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UNIVERSIDAD COMPLUTENSE
Fourier
Analysis
Javier Duoandikoetxea
T
R
American Mathematical Society
Mil
45
Providence, Rhode Island
Translated and
revised
by
David Cruz-Uribe, SFO
Graduate Studies
in Mathematics
Volume 29
o
Editorial Board
James Humphreys (Chair)
David Saltman
David Sattinger
Ronald Stern
ANALISIS DE FOURIER
by Javier Duoandikoetxea Zuazo
Published in Spanish by Addison-Wesley and Universidad Aut6noma de Madrid in 1995
Translated from the Spanish by David Cruz-Uribe, SFO
2000
Mathematics Subject Classification.
Primary 42B15, 42B20, 42B25.
ABSTRACT.
The purpose of this book is to develop Fourier analysis using the real variable methods
introduced by A. P. Calder& and A. Zygmund. It begins by reviewing the theory of Fourier series
and integrals, and introduces the Hardy-Littlewood maximal function. It then treats the Hilbert
transform and its higher dimensional analogues, singular integrals. In subsequent chapters it
discusses some more recent topics: H
I
and
BMO,
weighted norm inequalities, Littlewood-Paley
theory, and the T1 theorem. At the end of each chapter are extensive references and notes on
additional results.
Library of Congress Cataloging-in-Publication Data
Duoandikoetxea, Zuazo, Javier.
[Analisis de Fourier.
English]
Fourier analysis / Javier Duoandikoetxea ; translated and revised by David Cruz-Uribe.
p. cm. — (Graduate studies in mathematics ; v. 29)
Includes bibliographical references and index.
ISBN 0-8218-2172-5
1. Fourier analysis. I. Title. II. Series.
QA403.5 .D8313 2000
515'.2433—dc21
00-064301
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Requests for such
permission should be addressed to the Assistant to the Publisher, American Mathematical Society,
P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to
reprint -permiss ion@ams . org.
© 2001 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at URL:
http:
//www.ams org/
10 9 8 7 6 5 4 3 2 1
06 05 04 03 02 01
Dedicated to the memory of
Jose Luis Rubio de Francia, my teacher and friend,
who would have written a much better book than I have
Contents
Preface
xiii
Preliminaries
xvii
Chapter 1. Fourier Series and Integrals
1
§1.
Fourier coefficients and series
1
§2.
Criteria for pointwise convergence
2
§3.
Fourier series of continuous functions
6
§4.
Convergence in norm
8
§5.
Summability methods
9
§6.
The Fourier transform of L' functions
11
§7.
The Schwartz class and tempered distributions
12
§8.
The Fourier transform on
LP,
1
p <
2
15
§9.
The convergence and summability of Fourier integrals
17
§10.
Notes and further results
19
Chapter 2. The Hardy-Littlewood Maximal Function
25
§1.
Approximations of the identity
25
§2.
Weak-type inequalities and almost everywhere convergence 26
§3.
The Marcinkiewicz interpolation theorem
28
§4.
The Hardy-Littlewood maximal function
30
ix
x
Contents
§5.
The dyadic maximal function
32
§6.
The weak (1,1) inequality for the maximal function
35
§7.
A weighted norm inequality
37
§8.
Notes and further results
38
Chapter 3.
The Hilbert Transform
49
§1.
The conjugate Poisson kernel
49
§2.
The principal value of 1/x
50
§3.
The theorems of M. Riesz and Kolmogorov
51
§4.
Truncated integrals and pointwise convergence
55
§5.
Multipliers
58
§6.
Notes and further results
61
Chapter 4.
Singular Integrals (I)
69
§1.
Definition and examples
69
§2.
The Fourier transform of the kernel
70
§3.
The method of rotations
73
§4.
Singular integrals with even kernel
77
§5.
An operator algebra
80
§6.
Singular integrals with variable kernel
83
§7.
Notes and further results
85
Chapter 5.
Singular Integrals (II)
91
§ 1.
The CalderOn-Zygmund theorem
91
§2.
Truncated integrals and the principal value
94
§3.
Generalized CalderOn-Zygmund operators
98
§4.
CalderOn-Zygmund singular integrals
101
§5.
A vector-valued extension
105
§6.
Notes and further results
107
Chapter 6.
H
I-
and
BMO
115
§1.
The space atomic 1/
1
115
§2.
The space
BMO
117
Contents
xi
§3.
An interpolation result
121
§4.
The John-Nirenberg inequality
123
§5.
Notes and further results
126
Chapter 7.
Weighted Inequalities
133
§1.
The A
P
condition
133
§2.
Strong-type inequalities with weights
137
§3.
Al weights and an extrapolation theorem
140
§4.
Weighted inequalities for singular integrals
143
§
5
.
Notes and further results
147
Chapter 8.
Littlewood-Paley Theory and Multipliers
157
§1.
Some vector-valued inequalities
157
§2.
Littlewood-Paley theory
159
§3.
The Hi5rmander multiplier theorem
163
§4.
The Marcinkiewicz multiplier theorem
166
§5.
Bochner-Riesz multipliers
168
§6.
Return to singular integrals
172
§7.
The maximal function and the Hilbert transform along a
parabola
178
§8.
Notes and further results
184
Chapter 9.
The Ti Theorem
195
§1.
Cotlar's lemma
195
§2.
Carleson measures
197
§3.
Statement and applications of the T1 theorem
201
§4.
Proof of the Ti theorem
205
§5.
Notes and further results
212
Bibliography
217
Index
219
1
Preface
Fourier Analysis is a large branch of mathematics whose point of departure is
the study of Fourier series and integrals. However, it encompasses a variety
of perspectives and techniques, and so many different introductions with
that title are possible. The goal of this book is to study the real variable
methods introduced into Fourier analysis by A. P. Calder& and A. Zygmund
in the 1950's.
We begin in Chapter 1 with a review of Fourier series and integrals,
and then in Chapters 2 and 3 we introduce two operators which are basic
to the field: the Hardy-Littlewood maximal function and the Hilbert trans-
form. Even though they appeared before the techniques of Calder& and
Zygmund, we treat these operators from their point of view. The goal of
these techniques is to enable the study of analogs of the Hilbert transform
in higher dimensions; these are of great interest in applications. Such oper-
ators are known as singular integrals and are discussed in Chapters 4 and 5
along with their modern generalizations. We next consider two of the many
contributions to the field which appeared in the 1970's. In Chapter 6 we
study the relationship between Hl
,
BMO
and singular integrals, and in
Chapter 7 we present the elementary theory of weighted norm inequalities.
In Chapter 8 we discuss Littlewood-Paley theory; its origins date back to the
1930's, but it has had extensive later development which includes a number
of applications. Those presented in this chapter are useful in the study of
Fourier multipliers, which also uses the theory of weighted inequalities. We
end the book with an important result of the 80's, the so-called T1 theorem,
which has been of crucial importance to the field.
At the end of each chapter there is a section in which we try to give
some idea of further results which are not discussed in the text, and give
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