Martingale Hardy Spaces and their Applications in Fourier Analysis

227 Pages·1994·2.935 MB·English

by  Ferenc Weisz (auth.)

Lecture Notes in Mathematics 1568
Editors:
.A Dold, Heidelberg
B. Eckmann, Ztirich
E Takens, Groningen
Ferenc Weisz
elagnitraM ydraH Spaces
dna rieht snoitacilppA
in Fourier Analysis
Springer-Verlag
Berlin Heidelberg NewYork
London Paris Tokyo
Hong Kong Barcelona
Budapest
Author
Ferenc Weisz
Department of Numerical Analysis
E6tv6s L. University
Bogd~infy u. 10/B
H-1117 Budapest, Hungary
Mathematics Subject Classification (1991): 60G42, 60G46, 42B 30, 42C 10, 60G48,
46E30, 42A20, 42A50
ISBN 3-540-57623-1 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-57623-1 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data. Weisz, Ferenc, 1964-
Martingale Hardy spaces and their applications in Fourier analysis / Ferenc Weisz.
p. cm. - (Lecture notes in mathematics; 1568) Includes bibliographical references
and index. ISBN 0-387-57623-1
.1 Martingales (Mathematics) 2. Hardy spaces. 3. Fourier analysis. I. Title II. Series:
Lecture notes in mathematics (Springer-Verlag); 1568. QA3.L28 no.
1568 QA274.5 510 s-dc20 519.2'87 93-49415
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(cid:14)9 Springer-Verlag Berlin Heidelberg 1994
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TABLE OF CONTENTS
PREFACE .................................................................. vii
Chapter 1
PRELIMINARIES AND NOTATIONS ........................................ 1
1.1. Martingales .............................................................. 1
1.2. Vilenkin orthonormed systems ............................................ 8
Chapter 2
ONE-PARAMETER MARTINGALE HARDY SPACES ...................... 13
2.1. Atomic decompositions .................................................. 14
2.2. Martingale inequalities .................................................. 19
2.3. Duality theorems ........................................................ 33
2.4. BMO spaces and sharp functions ........................................ 51
2.5. Martingale transforms ................................................... 61
2.6. Conjugate transforms of Vilenkin martingales ............................ 67
Chapter 3
TWO-PARAMETER MARTINGALE HARDY SPACES ..................... 80
3.1. Atomic decompositions .................................................. 81
3.2. Martingale inequalities .................................................. 84
3.3. Duality theorems ....................................................... 100
3.4. Martingale transforms .................................................. 122
3.5. Two-parameter strong martingales ...................................... 124
Chapter 4
TREE MARTINGALES .................................................... 141
4.1. Inequalities ............................................................ 141
4.2. Convergence theorems .................................................. 158
vi
Chapter 5
REAL INTERPOLATION .................................................. 164
5.1. Interpolation between one-parameter martingale Hardy spaces .......... 171
5.2. Interpolation between two-parameter martingale Hardy spaces .......... 177
Chapter 6
INEQUALITIES FOR VILENKIN-FOURIER COEFFICIENTS ............. 183
6.1. Hardy type inequalities ................................................. 183
6.2. Dual inequalities ....................................................... 193
6.3. Convergence of Vilenkin-Fourier series .................................. 196
REFERENCES ............................................................. 204
SUBJECT INDEX .......................................................... 214
LIST OF NOTATIONS ..................................................... 216
PREFACE
The history of martingale theory goes back to the early fifties when Doob 16
pointed out the connection between martingales and analytic functions. On the basis
of Burkholder's scientific achievements (17, 20, 24) the theory of continuous
parameter martingales can perfectly well be applied in complex analysis and in
the theory of classical Hardy spaces. This connection is the main point of Durrett's
book 64. Attention was also drawn to martingale Hardy spaces when Burkholder
and Gundy 27 proved the inequality named after them ever since, which states
that the Lp norms of the maximal function and the quadratic variation of a one-
parameter martingale are equivalent for 1 < p < oo. Some years later this result was
extended to p = 1 by Davis 55. In 1973 the dual of the one-parameter martingale
Hardy space generated by the maximal function was characterized by Garsia 28
and Herz 94 as the space of the functions of bounded mean oscillation (BMO).
The beginning of the study of two-parameter martingales can be dated to 1970 when
Doob's inequality was shown by Cairoli 29. The Burkholder-Gundy inequality was
also proved later for two-parameter martingales by Metraux 125.
While the theory of the martingale Hardy spaces is discussed briefly, their ap-
plication in Fourier analysis has not yet been presented at all in books on martingale
theory (e.g. Neveu 142, Garsia 82, Dellacherie, Meyer 58). In the only book on
two-parameter martingales (see Imkeller 105) the theory of the martingale Hardy
spaces is not studied in great detail. Just a few papers have been published to
date on tree martingales. This book is intended to fill this gap, giving, on the one
hand, an exhaustive study of one- and two- (discrete-) parameter martingale Hardy
spaces, and on the other hand, demonstrating some of their applications in Fourier
analysis. Moreover, several new and still unpublished results are presented as well.
The methods of proof for one and two parameters are entirely different; in most
cases the theorems stated for two parameters are much more difficult to verify. A
method that can be applied both in the one- and in the two-parameter cases, the
so-called atomic decomposition method was improved by the author. Nevertheless,
in a simpler form, it has already appeared in the writings on this field (see e.g. Herz
94, Bernard, Maisonneuve 10). With the help of this method a new, common
construction of the theory of the one- and two-parameter martingale Hardy spaces
is presented. Most of the one- and two-parameter theorems as well as quite a few
of the theorems about tree martingales and of the applications can be traced back
to this method.
The book is structured as follows. In Chapter 1 the rudiments are summarized.
In Chapter 2 and 3 one- and two-parameter martingales are studied in a similar
structural way. The method of the atomic decomposition is described in Sections
2.1 and 3.1. Then some martingale inequalities follow; the Burkholder-Gundy in-
equality is proved in both cases while Davis's inequality is shown for arbitrary
one-parameter and for special (regular and strong) two-parameter martingales. The
latter is still unknown for general two-parameter martingales. In Sections 2.3 and
3.3 some duality theorems are verified. The equivalence between the BMO spaces
is examined and the John-Nirenberg theorem is proved. Furthermore, in these two
chapters martingale transforms are considered. The first three sections of Chapter
...
vnl
2 most resemble Garsia's book (82), however, their structure is much simpler and
several new theorems are given here.
In Chapter 4 tree martingales are dealt with. In Section 4.1, amongst other
things, the Burkholder-Gundy inequality together with one relative to martingale
transforms are shown. As an important application, one of the most difficult the-
orems in Fourier analysis, Carleson's theorem is obtained; it asserts that the one-
parameter Walsh-Fourier series and its generalization, the so-called Vilenkin-Fourier
series of a function in Lp (1 < p < oo), converge almost everywhere to the function
itself. In addition to this, the foregoing convergence in Lp norm is verified as well.
In Chapter 5 the still young theory of interpolation is applied and the interpo-
lation spaces of the martingale Hardy spaces are characterized. These results, which
are very interesting in themselves, will be used in the last chapter.
In Chapter 6 Hardy type inequalities are presented for Walsh-Fourier and
Vilenkin-Fourier coefficients. Applying these, one obtains Carleson's theorem for
two-parameter systems and for the functions of Lp (1 < p) with monotone Vilenkin-
Fourier coefficients.
Since this book deals with the relation between analysis and probability theory,
I have tried to write it in such a way that it is accessible to everyone familiar with
the fundamentals of any of the two fields.
First of all, I would like to express my thanks to my dear colleague, Prof. Ferenc
Schipp who supported me in the very first steps of my career, drew my attention to
this field; in addition, I am thankful for his valuable ideas to improve this work. I
am very grateful to Prof. Peter Imkeller who read through the manuscript and made
many suggestions. I would like to thank the "Deutscher Akademischer Austausch-
dienst" with whose financial support I had the opportunity to bring this book to its
final version at the Ludwig-Maximilians-Universit/it, Miinchen. My thanks are due
to the Hungarian Scientific Research Fund, as well as to the Hungarian Scientific
Foundation for supporting my research. Above all, I am particularly indebted to my
wife for reading through the manuscript and even more for her patience and love.
I would like to ask the readers to overlook the quality of the English in my
book.
CHAPTER 1
PRELIMINARIES AND NOTATIONS
1.1. MARTINGALES
Let us denote the set of integers, the set of non-negative integers, the set of
positive integers, the set of real numbers and the set of complex numbers by Z, N, P,
R and C, respectively. For a set X if- 0 let X 1 := X and X 2 be its Descartes product
X (cid:141) X taken with itself. To denote the sequences the notation z = (z~,, a E A) will be
used where A is an arbitrary index set. Thus the one- and two-parameter sequences
will be denoted by z = (zn, n E N) and z = (z,, n E N2), respectively.
A pair of non-negative integers from N 2 is denoted by (n, m), or, simply by n
depending on the text environment. We write na and nz for the first and the second
coordinate of a pair n E N 2, respectively. Let us introduce the following partial
ordering on N2: for n = (nl,n~), m = (ml,m2) E N 2 set n <_ m if nl <_ mx and
n2 < m2. We say that n < m ifn <_ m and n # m (n,m E N2). 0f course, n < cr
for all n E N 2. Moreover, n << m means that both the inequalities nl < ml and
n2 < m2 hold. For n = (hi,n2) E 1 := (n, - 1,n2 - 1).
N 2 we set n -
For two arbitrary sets H, G C N 2 consisting of incomparable number-pairs (i.e.
if n, m E H or G then neither of the inequalities n < m and m < n hold) we write
H << G resp. H < G if for all n E G there exists m E H, such that m << n
resp. m < n. An elementary computation shows that the relation < is reflexive and
transitive, tf the elements of H and G are incomparable then H < G and G < H
imply G = H. Denote by inf H the set of the number-pairs m E H for which there
does not exist any n E H such that n < m (H C N2). We shall use the convention
inf @ = cr For two arbitrary sets H, G C N 2 we write H << G resp. H < G if and
only ifinf H << infG resp. infH < infG. It is easy to see that for every H, G C N 2,
infH < H and that H C G implies G < H.
Let (12, .A, P) be a probability space and let .T = (.Tn, n E NJ) (j = 1, 2) be
a non-decreasing sequence of a-algebras with respect to the complete ordering on
N or to the partial ordering on N 2. The a-algebra generated by an arbitrary set
system i"7 wilI be denoted by a(7"/). Introduce the following a-algebras:
~oT. := a(U,ENj .T,) (j = 1,2),
.T,,,,o~ := a(Uk=o.T,,,k), .To~,n2 := a(Uk=0.Tk,,2) (n E N2).
For the sake of simplicity, suppose that .T~ = A. and let .T-1 := .To, .T-1,-1 := .T0,0,
.T-l,i := .TO,i and ~'i,--1 := .Ti,O (i E N).
The expectation operator and the conditional expectation operators relative
to 5rn (n E N j U {~},j = 1,2), .T,~,~ and .T~,,~= (n E N 2) are denoted by E,
E,, En,,o~ and Er162 respectively. We briefly write Lp instead of the real or
complex Lp(f~, ,4. P) space while the norm (or quasinorm) of this space is defined
by Ilfllp := (EIfIP) 1/" (0 < p _< oo). The space lp consists of those sequences
b - (b,,, n E Z j or Nil) of real or complex numbers for which
p,llblI :-- ( ~ Ib.lP) 1/p < oo.
n6ZJ
For simplicity, we assume that for a function f E L1 we have in the one-parameter
case Eof = 0 and in the two-parameter case E,f = 0 if 0 or 0. The
nx = n2 =
space A4(/3) is the set of B-measurable functions for an arbitrary a-algebra/3.
In the two-parameter case we also suppose the condition 4F introduced by
Cairoli and Walsh 31 for the stochastic basis f: the e-algebras ~.1,oo and ~-oo,.2
are conditionally independent with respect to the a-algebra .T'., i.e. every bounded
function f (cid:12)9 .Ad(~',,1,oo ) and g (cid:12)9 A/l(~oo,.2) satisfy the equation
(F4) E.(fg) = E.fE.g (n (cid:12)9 N2).
An equivalent condition to this (see Cairoli, Walsh 31 p.114) is the following: for
all n (cid:12)9 N 2 and for all bounded function f
(F4) Enf = E.~,o~(Eo~,.~f) = E~,~2(E~,,o~f).
Of course, the equalities between random variables mean P-almost everywhere equal-
ities in the whole book.
An integrable sequence f = (f~, n (cid:12)9 NJ) (j = 1,2) is said to be a martingale
if
(i) it is adapted, i.e. fn is F. measurable for all n (cid:12)9 N j
(ii) E.fm = f~ for all n <_ m.
If E.fm _> (<)fn for every n < m then f is called sub- (super-) martingale. For
simplicity, we always suppose that for a martingale f we have in the one-parameter
case f0 = 0 and in the two-parameter case fn = 0 if nl = 0 or n2 = 0. Of course, the
theorems that are to be proved later are true with a slightly modification without
this condition, too.
The stochastic basis ~- is said to be regular if there exists a number R > 0
such that
f. _< Rfn-~ (n (cid:12)9 N)
(for one index) and
f~l,n2 < Rfn,-1,~2, f.,,n2 -< Rfm,.2-1 (n (cid:12)9 N 2)
(for two indices) hold for all non-negative martingales (f.,n (cid:12)9 N j) (j = 1, 2). Some
examples for a regular sequence of a-algebras are given in Section 1.2.
The martingale f = (fn, n (cid:12)9 NJ) (j = 1, 2) is said to be Z -bounaed (0 < p _<
co) if f~ (cid:12)9 Lp (n (cid:12)9 NJ) and
flip := sup f~lp < co.
nENi
Iff (cid:12)9 L1 then it is easy to show that the sequence =~(Enf, n (cid:12)9 N j) (j = 1,2)
is a martingale. Moreover, if I < p < ~ and f (cid:12)9 Lp then f is Lp-bounded and
(1.1) lirn IIE, f - flip = o,
II/Ib
consequently, = Ilflb (see Neveu 142). In this book both the a.e. and the Lp
limit of a two-parameter sequence (zn, n (cid:12)9 N 2) axe taken in Pringsheim sense (see
e.g. M6ricz 137), namely, the a.e. resp. Lp limit of the sequence (z~) is z if for
IIz.
all e > 0 there exists an index N (cid:12)9 N 2 such that Izn - z I < e resp. -- zllp <,
whenever n > N (0 < p < oc). The converse of the lattest proposition holds also
if 1 < p < cr (see Neveu 142): for an arbitrary martingale f = (fn,n (cid:12)9 N j)
(j = 1, 2) there exists a function g (cid:12)9 Lp for which f, = E,g if and only if f is
Lp-bounded. If p = 1 then there exists a function g (cid:12)9 L1 of the preceding type if
and only if f is uniformly integrable (Neveu 142), namely, if
lim sup / If, l dP = .O
Y~ -eN, J{If-I>Y}
This representation of uniformly integrable martingales is valid for martingales in-
dexed with a directed index set, too (see Neveu 142).
Note that in case f (cid:12)9 Lp (1 < p < oo) besides the Lp convergence in (1.1)
the conditional expectation Enf converges also a.e. to f (n (cid:12)9 N j, j = 1,2). If
p = 1 then in the one-parameter case every Ll-bounded martingale converges a.e.
(but, of course, not necessarily in L1 norm), and in the two-paxarneter case every
Ll-bounded martingale converges in probability (Neveu 142).
Thus the map f +-~ f := (E,f,n (cid:12)9 N j) (j = 1,2) is isometric from Lp onto the
space of Lp-bounded martingales when 1 < p < ~a. Consequently, these two spaces
can be identified with each other. Similarly, the L1 space can be identified with the
space of uniformly integrable martingales. For this reason a function f (cid:12)9 L1 and
the corresponding martingale (Enf, n (cid:12)9 N j) (j = 1,2) will be denoted by the same
symbol f.
The concept of a stopping time will be of primary importance in the book. In
the one-parameter case a map v : ~f ----+ N U {oo} is called a stopping time relative
to (5",, n (cid:12)9 N) if
It is well known that the last condition is equivalent to the conditions
{v <_ n} e .7". (n e N)
and
V{ ___~ rt} (cid:12)9 1--n"~, (n e N).
Keeping these properties we generalize the concept of a stopping time for two
parameters (see Weisz 199). A function ~ which maps a into the set of subspaces
of N 2 U {or is said to be a stopping time relative to ()t'n, n E N 2) if the elements
of u(w) are incomparable for all w E f~, furthermore, if for an arbitrary n E N 2
{~ e a.~ e ~(~)} =: {~ c ~} e ~-..