14. Does a Banach space have CCP
i f it has KMP?
This question is more modest than question 24, although a negative answer to i t would also give a negative answer to question 24. Theorem 13 can also be used to give an alternate proof of Fact 9 because the hypothesis of Fact 9 rules out 6-trees. every closed bounded convex subset of
e,h
all
>
0. then X
does X
have
One surviving question is:
X has an CCP?
E
-
h
If
denting point for
How about the converse?
Finally we mention recent work of Ghoussoub. Godefroy, Maurey and Schachermayer [lS].
Say that a Banach space X
is strongly regular i f for
D of X and for every e > 0. there exist slices D and positive real numbers ul.u2.. . . .un whose sum is
every bounded subset S1.S2... . ,Sn of one such that
n diam(
1aiSi)<
e.
i=I They proved Theorem 15.
If X
is a strongly regular Banach space, then X
has CCP.
Our point in this section is simple: Our point is that the evidence suggests the strong possibility that CCP in the same way that
is an internal geometric property
RNP is. Still we must ask
Question 16. What is this internal geometric property? REFERENCES
[ 11
J. Bourgain, Dunford-Pettis operators on L1 and the Radon-Nikodym property, Israel J. Math. 37(1980).
[ 23
J. Bourgain and H. P. Rosenthal, Applications of the theory of semi-embeddings to Banach space theory, J. Funct. Anal. 52(1983).
[ 31 S. D. Chatterji. Martingale conversence and the Radon-Nikodym theorem in Banach spaces, Math. Scand. 22(1968). 21-41. [ 41 J.
A. Clarkson. Uniformly convex Banach spaces, Trans. h e r . Math.
SOC.
40( 1936). 396-414.
[ 51 W. J. Davis and R. R. Phelps. The Radon-Nikodym property and dentable sets in Banach spaces, Proc. Amer. Math. SOC. 45(1974), 119-122. [ 61 J. Diestel and J. J. Uhl. Jr. Vector measures, Math Surveys no. 15. Amer. Math. SOC. Providence 1977.
Differentiation in Banach Spaces [ 71
24 I
N. Dunford and A. P. Morse, Remarks on the preceding paper of James A . Clarkson, Trans. Amer. Math. SOC.40(1936). 415-420.
[ 81 N. Dunford and B. J. Pettis. Linear operations on summable functions, Trans. Amer. Math. Soc. 47(1940), 323-392. [ 91 N. Ghoussoub. G. Godefroy and B. Maurey. First class functions around sets and geometrically regular Banach spaces (preprint).
[lo]
N. E . Gretsky and J . J. Uhl, J r . Korotkov and Carleman operators on Banach spaces, Acta Sci. Math 43(1981). 207-219.
[ll]
R. E. Huff, Dentability and the Radon-Nikodym property in Banach spaces, Duke Math. J. 41(1976). 111-114.
[12] R. E . Huff and P. D. Morris, Dual spaces with the Krein-Milman property have the Radon-Nikodym property, Proc. Amer. Math. SOC. 49(1975). 104-108. [13] A. and C. Ionescua-Tulcea, Abstract ergodic theorems, Trans. Amer. Math. SOC. 107(1963), 107-124.
[14] B. J . Pettis. Differentiation in Banach spaces, Duke Math. J . 5(1939), 254-269.
[15] L. H. Riddle and J . J . Uhl. Jr., Martingales and the fine line between Asplund spaces and spaces not containing a copy of C1. Martingale theory in Harmonic Analysis and Banach spaces, Springer-Verlag Lecture Notes v. 939, 1982. [IS] M. A. Rieffel, Dentable subsets of Banach spaces with application to a Radon-Nikodym theorem, Proc. Conf. Irvine. Calif. 1966(B. R. Gelbaum. editor) Thompson, Washington D. C.. 71-77. [17]
K. Kunen and H . Rosenthal, Martingale proofs of some geometric results in Banach space theory, Pacific J. Math lOO(1982).
[lS] W. Schachernayer, The Radon-Nikodym property and Krein-Milnuan property, (preprint) 1986. [19] C. Stegall. The Radon-Nikodym property in conjugate Banach spaces, Trans. Amer. Math. SOC. 206(1975), 213-223.
This Page Intentionally Left Blank
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
243
"SPECTRAL SUBSETS'' OF Wm ASSOCIATED W I T H COMMUTING FAMILIES OF L I N E A R OPERATORS
Werner R I C K E R
*
S c h o o l of Mathematics and P h y s i c s Macquarie U n i v e r s i t y N o r t h Ryde, 2113 Australia
I n some r e c e n t p a p e r s [ 5 , 6 , 7 1 McIntosh and Pryde i n t r o d u c e d and gave some a p p l i c a t i o n s of a n o t i o n of " s p e c t r a l s e t " , y ( X ) , a s s o c i a t e d w i t h e a c h f i n i t e , commuting f a m i l y o f bounded l i n e a r o p e r a t o r s in a Banach s p a c e . U n l i k e most c o n c e p t s of j o i n t s p e c t r u m , t h e set y ( 2 ) i s p a r t of r e a l E u c l i d e a n s p a c e . A s s u c h , t h e r e a r i s e s t h e q u e s t i o n of t h e non-emptiness of y ( X ) . I t i s known t h a t i f t h e r e i s a n even numb e r of o p e r a t o r s i n v o l v e d o r i f e a c h o p e r a t o r T . c X h a s r e a l s p e c t r u m , t h e n n e c e s s a r i l y y(T) i s non-empty. I n t h i s n o t 2 i t i s shown t h a t y ( T ) i s non-empty f o r an a r b i t r a r y f i n i t e , commuting f a m i l y o f o p e r a t o r s ?F. ( c o n s i s t i n g of a t least two o p e r a t o r s ) whenever t h e underl y i n g Banach s p a c e i s f i n i t e d i m e n s i o n a l . I t i s s t i l l u n r e s o l v e d whether y ( 2 ) i s non-empty i n g e n e r a l .
L e t X be a complex Banach s p a c e and of c o n t i n u o u s l i n e a r o p e r a t o r s i n X ;
=
= {
(A1,.
.. , A m )
E
nm;
where B d e n o t e s t h e r e a l numbers
be a commuting m-tuple
t h e space of a l l such o p e r a t o r s i n X i s
d e n o t e d by L ( X ) . A j o i n t s p e c t r a l s e t y (2) , of
y (T)
. ., T m )
(Tl,.
j=l
C51.
z, i s d e f i n e d
by
( T . - A . ) 2 i. s n o t i n v e r t i b l e i n L ( X )
1
I
1,
T h i s n o t i o n h a s proved t o be u s e f u l i n
d e t e r m i n i n g f u n c t i o n a l c a l c u l i f o r c e r t a i n m-tuples
T
with applications t o find-
i n g estimates f o r t h e s o l u t i o n of l i n e a r 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 [5,6,71. I t i s known t h a t y(TJ
m i s always a compact s u b s e t o f iR c7; Theorem 4 . 1 1 .
The q u e s t i o n arises o f whether o r n o t y ( T ) is empty ? I f m = 1, t h e n it i s e a s y t o v e r i f y t h a t y(T) = u ( T ) n W
and hence,
t h i s case. F o r m 2 2 , i t i s known t h a t y ( 2 ) has of
real
2 [S;
i t may happen t h a t y ( T ) i s empty i n
# d i f each o p e r a t o r T .
1 _< j 5 m , 3' s p e c t r u m . I n d e e d , i n t h i s casey(X) c o i n c i d e s w i t h t h e T a y l o r s p e c t r u m Theorem
11
and so i s c e r t a i n l y non-empty.
F u r t h e r m o r e , McIntosh and
Pryde have shown, by a j u d i c i o u s u s e o f t h e t h e o r y of C l i f f o r d a n a l y s i s and monogenic f u n c t i o n s , t h a t y('I')
# d f o r a r b i t r a r y commuting m-tuples
2
whenever
m i s an even i n t e g e r [ 7 ; 531. The a i m of t h i s t a l k i s t o show t h a t y ( 2 ) # a r b i t r a r y commuting m-tuples
;and
for
i n t e g e r s m 2 2 whenever X i s f i n i t e dimen-
s i o n a l . I t i s s t i l l u n r e s o l v e d whether y ( T )
*
9
# d in general
( f o r m 2 2 ).
The a s s i s t a n c e p r o v i d e d by a r e s e a r c h t r a v e l g r a n t under t h e Queen E l i z a b e t h
11 F e l l o w s h i p Scheme is g r a t e f u l l y acknowledged.
W.Ricker
244
Let X be a complex Banach space, not necessarily finite dimensional. Let R denote the a-algebra of Borel subsets of the complex plane C. An operator
S
in
L(X) is a scalar-type spectral operator if there exists a spectral measure P:
+
L(X), supported by u(S), such that S =
jCzdP(z) =
ZdP ( 2 )
I
where the integral exists in the usual sense of integration with respect to a a-additive vector measure [3; p.19381. To say that P is a spectral measure means that P is o-additive when L(X) is equipped with the strong operator topology, P(E n F ) = P(E)P(F) for each E,F
E
8 and
P((1:) = P(o(S))
is the identity
operator in X. The measure P, necessarily unique c 3 ; XV Corollary 3.81, is called the r e s o l u t i o n of the i d e n t i t y of s. An operator T measure P: 8 of {P(E); E
-f
E
L(X) is a spectral operator if there exists a spectral
L(X), necessarily unique, such that T belongs to the commutant in L(X) and the spectrum of the restriction of T to each
E
closed invariant subspace P(E)X, E
B , is contained in the closure of E in
E
(c
[3; p.19301. This is equivalent to the existence of a scalar-type spectral operator S, whose resolution of the identity is necessarily P, and a quasinilpotent operator Q commuting with
S
such that T = S
+
Q [3; XV Theorem 4.51.
Furthermore, this decomposition is unique; S is called the scalar part of T and
Q the radical part of T. Let; = (T1,
...,Tm
)
be a commuting m-tuple in L(X)
m
consisting of spectral
operators. Let P . be the resolution of the identity of T . EjSm. Then define 1 If
(1)
P(EIX
N
lows that;
1'
Em) = P1(E1)P2(E2)
...Pm(Em)
,
...,
m
X... xEm in (1: , where each E E B , j = 1, m. 1 lSjSm, all commute [ 3 ; XV Corollary 3.71, it fol-
for each measurable rectangle E Noting that the measures P
...X
has an extension to a multiplicative, additive set function on the
algebra of s e t s , A , generated by the measurable rectangles c4; §41. The extension of
,p to A is again denoted by ,P. Observing that the a-algebra generated
by A is the Borel o-algebra,
Em,
of Cm it follows that ,P has an extension to
an unique spectral measure ,P:
-+ L(x) satisfying (l), in which case it is Bm called the j o i n t resozution of the i d e n t i t y o f 2 , if and only if, for each
x
E
x
the set {,P(E)x; E
Theorem 8 of [4],
E
A } is a relatively weakly compact subset of X; see
for example. If the Banach space X is weakly sequentially
complete, then the product measure ,P exists whenever {_P(E); E ed subset of L(X) c4;
E
A } is a boundE A } is
Theorem 91. In certain Banach spaces, {_P(E); E
automatically bounded for every
finite family of commuting spectral measures
p l ~ - - - , p mFor . example, this is known to be the case if X is finite dimensional or a Hilbert space or an LP-space: see pp.2098-2101 o f [31 for a more comprehensive list of such spaces. We remark that if the joint resolution of the
Spectral Subsets o f IRm f or Commuting Sets of Operators
245
identity of ,TI say ,PI exists, then there is available a multiplicative functional calculus based on the algebra of P-essentially bounded functions f:a m-a and defined via integration, namely f
-f
1
,
am f d :
[ 3 ; XVII 521.
In particular,
is the scalar part of T. l<j<m. The support of , P I de3 1 7 j 3' m noted by Supp(,P), is the complement of the largest open set U 5 such that am h.d,P(A) where
S. =
S
P(U) = 0.
N
2
THEOREM 1. Let X be a Banach space and
be a c o m t i n g m-tuple i n L ( X l r n
c o n s i s t i n g o f spectral operators. Suppose that t h e j o i n t r e s o l u t i o n o f t h e
2
i d e n t i t y of
exists. I f m
2 2,
then y Q ) is non-empty.
REMARKS. (1) Since every linear operator in a finite dimensional space is a spectral operator and the joint resolution of the identity always exists in finite dimensional spaces it follows from Theorem 1 that y(,T) is non-empty for every commuting m-tuple
2 in such spaces
(
m 2 2 ).
4 for every commuting
(2) If X is a Hilbert space and m 2 2, then y('T) #
m-tuple of normal operators. ( 3 ) If X = Lp(p)
,
where p is a u-finite measure and T , is the oper-
l
3
...,
m
E L (!.I), j = I, m, then the joint res7 olution of the identity of ,T exists and hence y ( 2 ) # 4.
ator in X of multiplication by
,)I
The proof of Theorem 1 is via a series of lemmas.
2
LEMMA 1. Let
be a commuting m-tuple i n L ( X l m c o n s i s t i n g o f s p e c t r a l oper-
a t o r s . I f S . i s t h e scalar p a r t o f Ti, 3
Proof. Let
Q
l s j s n , then y(_T) = y(,S).
be the radical part of T
j
j'
l<j<m. It follows from 1 3 ; XV
Corollary 3.71, the definition of spectral operator and the commutativity of the family {T.
l<j<m} that
all j and k. If
A
7;
form Q
X
E
+ z"1=1 (S.-X I
2
Rm , then it is easily seen that )2
Q = Q S . for J k k i (T.-A , ) is of the
and Q are commuting m-tuples and
where Q, commutes with
z"
N
z"
1=1
(S
7
S
7
and, being a finite
1=1
I Q,; l<j,k<m) with each
linear combination of products of operators from I S j l
product containing at least one element Q . ( for some j ) , Q, is quasinilpotent 7
(
cf. proof of XV Theorem 5.6 of C31). So, O ( T = ~ ( T ~ - , ~=NU(T=l(Sj-hj)2) )~)
[3; XV Lemma 4.41. Since ,X of
E
y ( 2 ) if and only if zero belongs to the spectrum
z"
(T.-A ) 2 it follows that y ( 2 ) = y(,S). I7 1=1 I j Lemma 1 shows that to establish Theorem 1 it suffices to do so for commuting
m-tuples of scalar-type spectral operators. For each 1 E R m define a subset
LEMMA 2. L e t
z
1
of
am
by
,S be a c o m u t i n g m-tuple in LO)''' c o n s i s t i n g of scalar-type
spectral operators. I f t h e j o i n t r e s o l u t i o n o f t h e i d e n t i t y o f S e x i s t s , say N
P, then N
Proof. If P . is the resolution of the identity of S 7
j'
l<j<m, then the def-
W.Ricker
246 i n i t i o n of
p
implies t h a t
p(IIm
m
o ( S . ) ) = ll. P , (u ( S . ) ) = I and hence Supp (p) 5 j=l 3 1=1 1 1 i s compactly supported. Accordingly, f o r e a c h f? E lRm
IIm o ( S . ) showing t h a t R j=1 3 t h e f u n c t i o n $ d e f i n e d by JI
k
q=l(~j-Aj)2, A
(A) =
M
E
Em, is R - e s s e n t i a l l y
bounded. I t f o l l o w s from t h e f u n c t i o n a l c a l c u l u s f o r
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 f u n c t i o n 1/$ ed [ 3 ; X V I I C o r o l l a r y 2.111.
cM
= 1/$
I?
f? (i)
and Supp(E) i s p o s i t i v e .
k
t h a t the operator
i s p _ - e s s e n t i a l l y bound-
= 0 i f and o n l y i f
i s compact, e i t h e r
a c l o s e d set and Supp(PJ between
But, jl
2
c
f?
A
E
5
P ( E ) = I and a > 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 .
This Page Intentionally Left Blank
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~(<)(l-~c~2).
S i n c e g(D) i s an a f f i n e t r a n s f o r m a t i o n of f ( D ) , i t f o l l o w s t h a t g a l s o belongs n
t o K. S u p p o s e g ( z ) = z + b2z
2
. S t r a i g h t f o r w a r d computations - I c I 2) f " ( r ; ) l f ' ( 5 ) - t .
+
b2 = ( $ ) ( I
show t h a t
From Theorem B , w e see t h a t lb21 C A2, and so /if"(S)/f'(i)
-
) < I 2 )1
21S12/(l -
S i n c e IwI C c c l e a r l y i m p l i e s Re{w)
> -
R e { 5 f " ( < ) / f V ( 5 ) }a 2 r ( r
-
C 2A2)51/(1
2
-
15) ) .
c , t h e above i n e q u a l i t y y i e l d s
-
A2)/(1
l
r2),
r.
=
Replacing 5 by z , we o b t a i n t h e d e s i r e d i n e q u a l i t y . n
Theorem 2 . 2 .
The r a d i u s of c o n v e x i t y f o r t h e c l a s s K i s
R
Proof.
Let f
E
i.
=
-
A2
-
(A;
1)'
-
0.4547.
From (2.1) and Lemma 2.1,
( z I = r o n t o a convex
f maps
curve i f r 2 - 2A r + I 2 0, and t h i s i s c l e a r l y nonnegative whenever
r C A2 R
2 A2
2 (A2
-
- (A:
2,
-
I)i.
-
Thus t h e r a d i u s of c o n v e x i t y , Rc,
satisfies
I)'.
On t h e o t h e r hand, l e t g(z) = [f((z+r)/(l + rz))
-
f(r)l/f'(r)(l
-
r
2
1,
where f i s t h e e x t r e m a l f u n c t i o n f o r t h e second c o e f f i c i e n t a s g i v e n i n Theorem B.
Formal computations show t h a t a t z = - r , 1 + zg"(z)/g'(z)
From ( 2 . 1 ) ,
=
(r2
-
2A2r + I ) / ( ]
every function f
E
o n t o convex c u r v e s .
2 (A2 - I ) ' ,
-
t h e c i r c l e s I z I = r , f o r r > A2
-
2
r ). a r e n o t mapped by - (A22 - I ) ' . This
T h e r e f o r e Rc = A2
completes t h e proof
3.
THE KOEBE D I S K The Koebe d i s k f o r K i s 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 A
c o n t a i n e d i n t h e r a n g e of each f u n c t i o n i n K.
The f o l l o w i n g theorem d e t e r m i n e s
t h e s i z e of t h i s d i s k . h
Theorem 3.1. {w : IwI < n / 8 1 .
The range of e v e r y f u n c t i o n f E K c o n t a i n s t h e d i s k The r a d i u s n / 8 i s b e s t p o s s i b l e . n
P r o o f . We w i l l show t h a t i f a f u n c t i o n f i n K o m i t s t h e p o i n t w , t h e n J w J 2 T/B.
Let
d = inf {lf(S)I : f
E
i,
IS/ =
11.
From Theorem A, i t s u f f i c e s t o minimize I f ( < ) I where f i s e i t h e r a h a l f - p l a n e h
mapping o r e l s e f i s t h e t r a n s f o r m of a s t r i p mapping. S i n c e K i s r o t a t i o n a l l y -I o r else f i s generated i n v a r i a n t , we may assume t h a t e i t h e r f ( z ) = z ( l - z ) by a v e r t i c a l s t r i p mapping. If f ( z ) = z(l - z)
-1
,
then min{l<(l
-
5)
-1
I
: 151 = 1 ) =
1.
M.A. Rosihan
252
Now suppose f i s o b t a i n e d from a v e r t i c a l s t r i p mapping.
These s t r i p
mappings i n K a r e given by
(3.1)
g(z,x)
=
[1/(2i s i n x)]log[(l + e
ix
=
m
T,
- g ( z , x ) l , where g ( n ) ,
r7
I , is finite.
0 < x <
z ) / ( l + e-ixz)],
and s o f t a k e s t h e form f ( z ) = g ( r l , x ) g ( z , x ) / [ g ( n , x ) Let =
inf {If
n(<)I
:
I
= lrll =
I, x
.
(0,n)
E
Then
{A,
d = min
(3.2)
ml.
Observe t h a t
m >,
inf inf I l g ( ~ , x ) g ( r l , x ) / [ g ( r l , x ) - g ( < , x ) I l : I < l =Iril = o<x<TI
which i m p l i e s t h a t m
-1
h(x)
x < TI, where
C sup h(x) : 0 < =
sup{ll/g(C,x)
11,
-
l/g(rl,x)I
:
lCl
=
= lrll
11.
Since g i s a v e r t i c a l s t r i p mapping, f o r each f i x e d x , t h e boundary o f t h e range of i t s r e c i p r o c a l c o n s i s t s of two c i r c l e s C 1 and C
2 ( s i n x ) / x and w2
and ( s i n x ) / ( T - x ) , and c e n t e r e d a t w 1 = respectively.
of r a d i i ( s i n x ) / x =
-(sin x)/(r - x),
These two c i r c l e s a r e symmetric w i t h r e s p e c t t o t h e r e a l a x i s
and they i n t e r s e c t a t t h e o r i g i n . Thus h ( x ) = d i a m e t e r of C 1 + d i a m e t e r of C2 =
l / g ( l , x ) - l/g(-l,x)
=
2T(sin x)/[x(n
The d e r i v a t i v e of L(x) = ( s i n x ) / [ x ( v Since R(n/2)
=
-
x ) ] i n (0,T)
4 / n 2 i s g r e a t e r t h a n l i m R(x)
x+o a t t a i n s i t s a b s o l u t e maximum a t x = r/2.
x)l.
= I/Ti =
vanishes only a t x = a / 2 .
l i m R ( x ) , we deduce t h a t R XylT
Thus h ( x ) 6 8 / n , t h a t i s , m 2 n / 8 .
However, i f we choose g t o be t h e symmetric v e r t i c a l s t r i p mapping i n ( 3 . 1 ) , that is,
(3.3)
g(z) = g ( z , n / Z ) = ( 1 / 2 i ) l o g [ ( l + i z ) / ( I
then g ( 1 ) = T / 4 and g ( - I ) point -T/8 a t z = -1.
=
-n/4,
iz)l,
-
g ( z ) ] omits t h e
T h i s shows t h a t m C n / 8 , and hence m = n / 8 .
From ( 3 . 2 ) we conclude t h a t d = n / 8 . g i v e s s h a r p n e s s of o u r r e s u l t .
4.
-
So f ( z ) = g ( l ) g ( z ) / [ g ( l )
The f u n c t i o n g i n ( 3 . 3 ) immediately
This completes t h e p r o o f .
A GROWTH THEOREM
For each convex f u n c t i o n f i n K , i t i s well-known -I r ( ~+ r ) - ' 6 J f ( z ) J c r ( l - r ) , J (4.1)
that
~ =J r
< I,
w i t h e q u a l i t y o c c u r r i n g o n l y f o r f u n c t i o n s which a r e h a l f - p l a n e m a p p i n g s , t h a t i s ,
MGbius Transformations
f u n c t i o n s of t h e form f ( z ) = z ( l
We d e r i v e s h a r p Theorem 4 . 1 .
- e i B z) - I , 0
253
E R.
upper and lower bounds f o r I f ( z )
0 < r < 1,
Let r,
h(x, r ) =
tx
be fixed.
-
I,
k.
f E
For x i n (0, I T ) , l e t
2arg(1 + reix)]
sin x
x a r g ( ~+ r e l x ) and H(x, r ) =
I
t x + 2 a r g ( 1 - re-1x)
sin x
x a r g ( ~- r e L x )
h
Then f o r each f E K , m(r) L I f ( z ) ( ,< M ( r ) ,
( z ( = r < 1,
where [M(r)]-'
=
min{h(x, r ) :
0 < x <
71)
< (1
-
r)r
-1
and
I
.
0 < x < 71) > ( I + r ) r -ia ia = m(r) o c c u r s f o r t h e f u n c t i o n s e f ( e z ) , a E R, where [rn(r)]-'
If(z)
-1
,
=
max{H(x, r ) :
f ( z ) = g ( l , x , ) g ( z , x )/[g(l, x , ) - g ( z , x , ) ] and g i s t h e v e r t i c a l s t r i p I mapping a s d e f i n e d i n ( 3 . l ) , and H a t t a i n s i t s maximum a t x l . S i m i l a r l y , If(z)
I
= M(r) o c c u r s f o r t h e f u n c t i o n s of t h e form given above, e x c e p t t h a t
g = g ( z , x 2 ) , where h a t t a i n s i t s minimum a t x
2' The proof i s r a t h e r l e n g t h y , s o we w i l l b r e a k i t i n t o s e v e r a l p a r t s .
t h a t from Theorem A, i t s u f f i c e s t o e x t r e m i z e I f ( z )
I
Notice
where f i s e i t h e r a h a l f -
p l a n e mapping o r e l s e f i s t h e t r a n s f o r m of a s t r i p mapping.
Although t h i s i s
c l e a r l y a major s t e p i n d e t e r m i n i n g t h e e x t r e m a l v a l u e s , a s o c c a s i o n a l l y happens i n t h e s e t y p e of problems, d e t e r m i n i n g t h e e x p l i c i t v a l u e s s t i l l r e q u i r e s some work. The bounds f o r t h e modulus of h a l f - p l a n e mappings a r e g i v e n by ( 4 . 1 ) .
Since
h
K i s r o t a t i o n a l l y i n v a r i a n t , we may assume t h a t f i s g e n e r a t e d by a v e r t i c a l s t r i p mapping.
(4.2)
Thus f h a s t h e form f ( z ) = g(rl, x ) g ( z , x ) / [ g ( r l , x)
-
g(z, x)l
where g ( Q , x ) , l r l l = I , i s f i n i t e and g i s g i v e n by ( 3 . 1 ) .
We f i r s t e s t a b l i s h
t h e f o l l o w i n g lemma. Lemma 4.2.
For e a c h f i x e d x ,
whenever Im{z} > 0 i n D. Here i t i s understood t h a t t h e argument f u n c t i o n v a n i s h e s a t z = 0.
Proof.
We w i l l prove t h e l e f t a s s e r t i o n ; t h e r i g h t a s s e r t i o n i s proved
analogously.
M. A. Rosihan
254
S p e c i f i c a l l y , we s h a l l show t h a t
-
m
(4.3)
< l i m sup a r g ( l 2'5
-
g(z, x)x-lsinx)
s l i m inf arg(zg'(z, x)/g(z, x)) < 'Z 5
m
f o r each p o i n t 5 on t h e boundary a D + , where D + i s t h e upper h a l f - d i s k .
An
a p p l i c a t i o n of t h e g e n e r a l i z e d maximum p r i n c i p l e f o r harmonic f u n c t i o n s
[ 3 , p.2541 w i l l then complete t h e p r o o f . For each f i x e d x i n ( 0 , I T ) , Re g < x / ( 2 s i n x ) i n D , so Re{l
4
Since g i s a convex f u n c t i o n , Re{zg'/g} > function arg( I
-
while arg(zg'/g)
[ 4 , p.731.
-
gx-lsinx} >
IT/^ and continuous i n
g x - l s i n x ) i s uniformly bounded by
1.
Thus t h e harmonic
i s bounded by IT/^ and continuous i n
except a t z = e
5,
i(n
_+
x)
T h e r e f o r e i t i s s u f f i c i e n t t o compare r a d i a l l i m i t s almost everywhere i n ( 4 . 3 ) .
-
I f 5 i s r e a l i n a D + , t h e n Cg'(5, x ) / g ( < , x) and 1
x)x-lsinx a r e both
g(<,
r e a l , so ( 4 . 3 ) holds. then l i m a r g ( l - g ( r e
I f < = ei(n-x), a r g ( z g ' / g ) > -T/2
i(r-x)
,
x)x
-I
sinx) =
IT/^.
Since
r+ 1 i n D, ( 4 . 3 ) again h o l d s .
I t remains t o show t h e v a l i d i t y of ( 4 . 3 ) f o r n o n r e a l 5 i n a D + w i t h i (T-x) , In t h i s case, i f
5 # e
log then f o r z = r e
iB
+
'.
ix
-lx
I + e
=
u(r,
e)
+ iv(r,
, l i m arg(zg'(z, x)/g(z, x)) = arctan r+ I
(4.4)
e),
z
m, u(l
0)
and
l i m arg(1 r+ I
(4.5)
-
-I g(z, x)x sinx) = arctan
2x
u(l, 0) - v ( ~ e, l *
The i d e n t i t y I + e i(e+x) i(e-x) I + e
- cos((x + e)/2) eix C O S ( ( X- 8 ) / 2 )
yields
(4.6. I)
v ( ~ e) , =
i
-(T-~)
u(i,e) < 0,
(4.6.2)
e
, e
E
, e
E
E
( 0 , IT-^)
TI-^, IT)
(0,~).
So from ( 4 . 4 ) and ( 4 . 5 ) , t h e v a l i d i t y of ( 4 . 3 ) f o r 5 = eie, 8
0 #
T
-
x, i s equivalent t o
E
( 0 , ~ )and
Mdbius Transformations
255
But ( 4 . 6 ) i m p l i e s t h e above i n e q u a l i t y , and c o m p l e t e s t h e p r o o f o f t h e lemma. We w i l l now u s e t h e above lemma t o p r o v e t h e f o l l o w i n g r e s u l t . L e t g b e d e f i n e d a s i n ( 3 . l ) , and f i x r , 0 < r < 1.
Lemma 4 . 3 . i0
z = r e
Then f o r
,Ocecn, -
1
sinx <
I
sinx
I
sinx
I
and I
Proof. __
sinx
A s b e f o r e , we w i l l o n l y p r o v e t h e f i r s t a s s e r t i o n s i n c e t h e o t h e r
assertion follows similarly.
F o r c o n v e n i e n c e , we w r i t e g ( z , x ) = g ( z ) .
0 < r < I,
For z = r e i e ,
a ae / h I 2 =
1 2 R e I i z h h ' ) w h e r e h ( z ) = ___ g(z)
sinx
-
__ x
.
Differentiation yields
a
Ihl
2
-2
=
Ig[ Im{zg'(z)/g(z)}
=
Igl-2e(o)
-
2 -1 . Im{zg'(z)/g ( z ) l x s l n x
-1
w h e r e k ( 0 ) = [ I-(x-'sinx)Reg(z)]Im~zg'(z)/g(z) S i n c e R e { z g ' / g } and l - ( x - l s i n x ) R e g
'
)+x
Im{zg'(z)/g(z)}
-(x-lsinx)Img(z)
Re zg ' ( ')
1 - ( x- I s i n x ) Reg ( z)
lg( ')
sinx(Img(z))Re{zg'(z)/g(z)}.
a r e positive, k(e) > O is equivalent t o = Im{I-g(z)x-lsinx}
'
Re { I -g ( z ) x- s i n x
T h i s i n e q u a l i t y is e q u i v a l e n t t o arg(zg'(z)/g(z) A p p l y i n g Lemma 4.2, we c o n c l u d e t h a t
> arg(l-g(z)x-lsinx).
L(A)
This completes t h e
> 0 i n (0,n).
proof We now p r o c e e d w i t h t h e p r o o f o f t h e t h e o r e m . P r o o f of Theorem 4 . 1 .
Let
E
6(r) = inf Ilf(z)l
: f
A(r)
: f E
ii,
=
i,
IzI = r}.
r},
and =
sup { l f ( z ) I
A s observed e a r l i e r , i t s u f f i c e s t o c o n s i d e r f where e i t h e r f ( z ) = z ( I
-I -2)
e l s e f i s g i v e n by ( 4 . 2 ) . I f f ( z ) = z ( l -z)-', t h e n f a t t a i n s i t s maximum a t z z = -r.
=
Thus r(l +
r1-I 5 -
lf(z)l
5
r ( ~ - r)
-1
.
r a n d minimum a t
or
M.A. Rosihan
256 Next l e t f be given by ( 4 . 2 ) .
Let
M(r) = sup { l f ( z ) I : I z I
=
rl,
and m(r) = i n f { l f ( z ) / : I z I = r l .
, we
Then by c o n s i d e r i n g t h e r e c i p r o c a l of f "(r)
(4.7)
deduce t h a t
if3
i n f min min { I ~ / g ( r e ,x) O<~<TI
I-' 2
e
- 1/g(ei',x) I 1,
and
(4.8)
[m(r)l-l
5
sup max max ocx<TI J,
e
I
I I / g ( r e if3 ,x) - l/g(ei',x) I I .
I n what f o l l o w s , we s h a l l show t h a t e q u a l i t y i s o b t a i n e d i n both ( 4 . 7 ) and (4.8),
and t h a t 6 ( r ) = min t r ( l + r )
-1
, m(r)l
-I
,
=m(r),
and A(r) = max { r ( l
-
r)
M(r)j=M(r),
As observed e a r l i e r , t h e boundary o f t h e range of l / g c o n s i s t s of two -1
c i r c l e s C I and C 2 w i t h c e n t e r s a t x - l s i n x , -(TI - x) s i n x , and of r a d i i x - l s i n x , (TI - x ) - l s i n x , r e s p e c t i v e l y . Moreover, t h e s e two c i r c l e s a r e symmetric w i t h respect t o the r e a l axis. Notice t h a t t h e range of l / g i s a l s o symmetric w i t h r e s p e c t t o t h e r e a l a x i s . i8 Since I / g ( z , x ) i s r e a l i f and o n l y i f z i s r e a l , i t s u f f i c e s t o c o n s i d e r z = r e f o r f3 i n [O,n].
For each f i x e d 0 , I / g ( r e i 8 , x ) l i e s o u t s i d e t h e c i r c l e s C l and i8 ,x) and C I i s given by d l ( f 3 , x ) ,
Thus t h e minimum d i s t a n c e between I / g ( r e
C2.
where dl(8,x) =
I [ g ( r eif3 , x ) I - l
w h i l e t h e minimum d i s t a n c e between I / g ( r e d2(f3,x)
=
I[g(re
if3
,x)l
So f o r a f i x e d 8 and x,
mini I [ I / g ( r e
if3 =
,x)
1
min{d
S i m i l a r l y , f o r a f i x e d 8 and x ,
where
-1
- x-lsinxl if3
-
x-lsinx,
, x ) and C p i s d ( e , x ) , where 2 -1 . -1 + (TI-x) s i n x l - (n-x) s i n x .
Mobius Transformations
257
and
I [ g ( r ei 8 , x ) l - '
D2(8. X) =
+ (n-x)-lsinxl
+ (TI-x)
-1
sinx.
Applying Lemma 4 . 3 , i t f o l l o w s t h a t f o r a f i x e d x , min { d , ( B , x) : 0
5 8 ( T I )=
d (0, x) = [ I / g ( r , x ) ] 1
-
[I/g(l, x)],
and min { d 2 ( e , x) : 0 ( 0
5 TI}=
d (TI, x) = [ l / g ( - l , 2
5 TI)=
D
-
x)]
[I/g(-r,
x)].
Also, f o r a fixed x max {Dl(e, x) : 0 ( 8
x) = [ I / g ( l , x ) ]
-
[l/g(-r, x)],
~ ~ ( x) 0 ,= [ I / g ( r , x ) l
-
[l/g(-l,
1
(71,
and max tD2(e, x) : 0
5
e 5 TI)=
Now g ( r , x) = a r g ( 1 + r e g(-r,
x) = - a r g ( l
-
g ( r , TI - x) = -g(-r,
ix
x)].
) / s i n x , g ( l , x) = x / 2 s i n x ,
re-lx)/sinx,
and g ( - I ,
x) = -(TI
-
x)/2sinx.
Since
x ) , we see t h a t
-
d (0, IT 1
x) = d (TI, x ) , 2
and D ( 0 , TI
2
-
x) = D
I
(TI,
x).
Thus i t s u f f i c e s t o minimize d ( 0 , x) and t o maximize D I ( n , x ) . 1
Specifically,
d (0, x) = h ( x , r ) and D l ( n , x) = H(x, r ) , where 1
h(x, r ) = [ x
-
2 a r g ( 1 + re
ix
) 1( s i n x ) / [ x a r g ( I + r e l x ) I ,
and ~ ( x r, ) = [ x + 2 a r g ( 1
-
re
-ix
) l ( s i n x ) / [ x arg(1
-
From ( 4 . 7 ) and ( 4 . 8 ) , [M(r)I-]
(4.9)
2
i n f { h ( x , r ) : 0 < x < TI),
and [ m ( r ) ~ - ' 5 s u p C H ( ~r ,) :
(4.10)
o
< x <
T I .
Applying L ' H o s p i t a l ' s r u l e , i t f o l l o w s t h a t h ( 0 , r ) = ( I
-
r)r
-I
=
A s t r a i g h t f o r w a r d c a l c u l u s argument shows t h a t h ( ~ 1 / 2 ,r ) = I / a r c t a n r less than ( I - r ) r
-1
h(n, r ) .
-
4/TI i s
; hence h a t t a i n s i t s minimum v a l u e i n ( 0 , TI).
Suppose h ( x ) = c1 i s t h e minimum v a l u e . The proof t h u s f a r shows t h a t t h e 2 r e c i p r o c a l of t h e f u n c t i o n f ( z ) = g ( 1 , x 2 ) g ( z , x 2 ) / [ g ( l , x2) - g ( z , x,)] assumes t h e v a l u e c1 a t z = r. [M(r)]-'
Combining t h i s w i t h ( 4 . 9 ) , we conclude t h a t = min { h ( x , r )
Similarly, H(0, r ) = (1 + r ) r
=
: 0 < x <
TI).
~ ( n ,r ) , and s i n c e
M. A. Rosihan
258
H(n/2, r)
=
4 71
t
>- 1 + r r '
arctan r
P r o c e e d i n g a n a l o g o u s l y a s b e f o r e , w e conclude
H assumes i t s maximum i n (0, 1~).
that [ m ( r ) l - ' = max I H ( ~ r, ) :
o
Finally, it i s c l e a r t h a t w e obtain sharpness given i n t h e s t a t e m e n t of t h e theorem.
< x <
711.
of o u r r e s u l t f o r t h o s e f a s
T h i s completes t h e p r o o f .
Let us t a k e a c l o s e r examination of t h e f u n c t i o n h a s given i n Theorem 4 . 1 . I t i s d i f f i c u l t t o determine e x p l i c i t y t h e p o i n t ( s ) i n ( 0 ,
assumes i t s minimum v a l u e . of ah/ax.
TI)
a t which h
So we would want t o a s c e r t a i n t h e number of z e r o s
Numerical e v i d e n c e seems t o s u g g e s t t h a t a h / a x and aH/ax, where H i s
a l s o given i n Theorem 4 . 1 , v a n i s h e x a c t l y once i n ( 0 ,
71).
Under t h i s
assumption, we g i v e below t h e approximate e x t r e m a l v a l u e s of h and H .
Note
t h a t x 1 and x2 d e n o t e t h e approximate z e r o t o aH/ax and a h / a x , r e s p e c t i v e l y . THE EXTREMAL VALUES OF H AND h
x2
H(xl , r )
h ( x 2 ,r )
2.024425
2.140862
11.351998
8.698759
1.969090
2.202705
6.376196
3.725457
1.862916
2.336489
3.922480
1.281990
1.761633
2.490874
3.132843
0.5 I0468
0.8
1 .66439 8
2 . 6 8646 1
2.757472
0.163417
0.9
1.6 17154
2 . 8 2 1787
2.638368
0 . 0 6 4 163
h
For e a c h f E K
Corollary 4.4.
where m(r) and M(r) a r e d e f i n e d by Theorem 4 . 1 .
Proof.
Let f
E
2
and 5 i n D b e f i x e d .
-
+ 52))
f , t h a t is, F(z) = [ f ( ( z + C ) / ( l
This r e s u l t i s sharp.
L e t F b e a Marty t r a n s f o r m a t i o n of
-
1LI2).
I 5 M(r)(I -
2 r )/r.
f(<)l/f'(<)(I
From Theorem
4. I , m(r)
5
IF(-<)\
6Nr),
151 = r .
Thus m(r)(l
-
2
r )/r
6
lf(C)/cf'(C)
By c o n s i d e r i n g t h e Marty t r a n s f o r m a t i o n of t h e e x t r e m a l f u n c t i o n s i n Theorem
4 . 1 , we o b t a i n s h a r p n e s s
of our r e s u l t .
M6bius Transformations
259
REFERENCES [11 [21
R.W. Barnard and G. Schober, "Mgbius Transformations of Convex Mappings", Complex Variables Theory Appl. 3 ( 1 9 8 4 ) , 55 - 6 9 . R.W. Barnard and G. Schober, "Mobius Transformations of Convex Mappings II", in print.
J.B. Conway, Functions of One Complex Variable, Znd, E d . , Springer-Verlag, New York, 1978. [ 4 1 P.L. Duren, Univalent Functions, Springer-Verlag, New York, 1983. [31
[51
Kirwan, "A Note on Extremal Problems for Certain Classes of Analytic Functions", Proc. h e r . Math. Soc. 1 7 (1966), 1028 - 1030. W.E.
This Page Intentionally Left Blank
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
26 1
UNIFORM ERGODIC THEOREMS FOR OPERATOR SEMIGROUPS
Sen-Yen Shaw Department of Mathematics, 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 , Chung-Li, Taiwan, Republic of China
The purpose of t h i s a r t i c l e i s t o examine c o n d i t i o n s f o r t h e uniform o p e r a t o r convergence of Cesaro means and Abel means o f a l o c a l l y i n t e g r a b l e semigroup of o p e r a t o r s on a Banach s p a c e X. We c o n s i d e r t h e c a s e t h a t X i s a g e n e r a l Banach s p a c e and t h e c a s e t h a t X i s a Grothendieck space with the Dunford-Pettis property.
1.
INTRODUCTION L e t X be a Banach s p a c e , and l e t { T ( t ) ; t > 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.
IT(t)
I I=o(t)
(t-)"
(see
3.
However, as shown by an example i n [6],
"0
"1
Usually
i t i s p o s s i b l e t h a t a
Our theorems i n [ 6 ] assume t h e weaker c o n d i t i o n
and show i n p a r t i c u l a r t h a t i n o r d e r T ( * ) t o be uniformly Cesaro-ergodic
ai t is necessary t h a t
I IT(t)R
I
(1) I = o ( t ) (t-),
but is not t h a t
I / T ( t )I I = o ( t )
(t-).
I n s e c t i o n 2 , we s h a l l f o r m u l a t e uniform e r g o d i c theorems i n which a n o t h e r necessary condition used.
''1
IT(t)S(u)
I I=o(t)
(t-)
Vu>O" i s u s e d .
T h i s c o n d i t i o n i s s l i g h t l y s t r o n g e r than t h a t
'(1
This condition i s
I T ( t ) S ( u ) x l I = o ( t ) (t-)
V d X , Vu>O," which i s n e c e s s a r y f o r t h e s t r o n g e r g o d i c i t y ( c f . [ 4 , C o r o l l a r y
S. - Y. Shaw
262
Then, under t h i s v e r y c o n d i t i o n t h e s t r o n g C e s a r o - e r g o d i c i t y i m p l i e s
3.71).
t h e uniform e r g o d i c i t y provided t h a t t h e ground s p a c e i s a Grothendieck space T h i s i s proved i n s e c t i o n 3 .
w i t h t h e Dunford-Pettis p r o p e r t y ,
2.
UNIFORM ERGODIC THEOREMS ON A GENERAL BANACH SPACE
For a l o c a l l y i n t e g r a b l e semigroup T(-) of o p e r a t o r s o n a Banach space X , w e have t h e f o l l o w i n g uniform e r g o d i c theorem. Theorem 1.
Assume t h a t u < O . Then t h e f o l l o w i n g s t a t e m e n t s are e q u i v a l e n t : a(1) T( * ) i s uniformly Cesaro-ergodic.
I / T ( t ) R s ( l ) I I = o ( t ) (t-), I I T ( t ) R s ( l ) I I = o ( t ) (t-), (4) I I T ( t ) S ( u ) 1 I = o ( t ) (t-) (2)
and T( - ) i s uniformly Abel-ergodic.
(3)
and R(Rs(l)-I)
i s closed.
f o r a l l u>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)<,m , form a uniformly c o n t i n u o u s group of A i s o m e t r i c a l isomorphisms of L" LO, 11, and g e n e r a t o r i s t h e m u l t i p l i c a t i o n
Uniform Ergodic Theorems for Operator Semigroups gAf.
o p e r a t o r AA:f
-+
I-- L m c O , l ] LA, 11
f o r A>O,
265
[ o , A ~ ~ [ O and ,lI
I t i s e a s y t o see t h a t N(AA)= 1
and N(Ao)={Ol and R ( A J m = { f E L L O , l ] ;
&I
R(AA)= f(s)=Ol.
S i n c e R(AA) i s c l o s e d f o r A > O , T i ( * ) i s u n i f o r m l y e r g o d i c t o t h e m u l t i c a t i o n by
I L ~ , ~S i]n c. e R(Ao) i s n o t c l o s e d , T o ( * ) i s n o t u n i f o r m l y e r g o d i c and hence -1 t In fact, L & I t / o T o ( s ) f d s e x i s t s i f and o n l y i f 1 9
not s t r o n g l y ergodic.
f ( s ) = O , and i n t h i s c a s e , t h e l i m i t i s t h e z e r o f u n c t i o n .
These f a c t s can a l s o
be v e r i f i e d by d i r e c t c o m p u t a t i o n .
REFERENCES El] H i l l e , E . and P h i l l i p s , R.S., F u n c t i o n a l A n a l y s i s and Semigroups (Amer. Math. SOC. C o l l o q . P u b l . , v o l . 31, Amer. Math. S O C . , P r o v i d e n c e , R . I . , 1957). 121 L i n , M . , On t h e u n i f o r m e r g o d i c theorem. 11, P r o c . Amer. Math. SOC. 46 ( 1 9 7 4 ) , 217-225. 131 L o t z , H.P., T a u b e r i a n theorems f o r o p e r a t o r s on L" and s i m i l a r s p a c e s , F u n c t i o n a l A n a l y s i s : Surveys and Recent R e s u l t s 111, ( 1 9 8 4 ) , 117-133. [4] Shaw, S . - Y . , E r g o d i c p r o p e r t i e s of o p e r a t o r semigroups i n g e n e r a l weak t o p o l o g i e s , J . F u n c t . Anal. 4 9 ( 1 9 8 2 ) , 152-169. [s] Shaw, S . - Y . , E r g o d i c theorems f o r semigroups of 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 , Proc. Janpan Acad. S e r . A 5 9 ( 1 9 8 3 ) , 132-135. 161 Shaw, S . - Y . , Uniform e r g o d i c theorems f o r l o c a l l y i n t e g r a b l e semigroups and p s e u d o - r e s o l v e n t s , P r o c . amer. Math. S O C . 9 8 ( 1 9 8 6 ) , 61-67. C7] Shaw, S.-Y., On w*-continuous c o s i n e o p e r a t o r f u n c t i o n s , J . F u n c t . Anal. 6 6 ( 1986), 73-95.
This Page Intentionally Left Blank
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
261
WEIGHTED NORM INEQUALITIES FOR SOME MAXIMAL FUNCTIONS Wang S i l e i Department of Mathematics, Hangzhou U n i v e r s i t y , Hangzhou, P e o p l e ' s R e p u b l i c of China The main purpose o f t h i s paper i s t o s t u d y t h e weighted norm i n e q u a l i t i e s f o r some maximal f u n c t i o n s . The r e s u l t s a r e sharp.
51.
INTRODUCTION.
L e t l ~ ~ , ~ ( f ) (denote x) t h e f o l l o w i n g maximal f u n c t i o n s d e f i n e d
where 1 <
x -<
y < =, x = (xl,
..., x n )
x2,
i n n-dimensional E u c l i d e a n space R n
and t = (tl,
, u(x,y)
i n t e g r a l of f E L p ( R n ) f o r some p 2 1.
= u(f)(x,y),
t2,
...,
tn) are points
y > 0 i s t h e Poisson
The purpose o f t h i s paper i s t o s t u d y
weighted norm i n e q u a l i t i e s f o r t h e s e maximal f u n c t i o n s which a r e t h e g e n e r a l i z a t i o n s of t h e "maximal f u n c t i o n " u X ( f ) ( x ) d e f i n e d by E. M. S t e i n [Sly p.2361 O r i g i n a l l y S t e i n i n t r o d u c e d t h e "maximal f u n c t i o n " u X ( f ) ( x ) UX(f)(X) =
SUP
Y>O
(
1
l u ( x - t , Y ) l 2 Y- n ( y ) n x d t ) 1 / 2
Itl+Y
Rn
i n t h e 1-dimensional p e r i o d i c set-up [ S Z ] , where these f u n c t i o n s p l a y an i m p o r t ant r o l e i n questions r e l a t e d t o Fourier series. B e f o r e we s t a t e o u r r e s u l t s , we c o n s i d e r t h e measures w i t h r e s p e c t t o which norms a r e t a k e n have t h e f o r m d u ( x ) = W(x)dx. condition A
f o r some p, 1 < p <
P f u n c t i o n which s a t i s f i e s
m
A w e i g h t W(x) i s s a i d t o s a t i s f y
i f W(x) i s nonnegative, l o c a l l y i n t e g r a b l e
f o r a l l n-dimensional cubes Q w i t h s i d e s p a r a l l e l t o t h e c o o r d i n a t e planes, I Q I b e i n g t h e volume o f Q, and C a c o n s t a n t independent o f Q. When p = 1, W(x) i s s a i d t o s a t i s f y c o n d i t i o n A1,
if W(x) i s nonnegative, l o c a l l y i n t e g r a b l e and
S.L. Wang
268
(A1)
(MW)(x)
C.W(x),
5
where (MW)(x) i s t h e H a r d y - L i t t l e w o o d maximal f u n c t i o n o f W, i . e . ,
-1
(MW)(x) = sup 1 Q3x I Q I We w i l l w r i t e W e A
P
, or
W(z)dz. Q
W e A1 f o r such W.
These c l a s s e s were i n t r o d u c e d i n
Muckenhoupt [ M I and i n an e q u i v a l e n t f o r m by Rosenblum [R]. F o r a measurable s e t S, we w i l l use t h e n o t a t i o n WISI =
I
W(x)dx
S
f o r t h e W-measure o f S.
The f o l l o w i n g w e l l known p r o p e r t i e s , which we s h a l l
a r e l i s t e d below. F o r p r o o f s see [ M I . P’ L e t ( M f ) ( x ) denote t h e H a r d y - L i t t l e w o o d maximal f u n c t i o n o f f; t h e n
need about A
[ (Mf(x))’
(1.2)
W(x) dx 5 C [ l f ( x ) I p W(x) dx
wn
Rn for 1 < p <
i f and o n l y i f W e A
The A c o n d i t i o n i s a l s o necessary and P’ P s u f f i c i e n t f o r t h e weak t y p e i n e q u a l i t y m
[ / f ( x ) l W(x) dx
Wt(Mf)(x) > a ) -< C
(1.3)
for 1 5 p <
Rn
-.
Moreover, W e A then (1.4)
W(X)
f o r some
E
P
implies t h a t W e A
9
( q t p ) ; c o n v e r s e l y , i f p > 1, W e A
P’
Ap-=
> 0 depending on W.
We now s t a t e o u r r e s u l t s . THEOREM 1. (i)
(1.5)
L e t 1 < A i. y <
m
I f f e LP(Wdx), W e Ap,po, Vxyy
and po = y / x , p >
p >
po.
t h e n t h e r e i s a c o n s t a n t C = CA,y,v,n
( f ) ( x ) s C ( ( M u f ) ( x ) ) l-Po/v
so t h a t
1l l v ( X I ,
where ( M p f ) ( x ) = ((Mlflp)(x))llu. (ii)
I f po < p <
m,
W e A
P I Po
and f e LP(Wdx), t h e n t h e mapping f
from LP(Wdx) t o LP(Wdx) i s o f s t r o n g t y p e (p,p),
i.e.,
+
p x Y y( f )
Weighted Norm Inequalities for some Maximal Functions
(1.6)
( ~ ~ , , ( f l ( x ) ) W(x) ~ dx 5
C
wn
I
269
( f ( x ) I P W(x) dx,
wn
t h e c o n s t a n t C b e i n g independent o f f . THEOREM 2. t h e mapping f
Let 1 < h +
i y <
-,
then
u A , y ( f ) i s o f weak t y p e (po,po) f r o m LpO (Wdx) t o LpO (Wdx),
1 .e.,
(1.7)
P
I f W e A1 and f e L '(Wdx),
po = .(/A.
Wt u A Y y ( f ) ( x ) >
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 < p <
-
and W e A
w i t h c o n s t a n t K, t h e n t h e r e i s a c o n s t a n t P C, depending o n l y on p and K, such t h a t f o r e v e r y cube Q and i t s complement QC, LEMMA 2.
Weighted Norm Inequalities for some Maximal Functions
T h i s lemma i s known i n t h e s p e c i a l case n = 1, see [HMW, does n o t work i n t h e g e n e r a l case n > 1.
17 1
p.2321.
Their proof
Here we s h a l l g i v e an a l t e r n a t i v e
p r o o f which covers b o t h t h e cases n = 1 and n > 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)
-<
C-a
j
J
If(x)IPo*W(x)dx
Rn
J
W(x)dx 5 C
(3.4)
a PO
f o r each
4
j'
F o r any cube Q . o f t h e c o l l e c t i o n , l e t (2Q.) be a cube w i t h t h e same
c e n t e r as Q
J
b u t w i t h t w i c e as l a r g e a s i d e .
j' t h a n N o f t h e cubes ( 2 Q j ) .
J
Then no p o i n t o f R n l i e s i n more
S. L. Wang
214
(3.5)
Two d i s t i n c t cubes Q,
o f ' { Q . 1 a r e s a i d t o touch, i f t h e i r
and Q,
J
boundaries have a common p o i n t . The f o l l o w i n g r e s u l t s a r e known. Suppose t h a t Furthermore, i f Q e { Q j 1 . Then t h e r e a t most N cubes i n i QJ. 1 which touch Q.
Q,
and Q,
touch, t h e n
-< diam(Ql) 5 C, diam(Q,).
C1 diam(Q,)
F o r p r o o f s see [GC,
p.1431 and [ S l ,
p.1691.
P L e t f e L o(Wdx), W B A1 and
PROOF OF THEOREM 2 .
cx >
0 be g i v e n .
F i r s t we
assume t h a t po > 1 and we have t o show t h a t (3.6)
wr\Ji2(f)(X)
>
c
1
5
I
c*Cr
( f ( x ) IPo*W(x)dx
( a > 0)
Rn w i t h C independent o f f and a. Apply t h e decomposition lemma t o f and
CI, t
cubes, s a t i s f y i n g (3.1) through ( 3 . 5 ) above.
S e t t i n g b(x) = f ( x )
-
o obtain a collection
{Q.) o f
J D e f i n e a f u n c t i o n g ( x ) on R n by
g ( x ) , we o b t a i n a decomposition w i t h t h e f o l l o w i n g
properties: (3.7)
I g ( x ) I -<
C-a
a.e.,
and
I
I g ( x ) I p o W(x)dx 5 C
Rn (3.8)
b(x)
I
I f ( x ) I P o W(x)dx;
Rn
i s s u p p o r t e d on n;
( j = l,2, . . . ) J
(3.10)
I
b ( x ) W(x)dx = 0
'j
L e t p > 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 , ...).
Weighted Norm Inequalities for some Maximal Functions
(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 .
Weighted Norm Inequalities for some Maximal Functions
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
-<
I
( b ( x ) I d x 5 C*a.y-nlQI,,
Q,
and (3.25)
J Q
lu,(t,y)ls
S
d t r; c * a
IQ~I
j
f o r any Q, which touches Q j , and 1 5 s I n fact,
-<
po.
(see
S. L. Wang
218
So (3.24)
f o l l o w s from (3.19).
On t h e o t h e r hand, t h e c o n d i t i o n W ( t ) (3.26)
ess sup t e Q
1 5
C 1Q(/
W(t)
B
A1 i m p l i e s t h a t
W(t)dt. 0
T h e r e f o r e , by H o l d e r ' s i n e q u a l i t y , ( 2 . 5 ) ,
and ( 1 . 2 ) ,
J
J
i n w h i c h we have u s e d t h e f o l l o w i n g i n e q u a l i t i e s
C 1 W{Q,}
5
WrQjl
5
C2 WIQ,]
w h i c h f o l l o w f r o m t h e c o n d i t i o n t h a t Q, a d o u b l i n g measure.
touches Q
1 ,U,(t,y)
so ( 3 . 5 ) h o l d s , and Wdx i s
T h i s c o m p l e t e s t h e p r o o f o f ( 3 . 2 4 ) and ( 3 . 2 5 ) . F i r s t note t h a t f o r t
Now we come t o t h e l a s t i n t e g r a l o f ( 3 . 2 3 ) . tsQ
j'
=
1 ,u,(t,y)
t j-Q
(t. being the center o f
J
8
9.). J
Q
j'
Weighted Nomi Inequalities for some Maximal Functions
1
Moreover, b y ( 3 . 5 ) t h e number o f t e r m s i n
u,(t,y)
219
a r e bounded f o r a l l
t 4 4 " J L
t . ( j = 1,2, ...). Hence, u s i n g ( 3 . 2 4 ) and ( 3 . 2 5 ) t h e l a s t i n t e g r a l i n ( 3 . 2 3 ) J l e s s than
is
Y+l-Y/Po
dt
dt
The l a s t t w o s t e p s f o l l o w from ( 3 . 5 ) a g a i n .
Thus, b y ( 3 . 2 3 ) and ( 3 . 2 7 ) we g e t
So i n o r d e r t o c o m p l e t e t h e p r o o f o f i n e q u a l i t y ( 3 . 2 2 ) ,
and w i t h i t , t h a t o f
Theorem 2, we h a v e o n l y t o p r o v e t h a t
B u t i f we i n v o k e Lemma 2, we i m m e d i a t e l y g e t
W(x)dx =
1 j
Rn- R
I
IQjI
*
Rn- R
x
i x - t p J
W( x)dx
S.L. Wang
280
5 C
1 W{Qj)
I f ( x ) l P o W(x) dx.
5 C*a-Po
j
Rn
T h i s completes t h e proof of (3.291, and w i t h it, t h a t o f Theorem 2.
REMARK.
We t h u s complete t h e p r o o f o f Theorem 2 i n t h e case p,
>
1.
When
Po = 1, t h e argument a l s o works if (3.26) i s used t o r e p l a c e H o l d e r ’ s i n e q u a l ity.
54. We b e g i n by a lemma. LEMMA 3.
Let 1 < A , y <
e x i s t s a constant C = C
n,x,y’
m
and l e t 0 < 1x1, 0 < yo 5 1x1/20.
Then t h e r e
depending o n l y on n, A , y such t h a t
PROOF OF LEMMA 3. F i r s t we observe t h a t i f It1 < 1x1/10, 0 < yo < 1x1/10, t h e n I x - t l + y < 61x1/5. Hence
T h e r e f o r e we have
On t h e o t h e r hand, d e n o t i n g B = B((O,yo), y0/2) t h e b a l l a t c e n t e r (O,yo)
e R:+’
w i t h r a d i u s yo/2,
by a lemma of H a r d y - L i t t l e w o o d [FS,
p.1721,
Weighted Norm Inequalities for some Maximal Functions
-ni = cnJ yo
J
sup YO/2
l u ( t , y ) I Y y”(’-’)
281
dt.
I t I < I x 1/10
From ( 4 . 2 ) and (4.3) we o b t a i n t h e r e s u l t ( 4 . 1 ) . We now come t o p r o v e Theorem 3.
PROOF OF THEOREM 3.
I t i s c l e a r t h a t we o n l y have t o p r o v e p a r t ( i i ) o f
L e t W(x) = I x I B .
E
A
Then i t can be e a s i l y v e r i f i e d t h a t f 6
E
Theorem 3.
Then W
P
i f and only i f
-
n < B < n(p-1).
Now we c o n s i d e r t h e f u n c t i o n
Suppose t h a t (4.5)
(4.6)
n(p/po-l) < B < n(p-1).
I
P
I f S ( x ) I W(x)dx =
Rn s i n c e 0 < 6 < (n+E()/p and On t h e o t h e r hand,
I
LP(Wdx), i . e . ,
I x ( - p 6 + 6 dx <
m
1x151
-
n
c
8 < n(p-1) imply t h a t
-
p6
+
8 >
-
n.
S.L. Wang
282
YO
By (4.1),
/
C
2
IxI-'dx
= C yo -6
IXl
we have
% .Y ( f 6 ) ( X ) 2 c 1x1
- n/ Po YO
Now i f (1.6) were t r u e f o r f g and W = 1x1' e A
P'
t h e r e would be
However, t h e l a t t e r i s e q u i v a l e n t t o (4.9)
B < n(p/po
#
Ap/po.
1)
I n o t h e r words, (1.6) w i l l never be t r u e , i f
which c o n t r a d i c t s ( 4 . 5 ) . 1x1'
-
On t h e o t h e r hand, Theorem 1 shows t h a t (1.6) i s t r u e i f T h i s completes t h e p r o o f o f Theorem 3.
95. PROOF OF THEOREM 4.
The same argument used t o p r o v e Theorem 3 can a l s o be
a p p l i e d t o prove Theorem 4, by s e t t i n g f = f 6 ( x ) w i t h 0 < 6 < (n+B)/po and W = 1x1
D
with 0 < B
<
n ( p o - l ) , p,
> 1.
The d e t a i l s a r e o m i t t e d .
56. PROOF OF THEOREM 5. W(x) = 1x1' w i t h
-
Set g ( x ) = f 6 ( x ) w i t h 0 < 6 < (ntp,)/p,,
n < 6 < 0.
Then W(x) E A1 and
i.e., g
E
P L '(Wdx).
and
Weighted Norm Inequalities for some Maximal Functions
283
On t h e o t h e r hand, a p p l y i n g ( 4 . 1 ) of Lemma 3, we have
( O
j
( ~ , , ~ ( g ) ( x ) )W(x)dx ~~ 2 C
IRn
I
Ixl-"'dx
= t
-
IXl
T h i s completes t h e p r o o f o f Theorem 5. PROOF OF THEOREM 6.
Let 1 s p
<
po = y/x.
Choose
n/po < ( n + @ ) / p . D e f i n e f u n c t i o n g ( x ) = f 6 ( x ) w i t h n/p Then c l e a r l y (6.4)
1x1 a
E A1 ( - n <
a
<
- n 0
<
a
< 6 <
<
0 such t h a t
(nt@/p.
0) and
g = f 6 E L p ( 1x1 'dx).
On t h e o t h e r hand, i f x # 0, by ( 4 . 1 )
i.e., (6.5)
= t =
I f x = 0, t h e n i t i s e a s i l y v e r i f i e d t h a t (6.6)
F I ~ , ~ ( ~ ) ( 2O )(Mg)(O) = +
-.
T h i s completes t h e p r o o f o f Theorem 6. REFERENCES Coifman, R. and Fefferman, C . , Weighted norm i n e q u a l i t i e s f o r maximal f u n c t i o n s and s i n g u l a r i n t e g r a l s , S t u d i a Math., 51(1974) 241-50. Fefferman, C . and S t e i n , E. M., Hp spaces o f s e v e r a l v a r i a b l e s , A c t a Math., 129 (1972), 137-193. Gercia-Cuerva, J . and Rubio de F r a n c i a , J . , Weighted norm i n e q u a l i t i e s and r e l a t e d t o p i c s , 1985, N o r t h H o l l a n d . Hunt, R., Muckenhoupt, B. and Wheeden, R., Weighted norm i n e q u a l i t i e s f o r t h e c o n j u g a t e f u n c t i o n and H i l b e r t t r a n s f o r m , Trans. Amer. Math. SOC. 176 (1973), 227-51. Muckenhoupt, B., Weighted norm i n e q u a l i t i e s f o r t h e Hardy maximal f u n c t i o n , Trans. Amer. Math. SOC., 165 ( 1 9 7 2 ) , 207-26. Rosenblum, M., S u m m a b i l i t y o f F o u r i e r s e r i e s i n L p ( d u ) , Trans. Amer. Math. SOC., 105 (1962), 32-42. S t e i n , E . M., S i n g u l a r i n t e g r a l s and d i f f e r e n t i a b i l i t y p r o p e r t i e s o f f u n c t i o n s , 1970, P r i n c e t o n . S t e i n , E. M., A maximal f u n c t i o n w i t h a p p l i c a t i o n s t o F o u r i e r s e r i e s , Ann. of Math., 68 (1958), 584-603.
This Page Intentionally Left Blank
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
285
THE SECOND DUALS OF THE NONABSOLUTE CESARO SEQUENCE SPACES Wu Bo-Er South China Normal U n i v e r s i t y , China L i u Yu-Qiang South China Normal U n i v e r s i t y , China Lee Peng-Yee N a t i o n a l U n i v e r s i t y o f Singapore, Singapore We c h a r a c t e r i z e c o m p l e t e l y t h e second d u a l s o f t h e n o n a b s o l u t e Cesaro sequence spaces.
1.
Introduction L e t X be a r e a l sequence space
We w r i t e x = t x k l and d e f i n e converges f o r a l l x e X } ,
xB
tYkk
= ty =
1
converges f o r a l l x E X }
XkYk
,
k=l
X B and Xy a r e c a l l e d t h e
The sequences spaces X", respectively. 8 o r y. z = {Zk
We denote (X')'',
where 5,
r(
= a,
I n what f o l l o w s , we s h a l l always w r i t e x = t x k l , y = I y k l and
follows:
x
w i t h a norm l l x l i = I l C x l l , cnk = l / n when 1:
,
B- and y- d u a l s o f X
L
The n o n - a b s o l u t e Cesaro sequence spaces X
X:
a-,
t h e second dual o f X, by X",
the
P' = {Y: i k y k l
a,
a r e d e f i n e d i n [2] as
I n [2],
Ng and Lee c h a r a c t e r i z e d
by E
em,
k(Yk-Yk+l)} E Lq},
I n [3], Wu and Lee determined t h e a- and 1 showing t h a t Xa = { y : { k y k } e i? 1 where - + P q P i c u l a r , X y = X;
15 p 5
= { x : c x E i? }, P P where C = ( c n k ) i s t h e Cesaro m a t r i x g i v e n by
P k 2 n and z e r o o t h e r w i s e .
dual o f X
x,"
,
P'
= XT = {y: tkyk}
e em}.
-1+ - 1 = l . ' P q for 1 5 p 5
I r p c -
y- dua s o f X
P 1 and t h a t X B = Xy - = 1, 9 P P'
m,
I n part-
B.-E. Wu e t al.
286
I n t h i s paper, we d i s c u s s t h e second d u a l s o f X
2.
The second d u a l s o f X
P'
P
We remark [l] that i)
X"
ii)
i f X I Y , t h e n X'; c Y';,
iii)
xc
iv)
X 5 = X'",
c x @ cx Y ;
for
X"
xRa =
f o r 5 = a , B or
y;
a, B o r y;
'; =
where 5 = a , 6 o r y ;
XY"
XYB c x B B
=xBY =
x""
xYY
= ~ " 6= x"Y.
According t o Remark ( v ) above, s i n c e X B = Xy we have P P
I t remains t o c o n s i d e r t h e f o u r second d u a l s
F i r s t , we have Theorem 1.
X";
{z: I z k / k l e
=
1
for
l s p s m
~~1
for
1 5 p <
P
The p r o o f i s easy.
XB" = I z : I z k / k } P
Theorem 2 .
E
m.
XB", t h e n 1 kzllykzkl i s convergent f o r each y e X B We t a k e P P' { y k } = t l / k } E X B 1 5 p < m. (Note t h a t { l / k l X!.) Thus 1 k r l l z k / k l i s conP' v e r g e n t . The r e v e r s e i n c l u s i o n i s obvious.
If z
Proof.
E
4
tz: I z k / k } e ~l
Corollary.
Xl5'
Theorem 3.
(i) XB!
=
= yX!
=
Xp = cesm,
(ii)
for 5,
= a , B o r Y.
0
xm-
where cesm = { x : C J x J E e r n } .
F i r s t n o t e t h a t f r o m [2],
Proof.
X!
= I y : kyk
-t
0 as k
-t
m
and
k=l Now kyk and yk
-f
-f
0 as k 0 as k
+ m
+
m,
i m p l i e s t h a t yk then
-t
0 as k
k ( yk -y k + l 1 <
+ m.
m
l k Y k l 5 j.1 = kk l Y j - Y j + l l
m}.
A l s o , i f lk~lklyk-yk+ll m
5
1
j = kj l Y j - Y j + l l
<
m
Nonabsolute Cesaro Sequence Spaces and, l e t t i n g k
kYk
m,
-f
-+
0.
X E = I y : yk
287
Thus, 0 as k
f
klyk-yktll < ml k=l A c c o r d i n g t o Remark ( i i i ) above, we have Xm c X E B .
(i)
+
-f
m
Conversely, we d e f i n e a norm o f y e X!
11
IIY
=
11 { Y k l l l c o t
11
and
by
{k(Yk-Yk+l)lllkl*
Then X E i s a BK space ( s e e [ 5 ] f o r d e f i n i t i o n o f Bk spaces). By Theorem 7.2.7
o f [5], which s t a t e s t h a t f o r a
f i n i t e sequences, Z B c
Zy = Z f where Z f
BK space Z c o n t a i n i n g a l l
= t i f ( ek ) } : f
e Z*] and e k i s t h e seq-
uence whose o n l y non-zero t e r m i s 1 i n t h e k - t h p l a c e , we o b t a i n
= xF4 x”f.
xB!
L e t f e (X!)*; g E k;
= em.
by Theorem 4 . 4 . 1 o f [5], f = F t g
Hence f ( y ) = F ( y )
f
f
k(yk-yk+l)
o
A w i t h F e C:
= l1 and
g k f o r y e X E and, i n p a r t -
k= 1 k k i c u l a r , f ( e ) = F(e ) and z[ = kgk-(k-l)gk-l, and {z;]
+
kgk
-
then f ( e k ) = z i + 21.
S i n c e F = I z ’ l E kl
k
T h e r e f o r e { f ( e k ) l e Xm and XBB c X @
E Xm.
k L e t z i = F(e )
f o r each k w i t h go = 0.
(k-l)gkvl
= XBf = X_.
=k
Xm
m c
The p r o o f i s
compl e t e . F i r s t , we show t h a t c e s m c X F .
(ii)
1 L e t z e cesm and s k = F
k Izi(, 1=1
t h e n i s k ) E em. Use Abel I s summation f o r m u l a , we g e t
I f y E X,!
t h e n kyk
t o z e r o as n
-f
m.
-t
0 as k
+ m
Since I k ( y k
-
and t h e l a s t t e r m i n t h e above e q u a l i t y tends yktl)1
e
k1
implies tk( lykl
-
/yktl/l
e el,
m
the series
1 lykzkl
i s convergent; t h i s y i e l d s z E X?.
k=l
To p r o v e t h e o t h e r i n c l u s i o n , l e t { z k l e
1
Xp b u t
tzd
k
{E 121 ,I Izi
ces_, t h e n m
11
.,L
Thus, t h e r e e x i s t s t c k l e
We may assume t h a t ck 2 0 f o r a l l k .
k1
such t h a t
1/ k=l
1 k C ~ I ( ~lzil) = 1=1
Rearranging t h e terms o f t h e s e r i e s and
m.
B.-E. Wu el al.
288 and t z k l
Note t h a t I c k / k j e and ty$
4 XE
E
Xr,
thus tykj
4 a.l.
i m p l i e s t h a t tK(yk-yktl)l
which c o n t r a d i c t s t h e f a c t t h a t t c k j
E
kl.
4 X.!
S i n c e yk
B u t k(yk-yktl)
+
0 as k
-+
m
= ck f o r a l l k ,
The p r o o f i s c o m p l e t e .
N e x t , we d e t e r m i n e ( X 5 ) * , t h e c o n t i n u o u s d u a l s o f X 5 where 5 = a , B o r y . P P' Theorem 4. The sequence space X" i s a Banach space f o r 1 5 p 5 m w i t h norm P ( I Y = 11 { k Y k l ( ( , , and (x")* i s i s o m e t r i c t o a. when 1 < p 5 m . F u r t h e r m o r e , P P' q (x",* i s i s o m e t r i c t o .:P
I/
We c a n p r o v e t h a t t h e mapping T: X" + a. g i v e n by t y k l P q' l i n e a r i s o m e t r y . Thus X" a. and t h e c o n c l u s i o n f o l l o w s . P 9
Proof.
N o t e t h a t s i n c e X y = X!
=
-+
tkykl i s a
X T , we h a v e a l s o f o u n d t h e c o n t i n u o u s d u a l s o f X y
and X i . Theorem 5.
L e t X B ( = X y ) h a v e norm P P (i)
X!
I/ y 11
I( .
= l l i k ( ~ ~ - y ~ + ~ Then ) l
i s a Banach space a n d ( X E ) *
q i s equivalent t o
em.
xu
X B i s n o t c o m p l e t e f o r 1 < p < -; t h e c o m p l e t i o n o f X B i s t h e space o f a l l P P P 1 1 n u l l sequences y such t h a t I k(yk-yktl)} E eq where 1 < p < m and - t - = 1. P 9 ( x B ) * i s i s o m e t r i c t o L f o r 1 < p < m . (ii) P P 1 1 L e t 1 < p 5 m , + = 1 and Y = t y : yk + 0 as k + m, Proof. (i) P 9 g i v e n by F i r s t , we p r o v e t h a t T1 : Y P E L I. {k(yk-yktl)l 9 9' i y k l + tk(yk-yktl)l, i s a l i n e a r i s o m e t r y . The map T1 i s c l e a r l y l i n e a r and -+
S i n c e t h e mapping i s i n j e c t i v e , we need o n l y t o show t h a t i t
norm p r e s e r v i n g . i s surjective.
If z E a
then the series q'
L e t y = t y k 1 be d e f i n e d by y k =
lizk
1 zk/k,
c o n v e r g e s s i n c e t l / k l E .P
k zi/i
P'
f o r each k, t h e n y e Y a n d T1y = z .
We c o n c l u d e t h a t T1 i s a n i s o m e t r i c i s o m o r p h i s m o f Y o n t o P and t h u s Y i s a q Banach space. N o t e t h a t Y = Xmii when q = 1.
T h e r e f o r e X E i s a Banach space a n d (X:)*
is
e q u i v a l e n t t o L_. To show t h a t Y i s t h e c o m p l e t i o n o f X a f o r 1 < p =, i t i s enough t o show P 1 t h a t X', Y and X a i s dense i n Y . Choose t such t h a t 1 t - < t < 2 and l e t P P 9 y k = Ci=k i-tt h e n yk
+
0 as k
-+
-
and
Nonabsolute Cesaro Sequence Spaces
289
4 X BP '
T h i s i m p l i e s t h a t X 4 c Y and X F # Y . P To f i n i s h t h e p r o o f we show t h a t X e i s dense i n Y . F o r any y = t y k } E Y , P there exists y(N) = ,..., yN-yNtl,O.O ,...) e X B such t h a t P'
and hence y
The p r o o f i s complete. (ii)
(iE)*
By ( i ) , X a i s congruent t o
P
e for 1 9
<
p
<
m
and
1
-
1
t - = 1.
Thus
9*
f o r ( 1 < p < m ) . To p r o v e t h a t ( X B ) * I ( i o ) , we r e P P P L c a l l t h a t if M i s a subspace o f X and M i s t h e a n n i h i l a t o r o f M, t h e n M I i s a i s congruent t o B
c l o s e d subspace o f X* and t h e c o n j u g a t e space M* i s congruent t o X
* I
/M
.
(see
t h e a n n i h i l a t o r o f X' i s t h e ~ 9 3 1 ) . S i n c e X B i s a dense subspace o f iD P P z e r o v e c t o r i n ( f a ) * . T h e r e f o r e , ( X B ) * z (1:') / ( X a ) * z P P P P
[6;
(iF)*.
Theorem 6.
Proof.
X B a = XBy = Xaa P P P
t
X
for 1
P
<
p <
m.
Xaa ,X4'
and X c X@@, we have X4a t X c X B o . Thus, P P P P P P i t i s enough t o show t h a t XBv c XBa + X P P' P L e t A = (ank) be d e f i n e d by ank = n when n = k, ank = - n when n = k - 1, and Note t h a t s i n c e
ank = 0 o t h e r w i s e .
P
Then X
D
P
i s r e p r e s e n t e d as an i n t e r s e c t i o n o f two sequence
spaces, i . e . XB = P
xy
fl iy : Ay e "1
We d e f i n e a norm o f y E X B by P IIY11 = /likYkl Then X B i s a BK space. P Again, by Theorem 7.2.7
x;o
c
Ilk_'
IIrk(Yk
Yk+l))l/k
9' o f [5], xby c
"f.
P by Theorem 4.4.1 o f [5],
Let f E (XR)*; P Hence f ( y ) = F ( y ) t L: = eP.
- ic u l a r ,
-
1k= 1k ( y k
f = F + g
-
yktl)gk
0
A with F
E
( X y ) * and
f o r y E X B and i n p a r t P
B.-E. Wu et al.
290
f(e f o r each k w i t h go = 0. where F E (X;)* k:
k
k
T - l E P*_.
p. 426]),
Thus, F ( e k ) = Uk f o r each k .
1
xOy,
X O f c XOa
+x
P P P P P The p r o o f i s cornpl e t e .
kgk
-
(k-1)gk-l
k k k Hence F ( e ) = Fl(ke ) = kFl(e ) .
0
T,
Since
F1 = u + v w i t h u = { u k l E il,v = i v k 1 E cL0 ' k L e t z;( = kF ( e ) = kuk and 1
t h e n f ( e k ) = z k + z i where z ' = { z i }
xea,
+
I n v i e w o f t h e p r o o f o f Theorem 4, we have F = F1
and F1 = F
= t1 @ c t I( s e e [6;
= F(e
8
X" P
2;
and z " = t z j 0 E X
= kgk
P'
-
(k-l)gkml;
Therefore
'
REFERENCES
1. 2. 3. 4.
5. 6.
D.J.H. G a r l i n g , The a- and y - d u a l i t y o f sequence spaces, Proc. Camb. P h i l . SOC. 63( 1967) 963-981. Ng Peng-Nung and Lee Peng Yee, Cesaro sequence spaces o f a n o n a b s o l u t e type, Comm. Math. 20 (1978), 429-433. Lee Peng Yee, Cesaro sequence spaces, Math. C h r o n i c l e 13 (1984), 29-45. Wu Bo E r and Lee Peng Yee, The d u a l s o f some sequence spaces of a nona b s o l u t e t y p e , SEA B u l l . Math. 9 ( 1 9 8 5 ) , 77-80. A l b e r t W i l a n s k y , Sumrnability t h r o u g h f u n c t i o n a l a n a l y s i s ( 1 9 8 4 ) . G o t t f r i e d Kb'the, T o p o l o g i c a l V e c t o r Spaces I ( 1 9 8 3 ) .
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
29 1
RANACH R E U U C I B I L I T Y OF UECOHPOSABLE OPERATORS Xu Ferig N o r t h e a s t Normal U n i v e r s i t y , Chanychun, C h i n a Zou Chenzu
J i l i n I l n i v e r s i t y , Changchun, C h i n a
I n t h i s p a p e r , we c o n t i n u e t h e s t u d y o f Hanach r e d u c i b i l i t y o f decomposable o p e r a t o r s . The n o t i o n o f c o m p l e t e Hanach r e d u c i b i l i t y f o r bounded o p e r a t o r s i s i n t r o d u c e d , and v a r i o u s f a c t s a r e p r o v e d a b o u t c o m p l e t e l y Hanach r e d u c i b l e weak decomposable o p e r a t o r s . T h r o u y h o u t t h i s p a p e r , X i s a complex Hanach space and B ( X ) i s t h e Hanach a l g e b r a o f a l l hounded l i n e a r o p e r a t o r s on X.
It T
6 H(X)
has t h e s i n g l e -
v a l u e d e x t e n s i o n p r o p e r t y , ( J ~ ( x d) e n o t e s t h e l o c a l s p e c t r u m o f T a t x. d e n o t e s t h e cornplex p l a n e .
6:
For e v e r y E c C , X T ( E ) = { x l u T ( x ) C E } d e n o t e s a
s p e c t r a l m a n i f o l d o f T. I n o u r p r e v i o u s p a p e r [ I ] , we have i n v e s t i g a t e d t h e Banach r e d u c i b i l i t y o f I n t h i s p a p e r we w i l l i n v e s t i g a t e f u r t h e r t h e Banach
decomposable o p e r a t o r s .
r e d u c i b i 1 it y o f deconiposabl e o p e r a t o r s .
D E F I N I T I O N 1.
T e H ( X ) i s s a i d t o have c o m p l e t e Ranach r e d u c i b i l i t y i f f o r
e v e r y i n v a r i a n t subspace P I o f T , t h e r e e x i s t s a n o t h e r i n v a r i a n t subspace I1 o f T such t h a t X = Pl*lY, K.
where
Tanahashi [ 2 ]
* denotes
t h e d i r e c t sum.
p r o v e d t h a t : i f H i s a complex H i l b e r t space, T i s a
bounded l i n e a r o p e r a t o r on X w i t h t h e s i n y l e - v a l u e d e x t e n s i o n p r o p e r t y and E C &, t h e n o ( T ( H T ( E ) )
c E
i f and o n l y i f H1(E) i s c l o s e d .
C l e a r l y , t h e above
r e s u l t i s r i y h t when H i s a Hanach space.
PROPOSITION.
Let T
6
H ( X ) have t h e s i n y l e - v a l u e d e x t e n s i o n p r o p e r t y and E
C
C.
Then
a(TIXT(E)) c E i f and o n l y i f X T ( E ) i s c l o s e d .
LEHMA.
L e t T e H ( X ) have t h e s i n g l e - v a l u e d e x t e n s i o n p r o p e r t y and c o m p l e t e
Banach r e d u c i b i l i t y .
PROOF.
Then, f o r e v e r y c l o s e d s e t F o f E ,
-
S i n c e X T ( F ) i s a r e d u c i b l e subspace f o r T , t h e r e e x i s t s Z,
subspace o f T , such t h a t
an i n v a r i a n t
E X u and C Zou
292
x=xT(F)iz. Suppose P i s t h e p r o j e c t i o n o f X o n t o
xT(F) a l o n y
Z;
t h e n TP = PT.
Obviously,
Now f o r a r b i t r a r y x
-
X T ( F ) , we show U , ( X ) C
f
p T ( x ) + X be a n a l y t i c .
u
Then
T
1 %ym( X I .
Let f ( A )
:
so
The above f a c t shows
THEOREM 1.
Let T
PROOF.
8
H ( X ) be a c o m p l e t e l y Hanach r e d u c i b l e weak decomposable
Then T i s a q u a s i - d e c o m p o s a b l e o p e r a t o r .
operator.
S i n c e T i s a weak decomposable o p e r a t o r , T has t h e s i n g l e - v a l u e d
extensiori property.
F i r s t , we show t h a t f o r an a r b i t r a r y c l o s e d s u b s e t F o f C, u ( T ( V )
L e t A" $ F.
Then { G 1 ,
Then t h e r e e x i s t s an
E
C
F.
> 0 such t h a t
G2} f o r m s an open c o v e r i n g o f u ( T ) .
S i n c e T i s a weak decomposable
o p e r a t o r , t h e r e e x i s t s p e c t r a l maximal subspaces Y1 and Y2
x Eloreover,T
=
y1+y2,
~ ( T I Y =tii, ~ )
Of
T such t h a t
i = 1,~.
i s a c o m p l e t e l y Banach r e d u c i b l e o p e r a t o r and
i s an i n v a r i a n t
Banach Reducibility of Decomposable Operators subspace o f T,
293
so t h e r e e x i s t s a p r o j e c t i o n P or1 X sirch t h a t
__
PX = X T ( F )
and
T P = PT.
=
PY1 + PY2.
Iie o b t a i n
PX = P(Y1+Yz) lhviously,
-.
so PYl c Y1 and h e n c e
Y1 i s a h y p e r - i n v a r i a n t s u b s p a c e o f T,
P Y 1 c Y1 fl X T ( F ) . S i r i c e T i s a c o m p l e t e l y Hanach r e d u c i b l e o p e r a t o r , t h e r e e x i s t s an i n v d r i a n t
* Z1.
subspace Z 1 o f T sucti t h a t X = Y1 a l o n y Z1.
Then T P l = PIT.
Thus,
he t h e p r o j e c t i o n o f X o n t o Y1
L e t PI
f o r a r h i t r a r y x 8 X T ( F ) , we h a v e u T ( x )
=
F
and
n
u T ( P 1 x ) c u ~ ( x )n o ( T ( Y 1 ) c F Thirs, o T ( P 1 x ) c F
n
Gl =
+,
and hence P1x = 0.
Pl(Fp)
G1
=
8.
Therefore
= 0.
Then PY1 = 0 , arid hence
xT(F) = PYl
+ PY2
-
= PYz=
Yz.
-
By t h e lemma, f o r e v e r y x € X T ( F ) , rJ
___ ( x ) = u T ( X ) TIx~(F)
C
o ( T I Y Z ) c tiz.
Therefore o ( ~ ( ~ T c ( ~L ) ~) .
--
Lie h a v e l o
u ( T ( x ~ ( F ) ) ,t t i i r s
o(T\XT(F)) C By t h e p r o p o s i t i o n ,
THEOREM 2. operator.
F.
X T ( F ) i s c l o s e d , hence T i s a q u a s i - d e c o m p o s a b l e o p e r a t o r .
L e t T 6 H ( X ) be a c o m p l e t e l y Kanach r e d u c i b l e weak decornposable Then T i s a s t r o n y l y i d e n t i c a l l y d e c o m ~ ~ o s a bo~pee r a t o r .
PROOF. By Theoreni 1, f o r a c l o s e d s u h s e t F c (;, X T ( F ) i s an i n v a r i a n t s i i b s p a c e o f T. Un t h e o t h e r hand, l e t G 3 F he an open s u h s e t . Then X T ( F ) c X T ( c ) , and t h e r e e x i s t s an i n v a r i a n t s u b s p a c e N o f T s u c h t h a t
X = XT(c)
N.
S i n c e X / X T ( z ) and N a r e t o p o l o y i c a l l y i s o m o r p h i c by K.
u(T xT(c;) where T
xTm ) C
Lange c31, we h a v e
,
d e n o t e s t h e c o i n d u c e d o p e r a t o r b y T on t h e q u o t i e n t s p a c e
E Xuand C Zou
294 X/XT(c).
Hence ,,(TIN) =
,,(T
Therefore
XTG) c
[ice F'.
--
N c XT(Fc),
___ C
S i n c e XT(F ) i s an i n v a r i a n t subspace o f T, t h e r e e x i s t s a n o t h e r i n v a r i a n t subspace Z o t T such t h a t X = XT(Fc)
Thus, Z c X
5
(c),so
* Z.
we have
z = n
ti =F
xT(%) = x ~ ( F ) ,
dnd hence
X = X T ( F ) + XT(F - C ).
Hy
[4], T i s a s t r o n y l y i d e n t i c a l l y decomposable o p e r a t o r .
THEOKEM 3.
L e t X be a i r e a k l y c o m p l e t e Hanach space and T
6
K(X) be a
c o m p l e t e l y Hariach r e d u c i b l e weak decoinposable o p e r a t o r w i t h t h e p r o p e r t y ( H ) . Then T i s a s p e c t r a l o p e r a t o r . PROOF.
By Theoreiri 2, a n y c l o s e d s u b s e t
(A, H , C ) ( s e e
[51),
therefore F
f a m i l y o t Bore1 s u b s e t s o f C.
€
bl(T).
F
8
gl(T),
and s i n c e T has t h e p r o p e r t y
Thus, we o h t a i n t h a t g ( T ) i s t h e
By C51, T i s a s p e c t r a l o p e r a t o r , where $l(T)
and + ( T ) were i n t r o d u c e d hy D u n f o r d and Schwartz [ 5 ] .
KEFERENCES
[l] [2]
[3]
[4] C5l
Xu Feng and Zou Chenzu, Hanach r e d i r c i b i 1 it y o t decornposabl e o p e r a t o r s , Uonybei Shida Xuehao 4 ( 1 9 8 3 ) , 61-69. K . Tanahashi, K e d u c t i v e weak decomposable o p e r a t o r s a r e s p e c t r a l , Proc. Amer. r l a t h . SOC., 87 (1Y83), 44-46. K. Lanye, E q u i v a l e n t c o n d i t i o n s f o r decomposable o p e r a t o r s , Proc. Arner. Math. SOC., 82( 1981), 4U1-4U6. L i r i tiuanyu, Uecomposable o p e r a t o r s w i t h t i n i t e o r d e r and t h e i r p r o p e r t i e s , J. N a n j i n y Ilniv., 3 ( 1 Y 8 2 ) , 598-6%. N. U u n t o r d and J . Schrrartz, L i n e a r O p e r a t o r s ( I I I ) , New York, 19/1.
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
295
THERE CAN BE NO LIPSCHITZ VERSION OF MICHAEL'S SELECTION THEOREM DAVID YOST Mathematics Department, I.A.S., Australian National University, G.P.O. Box 4, Canberra, A.C.T. 2601, AUSTRALIA
Given a real Banach space E, let H(E) denote the family of closed, bounded, convex nonempty subsets of E . We equip H(E) with the Hausdorff metric : for A,B E H(E), set dH(A,B) = sup ((d(x,A) : x E B) U (d(x,B): x E A ) ) . Let X be a metric space and w : X + H(E) a continuous map. Then, as a special case of Michael's selection theorem [M, Theorem 3.2"] , w admits a continuous selection. This means that there is a continuous map f : X -+ E satisfying f(x) E ~ ( x for ) all x in X. Michael's theorem is actually somewhat more general. It states that w admits a continuous selection, assuming only that w is lower semicontinuous, and allowing the values of to be all the nonempty, closed, convex subsets of E (not necessarily bounded). We will not concern ourselves with the more general result, or with the various other selection theorems proved by Michael ([MI, and the references therein). Michael's theorem has had applications to numerous areas of mathematics, including infinite-dimensional topology [E,J], functional analysis [C,N,V,W], approximation theory [D,O,X], mathematical economics [GI, differential equations and inclusions [H,K], optimization theory [R] and vector bundles [TI. This list of references is not meant to be comprehensive; it is just an arbitrary selection. (Not all of these papers require the full strength of Michael's selection theorem. For the topological properties of H(E), at least in the finite dimensional case, we refer to [QI.) Naturally, various mathematicians (see, for example [I,Y]) have thought about the following problem : if w : X + H(E) is Lipschitz continuous, is it possible to choose f : X + E to be Lipschitz continuous also? This is easily seen to be equivalent to the following problem : does there exist, for a given Banach space E , a Lipschitz map f : H(E) -+ E satisfying the identity f(A) E A ? Bressan [B] and Przeslawski [PI showed that this is possible if E is finite dimensional. (Thanks are due to Alicja Sterna-Karwat for bringing [PI to our attention.) The existence of Lipschitz selections in some other special cases is proved in [L],[U] and [Z]. But in general, as we now show, the preceding problems have a negative solution. For a counterexample we take E = D, the set of bounded functions from [0,1] into IR which are continuous at every irrational point in (O,l), and right continuous
w
296
D. Yost
with left limits at every rational point. Equipped with the supremum norm, D is a separable Banach space. The nonexistence of a Lipschitz selection from H(D) to D follows by combining some results from [A] and [XI . However, we think it is worthwhile to give a self-contained argument. For f E D ,denote by Pf the set of best approximants to f in the subspace C of continuous functions. That is, Pf = ( g E C : )I f - g )I = d (f,C)) . Let Jf : [0,1] + IR denote the "jump function" of f, Jf(t) = f(t) - f(t-) . (For convenience we may set f(0-) = f(O).) Obviously Jf is a bounded function. Our next result is a special case of [S ,7.5.6]. Lemma I For each f E D , we have Pf E H(C) and d(f,C) = '/2 11 Jf 1 . Proof Obviously Pf is a closed, bounded, convex subset of C ; we must show it is nonempty. For any g E C and any t ,we have either I g(t) - f(t) I 2 1/2 I f(t) - f(t-) I or I g(t-) - f(t-) 1 2 l/2 I f(t) - f(t-) I . Taking the supremum over t , we conclude that II S-f II 2 l/2 II Jf II * Next we will exhibit a g E C satisfying 11 g - f 11 _< 1/211 Jf 11 . This will simultaneously show that d(f,C) = l/2 11 Jf 11 and that g E Pf . Define fo,fl :[0,1] -+IR by fo(t) = max{f(t), f(t-)) and fl(t) = min(f(t), f(t-)) . Clearly, fo is upper semicontinous, f1 is lower semicontinuous, and fo(t) - 1/2 11 Jf 11 I fl(t) + l/2 11 Jf 11 for all t . From Michael's theorem, we obtain a g E C which satisfies the identity fo(t) - l/2 11 Jf 11 I g(t) I fl(t) + l/2 11 Jf 11 . Clearly 11 g - f 11 I l/2 11 Jf ( 1 . Now let us define Qf = f - Pf for each f E D. It is clear that Qf E H(D) and that Qf = Qg <=> f - g E C .The next result could be deduced from [X, Lemma 1.1 and Corollary 2.31, but the following argument is simpler. Lemma 2 For all f,g E D, we have (i) dH(Pf, Pg) I 2 11 f - g 11 and (ii) d(f - g,C) I dH(Qf, Qg) I 3 d ( f - g,C) . Proof (i) Set y = 11 f - g 11 and 6 = d(g,C) . By symmetry, it suffices to show that, given a E P f , wecanfind b E Pg with ) I a - b ) ) 1 2 6 .Now I l g - a I l I l l g - f l l + d ( f , C ) I 2 I l g - f l l + d(f,C)=2y+6. Thus g - a - 6 5 2 7 and - 2 y I g - a + 6 . Lemma 1 now gives us some c E P(g-a) = C n B(g-a,6). Thus g-a-6 5 c I g-a + 6 . If we put b(t) = max (-2y, min (2y, c(t)) ) + a(t) ,then b(t) = min (2y, max (-2y, c(t)] ] + a(t) and so g-a-6 I b-a I g-a + 6 . Clearly b E C and 11 g-b 11 5 6 , whence b E Pg . Obviously 11 b-a 11 I 2y. (ii) For any h E Pf, we have d(f-g,C) = d(f-h,g+C) I d(f-h,Qg) IdH(Qf,Qg). This proves the first inequality. To prove the second inequality, note that, for any h E C, dH(Qf,Qg) = d,(f + h - P(f + h), g - Pg) I 11 f + h - g 1 + dH( P( f + h ) , Pg) I 3 1 f - g + h 11. Lemma 2 tells us that the quotient space D/C is Lipschitz equivalent to a subset of H(D) . Aharoni and Lindenstrauss [A, Remark (ii)] showed that there is no Lipschitz lifting from D/C to D . This gives us our main result.
Michael's Selection Theorem
291
Theorem 1 There is no Lipschitz continuous selection from H(D) to D . Proof (after [A]) Suppose that such a selection exists. Then there certainly exists a Lipschitz selection w : Q(D) + D, where Q(D) = (Qf : f E D ) C H(D). Let us equip Q(D) with the equivalent metric
By Lemma 2 , w is also Lipschitz with respect to d , so we may work exclusively with d henceforth. Let K be the best Lipschitz constant for w . Choose fl and f2 in D sothat
Since w is Lipschitz, we assume without loss of generality that d(Qf1, Qf2) = 1 . Choose an irrational to so that
By continuity at to , there is an open interval I and a rational q E I such that
1 / 2 ( ~ Q f+l yQf2) varies by less than l/5 on I , and
Let s = 1 1 be the sign of J(f, + f2)(q) , and put g = */2(fl + f2 + sx ~ 0 , ~.) )Here, of course, X[o,ql denotes the characteristic function of the interval [O,q) . Now for any t z q , wehave
d(Qf1 , Qg) = 1/2 11 J(fl - g) 11 5 1/2 . Similarly
Then, for every t E I ,we have
D. Yost
298
Simple calculations then show that WQg varies by less than 2/5 on I . In particular I (JWQg)(q) I 2/5 . But VQg E g - Pg C g + C , whence JWQg = Jg. Since I (Jg)(q) I = l/2 I J(f, + f2)(q) + s I 2 l/2 , we have a contradiction.
I
If F is a closed linear subspace of a Banach space E ,and there is no Lipschitz selection H(F) + F , then clearly there is no Lipschitz selection H(E) + E . In particular, there is no Lipschitz selection when E = C , as C contains an isometric copy of every separable Banach space. Denote by IRn Euclidean n-space, and by Kn the smallest Lipschitz constant for selections f : H(IRn) + IR" . It is known [B,P] that K, 5 n . Thus, Lipschitz selections exist, but the Lipschitz constants might not be uniformly bounded as n becomes larger. Quite recently, Krysztof Przeslawski [personal communication] proved that Kn does go to infinity with n , and hence that there is no Lipschitz selection H(E) + E whenever E is an infinite dimensional Hilbert space. But now, let us recall Dvoretzky's theorem [F] : for any infinite dimensional Banach space E, any n E N , and any E > 0 , there is an n-dimensional subspace of E which is &-isometricto IRn . From this, and Przeslawski's result, the following can be proved. Theorem 2 There exists a Lipschitz selection from H(E) to E if and only if the Banach space E is finite dimensional. Theorem 2 was stated as a conjecture when this paper was originally presented. It lies much deeper than Theorem 1, and full details will appear elsewhere. REFERENCES [A] LAharoni and J.Lindenstrauss, Uniform equivalence between Banach spaces, Bull. Amer. Math. SOC. 84 (1978) 281-283. [B] A.Bressan, Misure di curvatura e selezioni lipschitziane, preprint. [C] A.Clausing and S.Papadopolou, Stable convex sets and extremal operators, Math. Ann. 231 (1978) 193-203. [D] F.Deutsch, A survey of metric sdections, Contemporary Math. 18 (1983) 49-71. [El K.D.Elworthy, Embeddings, isotopy and srabiliry of Banach spaces, Compositio Math. 24 (1972) 175-226. [F] T.Figie1,A shorrproof of Dvoretzky's theorem on almost spherical sections of convex bodies, Compositio Math. 33 (1976) 297-301. [GI R.Guesnerie and J.-J.Laffont, Advantageous reallocation of initial resources, Economemca 46 (1978) 835-841.
Michael’s Selrctiori Theorem
299
[HI G.Haddad and J.M.Lasry, Periodic solutions of functional differential inclusions and$xed points of o-selectionable correspondences, J. Math. Anal. Appl. 96 (1983) 295-312. [I] A.D.Ioffe, Single-valued representation of set-valued mappings 11; Applications to differential inclusions, SIAM J. Control. Optirn 21 (1983) 641-651. [J] K.John and V.Zizler, Weak compact generating in duality, Studia Math 55 (1976) 1-20. [K] W.G.Kelley, Periodic solrrtions of generalized differential equations, SIAM J. Appl. Math. 30 (1976) 70-74. [L] S.Lojasiewicz Jr, A.Plis and R.Suarez, Necessary conditions for a nonlinear control system, J. Diff. Eqns. 59 (1985) 257-265. [MI E.Michae1, Continiroirs selections. I, Ann. of Math. 63 (1956) 361-382. [N] R.R.Nelson, Pointwise evaliution of Bochner integrals in Marcinkiewcz spaces, Nederl. Akad. Wetensch. Proc. Ser. A. 85 (1982) 365-379. [O] C.Olech, Approximation of set-valuedfiinctions by continiioiis fiinctions, Colloq. Math. 19 (1968) 285-293. [PI K.Przeslawski, Linear and Lipschitz continuous selectors for the family of convex sets in Euclidean vector spaces, Bull. Pol. Acad. Sci. 33 (1985) 31-33. [Q] J.Quinn, S.B.Nadler Jr and N.M.Stavrakas, Hyperspaces of compact, convex sets, Pacific J. Math 39 (1971) 439-469. [R] R.T.Rockafellar, Integrals which are convex functionals, 11, Pacific J. Math. 39 (1971) 439-469. [S] ZSernadeni, Banach spaces of continuous functions, PWN, Warsaw (1971). [TI A.J.Tromba, The Eider characteristic of vectorfields on Banach manifolds and a globalization of Leray-Schauder degree, Adv. in Math. 28 (1978) 148-173. [U] V.A.Ubhaya, Lipschitz condition in minimum norm problems on boundedficnctions, J. Approx. Theory 45 (1985) 201-218. [V] M.Valadier, Closedness in the weak topology of the dildpair L1, c,J. Math. Anal. Appl. 69 (1979) 17-34. [W] D.Werner, Extreme points in funcrion spaces, Proc. Arner. Math. Soc. 89 (1983) 598-600. [XI D.Yost, Best approximation and intersections ofballs in Banach spaces, Bull. Austral. Math. SOC.19 (1979) 285-300. [Y] D.Yost, Best approximation operators infitnctional analysis, Proc. Centre Math. Anal. Austral. Nat. Univ. 8 (1984) 249-270. [Z] P.Zecca and GStefani, M~rlrivaliceddifferential equations on manifolds with application to control theory, Illinois J. Math. 24 (1980) 560-575.
This Page Intentionally Left Blank
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
301
A NEW SMOOTHNESS OF BANACH SPACES Wenyao ZHANG Department of Mathematics Liaoning Normal University Dalian, Liaoning Chi nay
If a Banach space monotone basis on
X X
is weakly very smooth, then every is shrinking.
Throughout this paper,
X
is a Banach space,
s*
11x11 = 1 1 ,
* x* , I I ~=* I1 1I.
= {x E
Let
x E X, x
S = {x 6 ji
0,
X,
if
f E S*, f(x) = llxll, then f is called a supporting functiona of x. If each point in S has a unique supporting functional then X is called smooth. It is known (see, e.g., [ l ] ) that X is smooth if and only i f for any x E S , f n E S * , fn(x) + 1 implies that
{f,}
converges weak* to a supporting functional o f
x. A Banach space X is very smooth i f for any x 6 S , Jx has *** . I t was proved [I] that X a unique supporting functional in X is very smooth i f and only i f for any x E S , fn E s*, fn(x) + 1
{f,}
implies that
converges weakly to a supporting
functional of x. Now we introduce a concept of smoothness which X be a Banach space, i f is weaker than very smoothness. Let for any
x E S,
fn
E S*,
fn(x)
+
1
subsequence which converges weakly in
implies that
{fn}
X*,
is called
then
X
has a
weakly very smooth. Obviously a very smooth Banach space is weakly very smooth, and every reflexive Banach space is weakly very smooth, thus there a r e weakly very smooth Banach spaces which are not very smooth. On the other hand, i t is routine to show that i f a weakly very smooth Banach space is smooth, then i t is very smooth.
It is known that i f
X*
is very smooth, then
X
is reflexive.
By the well known James‘ characterization of reflexive spaces i t follows immediately that i f X* is weakly very smooth, then X *Present address: Department of Mathematics, The University of Iowa, Iowa City, Iowa, 52242, U.S.A.
W.Zhang
302 is reflexive.
X
Let
be a Banach space with a basis
1 aiei]
Pn[
{en},
we define
__
n
m
1a i e i
=
i=1
the natural projections associated to
{en}
i=1 m
and
1 aiei] = an
fn[
the biorthogonal functional associated t o
i=1
* If for any
{en}.
x
x*,
E
llxll < I} = 0, then
rn
I
lim sup{lx*(x) n
:
1
x =
aiei'
i =n+ 1
For other
is called a shrinking basis.
{en}
concepts and results appearing in this paper, we refer to [ 2 ] and
c31. X
THEOREM: Let
X
if
{fn}
{en}. that
{Pn}
span{fn}
IlX*ll}
Let
be the associated natural projections of
{f,}
To show that
show that
{x
*
* X .
=
E X*
is a basis of
XI,
we need to show
By Bishop-Phelps Theorem [ 4 ] , there exists an
:
x E S
i t remains t o
such that
x*(x)
c span{fn}.
*
*
*
x 0 (x0 ) = IlxoII
xo E X*,
generality, we assume that
IIPnll = 1
for every
hand, for any that
{en},
is weakly very smooth, then the biorthogonal functional associated to {en} is a basis of X*.
Proof: Let
=
be a Banach space with a monotone basis
n,
* IlxoII
S, Without loss of
with
xo
= 1.
Since
thus we have
€
* IIPnII
{en} = 1.
is monotone, On the other
N,
e > 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.