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Graduate Texts in Mathematics
1 TAKEUTI /Z ARING. Introduction to Axiomatic
Set Theory. 2nd ed.
2 O XTOBY. Measure and Category. 2nd ed.
3 S CHAEFER . Topological Vector Spaces.
2nd ed.
4 H ILTON/S TAMMBACH. A Course in
Homological Algebra. 2nd ed.
5 M AC L ANE. Categories for the Working
Mathematician. 2nd ed.
6 H UGHES /P IPER. Projective Planes.
7 J.-P. S ERRE. A Course in Arithmetic.
8 TAKEUTI /Z ARING. Axiomatic Set Theory.
9 H UMPHREYS . Introduction to Lie Algebras
and Representation Theory.
10 C OHEN. A Course in Simple Homotopy
Theory.
11 C ONWAY. Functions of One Complex
Variable I. 2nd ed.
12 B EALS . Advanced Mathematical Analysis.
13 A NDERSON/F ULLER. Rings and Categories
of Modules. 2nd ed.
14 G OLUBITSKY/G UILLEMIN. Stable Mappings
and Their Singularities.
15 B ERBERIAN. Lectures in Functional Analysis
and Operator Theory.
16 W INTER. The Structure of Fields.
17 ROSENBLATT. Random Processes. 2nd ed.
18 H ALMOS . Measure Theory.
19 H ALMOS . A Hilbert Space Problem Book.
2nd ed.
20 H USEMOLLER. Fibre Bundles. 3rd ed.
21 H UMPHREYS . Linear Algebraic Groups.
22 BARNES /M ACK. An Algebraic Introduction
to Mathematical Logic.
23 G REUB. Linear Algebra. 4th ed.
24 H OLMES . Geometric Functional Analysis and
Its Applications.
25 H EWITT/S TROMBERG. Real and Abstract
Analysis.
26 M ANES . Algebraic Theories.
27 K ELLEY. General Topology.
28 Z ARISKI /S AMUEL. CommutativeAlgebra.
Vol. I.
29 Z ARISKI /S AMUEL. Commutative Algebra.
Vol. II.
30 JACOBSON. Lectures in Abstract Algebra I.
Basic Concepts.
31 JACOBSON. Lectures in Abstract Algebra II.
Linear Algebra.
32 JACOBSON. Lectures in Abstract Algebra III.
Theory of Fields and Galois Theory.
33 H IRSCH. Differential Topology.
34 S PITZER. Principles of Random Walk. 2nd ed.
35 A LEXANDER/W ERMER. Several Complex
Variables and Banach Algebras. 3rd ed.
36 K ELLEY/N AMIOKA ET AL. Linear
Topological Spaces.
37 M ONK. Mathematical Logic.
38 G RAUERT/F RITZSCHE. Several Complex
Variables.
39 A RVESON. An Invitation to C-Algebras.
40 K EMENY/S NELL /K NAPP. Denumerable
Markov Chains. 2nd ed.
41 A POSTOL. Modular Functions and Dirichlet
Series in Number Theory. 2nd ed.
42 J.-P. S ERRE. Linear Representations of Finite
Groups.
43 G ILLMAN/J ERISON. Rings of Continuous
Functions.
44 K ENDIG. Elementary Algebraic Geometry.
45 L O E` VE. Probability Theory I. 4th ed.
46 L O E` VE. Probability Theory II. 4th ed.
47 M OISE. Geometric Topology in Dimensions 2
and 3.
48 S ACHS /W U. General Relativity for
Mathematicians.
49 G RUENBERG/W EIR. Linear Geometry.
2nd ed.
50 E DWARDS . Fermat’s Last Theorem.
51 K LINGENBERG. A Course in Differential
Geometry.
52 H ARTSHORNE. Algebraic Geometry.
53 M ANIN. A Course in Mathematical Logic.
54 G RAVER/WATKINS. Combinatorics with
Emphasis on the Theory of Graphs.
55 B ROWN/P EARCY. Introduction to Operator
Theory I: Elements of Functional Analysis.
56 M ASSEY. Algebraic Topology: An
Introduction.
57 C ROWELL/F OX. Introduction to Knot Theory.
58 K OBLITZ. p-adic Numbers, p-adic Analysis,
and Zeta-Functions. 2nd ed.
59 L ANG. Cyclotomic Fields.
60 A RNOLD. Mathematical Methods in Classical
Mechanics. 2nd ed.
61 W HITEHEAD. Elements of Homotopy Theory.
62 K ARGAPOLOV/M ERIZJAKOV. Fundamentals
of the Theory of Groups.
63 B OLLOBAS . Graph Theory.
64 E DWARDS . Fourier Series. Vol. I. 2nd ed.
65 W ELLS . Differential Analysis on Complex
Manifolds. 2nd ed.
66 WATERHOUSE. Introduction to Affine Group
Schemes.
67 S ERRE. Local Fields.
68 W EIDMANN. Linear Operators in Hilbert
Spaces.
69 L ANG. Cyclotomic Fields II.
70 M ASSEY. Singular Homology Theory.
71 FARKAS /K RA. Riemann Surfaces. 2nd ed.
72 S TILLWELL. Classical Topology and
Combinatorial Group Theory. 2nd ed.
73 H UNGERFORD . Algebra.
74 D AVENPORT. Multiplicative Number Theory.
3rd ed.
(continued after index)
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Loukas Grafakos
Classical Fourier Analysis
Second Edition
123
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Loukas Grafakos
Department of Mathematics
University of Missouri
Columbia, MO 65211
USA
Editorial Board
S. Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132
USA
K.A. Ribet
Mathematics Department
University of California at Berkeley
Berkeley, CA 94720-3840
USA
ISSN: 0072-5285
ISBN: 978-0-387-09431-1
e-ISBN: 978-0-387-09432-8
DOI: 10.1007/978-0-387-09432-8
Library of Congress Control Number: 2008933456
Mathematics Subject Classification (2000): 42-xx 42-02
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To Suzanne
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Preface
The great response to the publication of the book Classical and Modern Fourier
Analysis has been very gratifying. I am delighted that Springer has offered to publish
the second edition of this book in two volumes: Classical Fourier Analysis, 2nd
Edition, and Modern Fourier Analysis, 2nd Edition.
These volumes are mainly addressed to graduate students who wish to study
Fourier analysis. This first volume is intended to serve as a text for a one-semester
course in the subject. The prerequisite for understanding the material herein is satisfactory completion of courses in measure theory, Lebesgue integration, and complex
variables.
The details included in the proofs make the exposition longer. Although it will
behoove many readers to skim through the more technical aspects of the presentation and concentrate on the flow of ideas, the fact that details are present will be
comforting to some. The exercises at the end of each section enrich the material
of the corresponding section and provide an opportunity to develop additional intuition and deeper comprehension. The historical notes of each chapter are intended to
provide an account of past research but also to suggest directions for further investigation. The appendix includes miscellaneous auxiliary material needed throughout
the text.
A web site for the book is maintained at
/>I am solely responsible for any misprints, mistakes, and historical omissions in
this book. Please contact me directly () if you have corrections, comments, suggestions for improvements, or questions.
Columbia, Missouri,
April 2008
Loukas Grafakos
vii
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Acknowledgments
I am very fortunate that several people have pointed out errors, misprints, and omissions in the first edition of this book. Others have clarified issues I raised concerning
the material it contains. All these individuals have provided me with invaluable help
that resulted in the improved exposition of the present second edition. For these
reasons, I would like to express my deep appreciation and sincere gratitude to the
following people:
Marco Annoni, Pascal Auscher, Andrew Bailey, Dmitriy Bilyk, Marcin Bownik,
Leonardo Colzani, Simon Cowell, Mita Das, Geoffrey Diestel, Yong Ding, Jacek
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Ana Jim´enez del Toro, John Kahl, Cornelia Kaiser, Nigel Kalton, Kim Jin Myong, Doowon Koh, Elena Koutcherik, Enrico Laeng, Sungyun Lee, Qifan Li, ChinCheng Lin, Liguang Liu, Stig-Olof Londen, Diego Maldonado, Jos´e Mar´ıa Martell,
Mieczyslaw Mastylo, Parasar Mohanty, Carlo Morpurgo, Andrew Morris, Mihail
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Carlos P´erez, Humberto Rafeiro, Maria Carmen Reguera Rodr´ıguez, Alexander
Samborskiy, Andreas Seeger, Steven Senger, Sumi Seo, Christopher Shane, Shu
Shen, Yoshihiro Sawano, Vladimir Stepanov, Erin Terwilleger, Rodolfo Torres,
Suzanne Tourville, Ignacio Uriarte-Tuero, Kunyang Wang, Huoxiong Wu, Takashi
Yamamoto, and Dachun Yang.
For their valuable suggestions, corrections, and other important assistance at different stages in the preparation of the first edition of this book, I would like to offer
my deepest gratitude to the following individuals:
Georges Alexopoulos, Nakhl´e Asmar, Bruno Calado, Carmen Chicone, David
Cramer, Geoffrey Diestel, Jakub Duda, Brenda Frazier, Derrick Hart, Mark Hoffmann, Steven Hofmann, Helge Holden, Brian Hollenbeck, Petr Honz´ık, Alexander
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ix
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Acknowledgments
Roman Shvidkoy, Elias Stein, Atanas Stefanov, Terence Tao, Erin Terwilleger,
Christoph Thiele, Rodolfo Torres, Deanie Tourville, Nikolaos Tzirakis Don Vaught,
Igor Verbitsky, Brett Wick, James Wright, and Linqiao Zhao.
I would also like to thank all reviewers who provided me with an abundance
of meaningful remarks, corrections, and suggestions for improvements. Finally, I
would like to thank Springer editor Mark Spencer, Springer’s digital product support
personnel Frank Ganz and Frank McGuckin, and copyeditor David Kramer for their
invaluable assistance during the preparation of this edition.
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Contents
1
L p Spaces and Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 L p and Weak L p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 The Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Convergence in Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 A First Glimpse at Interpolation . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Convolution and Approximate Identities . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Examples of Topological Groups . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Basic Convolution Inequalities . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Approximate Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Real Method: The Marcinkiewicz Interpolation Theorem . . .
1.3.2 Complex Method: The Riesz–Thorin Interpolation Theorem . .
1.3.3 Interpolation of Analytic Families of Operators . . . . . . . . . . .
1.3.4 Proofs of Lemmas 1.3.5 and 1.3.8 . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Decreasing Rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Duals of Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4 The Off-Diagonal Marcinkiewicz Interpolation Theorem . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
5
8
10
16
16
18
19
24
28
30
31
34
37
39
42
44
44
48
51
55
63
2
Maximal Functions, Fourier Transform, and Distributions . . . . . . . . . .
2.1 Maximal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The Hardy–Littlewood Maximal Operator . . . . . . . . . . . . . . .
2.1.2 Control of Other Maximal Operators . . . . . . . . . . . . . . . . . . . .
2.1.3 Applications to Differentiation Theory . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
78
78
82
85
89
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Contents
2.2
2.3
2.4
2.5
2.6
3
The Schwartz Class and the Fourier Transform . . . . . . . . . . . . . . . . . . 94
2.2.1 The Class of Schwartz Functions . . . . . . . . . . . . . . . . . . . . . . . 95
2.2.2 The Fourier Transform of a Schwartz Function . . . . . . . . . . . 98
2.2.3 The Inverse Fourier Transform and Fourier Inversion . . . . . . 102
2.2.4 The Fourier Transform on L1 + L2 . . . . . . . . . . . . . . . . . . . . . . 103
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
The Class of Tempered Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.3.1 Spaces of Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.3.2 Spaces of Functionals on Test Functions . . . . . . . . . . . . . . . . . 110
2.3.3 The Space of Tempered Distributions . . . . . . . . . . . . . . . . . . . 112
2.3.4 The Space of Tempered Distributions Modulo Polynomials . . 121
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
More About Distributions and the Fourier Transform . . . . . . . . . . . . . 124
2.4.1 Distributions Supported at a Point . . . . . . . . . . . . . . . . . . . . . . 124
2.4.2 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
2.4.3 Homogeneous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Convolution Operators on L p Spaces and Multipliers . . . . . . . . . . . . . 135
2.5.1 Operators That Commute with Translations . . . . . . . . . . . . . . 135
2.5.2 The Transpose and the Adjoint of a Linear Operator . . . . . . . 138
2.5.3 The Spaces M p,q (Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.5.4 Characterizations of M 1,1 (Rn ) and M 2,2 (Rn ) . . . . . . . . . . . . 141
2.5.5 The Space of Fourier Multipliers M p (Rn ) . . . . . . . . . . . . . . . 143
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Oscillatory Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
2.6.1 Phases with No Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . 149
2.6.2 Sublevel Set Estimates and the Van der Corput Lemma . . . . . 151
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Fourier Analysis on the Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
3.1 Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
3.1.1 The n-Torus Tn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.1.2 Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
3.1.3 The Dirichlet and Fej´er Kernels . . . . . . . . . . . . . . . . . . . . . . . . 165
3.1.4 Reproduction of Functions from Their Fourier Coefficients . . 168
3.1.5 The Poisson Summation Formula . . . . . . . . . . . . . . . . . . . . . . . 171
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3.2 Decay of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.2.1 Decay of Fourier Coefficients of Arbitrary Integrable
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.2.2 Decay of Fourier Coefficients of Smooth Functions . . . . . . . . 179
3.2.3 Functions with Absolutely Summable Fourier Coefficients . . 183
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
3.3 Pointwise Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 186
3.3.1 Pointwise Convergence of the Fej´er Means . . . . . . . . . . . . . . . 186
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3.3.2
3.3.3
3.3.4
3.4
3.5
3.6
3.7
4
Almost Everywhere Convergence of the Fej´er Means . . . . . . 188
Pointwise Divergence of the Dirichlet Means . . . . . . . . . . . . . 191
Pointwise Convergence of the Dirichlet Means . . . . . . . . . . . 192
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Divergence of Fourier and Bochner–Riesz Summability . . . . . . . . . . . . . 195
3.4.1 Motivation for Bochner–Riesz Summability . . . . . . . . . . . . . . 195
3.4.2 Divergence of Fourier Series of Integrable Functions . . . . . . 198
3.4.3 Divergence of Bochner–Riesz Means of Integrable
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
The Conjugate Function and Convergence in Norm . . . . . . . . . . . . . . 211
3.5.1 Equivalent Formulations of Convergence in Norm . . . . . . . . . 211
3.5.2 The L p Boundedness of the Conjugate Function . . . . . . . . . . . 215
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Multipliers, Transference, and Almost Everywhere Convergence . . . . 220
3.6.1 Multipliers on the Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
3.6.2 Transference of Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
3.6.3 Applications of Transference . . . . . . . . . . . . . . . . . . . . . . . . . . 228
3.6.4 Transference of Maximal Multipliers . . . . . . . . . . . . . . . . . . . . 228
3.6.5 Transference and Almost Everywhere Convergence . . . . . . . 232
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Lacunary Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
3.7.1 Definition and Basic Properties of Lacunary Series . . . . . . . . 238
3.7.2 Equivalence of L p Norms of Lacunary Series . . . . . . . . . . . . . 240
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Singular Integrals of Convolution Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
4.1 The Hilbert Transform and the Riesz Transforms . . . . . . . . . . . . . . . . 249
4.1.1 Definition and Basic Properties of the Hilbert Transform . . . 250
4.1.2 Connections with Analytic Functions . . . . . . . . . . . . . . . . . . . 253
4.1.3 L p Boundedness of the Hilbert Transform . . . . . . . . . . . . . . . . 255
4.1.4 The Riesz Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
4.2 Homogeneous Singular Integrals and the Method of Rotations . . . . . 267
4.2.1 Homogeneous Singular and Maximal Singular Integrals . . . . 267
4.2.2 L2 Boundedness of Homogeneous Singular Integrals . . . . . . 269
4.2.3 The Method of Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
4.2.4 Singular Integrals with Even Kernels . . . . . . . . . . . . . . . . . . . . 274
4.2.5 Maximal Singular Integrals with Even Kernels . . . . . . . . . . . . 278
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
´
4.3 The Calderon–Zygmund
Decomposition and Singular Integrals . . . . 286
4.3.1 The Calder´on–Zygmund Decomposition . . . . . . . . . . . . . . . . . 286
4.3.2 General Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
4.3.3 Lr Boundedness Implies Weak Type (1, 1) Boundedness . . . 290
4.3.4 Discussion on Maximal Singular Integrals . . . . . . . . . . . . . . . 293
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Contents
4.3.5
Boundedness for Maximal Singular Integrals Implies
Weak Type (1, 1) Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . 297
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
4.4 Sufficient Conditions for L p Boundedness . . . . . . . . . . . . . . . . . . . . . . 305
4.4.1 Sufficient Conditions for L p Boundedness of Singular
Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
4.4.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
4.4.3 Necessity of the Cancellation Condition . . . . . . . . . . . . . . . . . 309
4.4.4 Sufficient Conditions for L p Boundedness of Maximal
Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
4.5 Vector-Valued Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
4.5.1 ℓ2 -Valued Extensions of Linear Operators . . . . . . . . . . . . . . . . 316
4.5.2 Applications and ℓ r-Valued Extensions of Linear Operators . . 319
4.5.3 General Banach-Valued Extensions . . . . . . . . . . . . . . . . . . . . . 321
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
4.6 Vector-Valued Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
4.6.1 Banach-Valued Singular Integral Operators . . . . . . . . . . . . . . 329
4.6.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
4.6.3 Vector-Valued Estimates for Maximal Functions . . . . . . . . . . 334
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
5
Littlewood–Paley Theory and Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . 341
5.1 Littlewood–Paley Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
5.1.1 The Littlewood–Paley Theorem . . . . . . . . . . . . . . . . . . . . . . . . 342
5.1.2 Vector-Valued Analogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
5.1.3 L p Estimates for Square Functions Associated with Dyadic
Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
5.1.4 Lack of Orthogonality on L p . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
5.2 Two Multiplier Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
5.2.1 The Marcinkiewicz Multiplier Theorem on R . . . . . . . . . . . . . 360
5.2.2 The Marcinkiewicz Multiplier Theorem on Rn . . . . . . . . . . . . 363
5.2.3 The HăormanderMihlin Multiplier Theorem on Rn . . . . . . . . 366
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
5.3 Applications of Littlewood–Paley Theory . . . . . . . . . . . . . . . . . . . . . . 373
5.3.1 Estimates for Maximal Operators . . . . . . . . . . . . . . . . . . . . . . . 373
5.3.2 Estimates for Singular Integrals with Rough Kernels . . . . . . . 375
5.3.3 An Almost Orthogonality Principle on L p . . . . . . . . . . . . . . . . 379
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
5.4 The Haar System, Conditional Expectation, and Martingales . . . . . . 383
5.4.1 Conditional Expectation and Dyadic Martingale Differences . . 384
5.4.2 Relation Between Dyadic Martingale Differences and
Haar Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
5.4.3 The Dyadic Martingale Square Function . . . . . . . . . . . . . . . . . 388
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5.4.4
Almost Orthogonality Between the Littlewood–Paley
Operators and the Dyadic Martingale Difference Operators . . . 391
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
5.5 The Spherical Maximal Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
5.5.1 Introduction of the Spherical Maximal Function . . . . . . . . . . 395
5.5.2 The First Key Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
5.5.3 The Second Key Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
5.5.4 Completion of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
5.6 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
5.6.1 Some Preliminary Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
5.6.2 Construction of a Nonsmooth Wavelet . . . . . . . . . . . . . . . . . . . 404
5.6.3 Construction of a Smooth Wavelet . . . . . . . . . . . . . . . . . . . . . . 406
5.6.4 A Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
A
Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
A.1 A Useful Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
A.2 Definitions of Γ (z) and B(z, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
A.3 Volume of the Unit Ball and Surface of the Unit Sphere . . . . . . . . . . . 418
A.4 Computation of Integrals Using Gamma Functions . . . . . . . . . . . . . . . 419
A.5 Meromorphic Extensions of B(z, w) and Γ (z) . . . . . . . . . . . . . . . . . . . 420
A.6 Asymptotics of Γ (x) as x → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
A.7 Euler’s Limit Formula for the Gamma Function . . . . . . . . . . . . . . . . . 421
A.8 Reflection and Duplication Formulas for the Gamma Function . . . . . 424
B
Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
B.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
B.2 Some Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
B.3 An Interesting Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
B.4 The Fourier Transform of Surface Measure on Sn−1 . . . . . . . . . . . . . . 428
B.5 The Fourier Transform of a Radial Function on Rn . . . . . . . . . . . . . . . 428
B.6 Bessel Functions of Small Arguments . . . . . . . . . . . . . . . . . . . . . . . . . 429
B.7 Bessel Functions of Large Arguments . . . . . . . . . . . . . . . . . . . . . . . . . 430
B.8 Asymptotics of Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
C
Rademacher Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
C.1 Definition of the Rademacher Functions . . . . . . . . . . . . . . . . . . . . . . . . 435
C.2 Khintchine’s Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
C.3 Derivation of Khintchine’s Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 436
C.4 Khintchine’s Inequalities for Weak Type Spaces . . . . . . . . . . . . . . . . . 438
C.5 Extension to Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
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Contents
D
Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
D.1 Spherical Coordinate Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
D.2 A Useful Change of Variables Formula . . . . . . . . . . . . . . . . . . . . . . . . 441
D.3 Computation of an Integral over the Sphere . . . . . . . . . . . . . . . . . . . . . 442
D.4 The Computation of Another Integral over the Sphere . . . . . . . . . . . . 443
D.5 Integration over a General Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
D.6 The Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
E
Some Trigonometric Identities and Inequalities . . . . . . . . . . . . . . . . . . . . 447
F
Summation by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
G
Basic Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
H
The Minimax Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
I
The Schur Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
I.1 The Classical Schur Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
I.2 Schur’s Lemma for Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . 457
I.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
J
The Whitney Decomposition of Open Sets in Rn . . . . . . . . . . . . . . . . . . . 463
K
Smoothness and Vanishing Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
K.1 The Case of No Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
K.2 The Case of Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
K.3 The Case of Three Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
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Chapter 1
L p Spaces and Interpolation
Many quantitative properties of functions are expressed in terms of their integrability to a power. For this reason it is desirable to acquire a good understanding
of spaces of functions whose modulus to a power p is integrable. These are called
Lebesgue spaces and are denoted by L p . Although an in-depth study of Lebesgue
spaces falls outside the scope of this book, it seems appropriate to devote a chapter
to reviewing some of their fundamental properties.
The emphasis of this review is basic interpolation between Lebesgue spaces.
Many problems in Fourier analysis concern boundedness of operators on Lebesgue
spaces, and interpolation provides a framework that often simplifies this study. For
instance, in order to show that a linear operator maps L p to itself for all 1 < p < ∞,
it is sufficient to show that it maps the (smaller) Lorentz space L p,1 into the (larger)
Lorentz space L p,∞ for the same range of p’s. Moreover, some further reductions can
be made in terms of the Lorentz space L p,1 . This and other considerations indicate
that interpolation is a powerful tool in the study of boundedness of operators.
Although we are mainly concerned with L p subspaces of Euclidean spaces, we
discuss in this chapter L p spaces of arbitrary measure spaces, since they represent a
useful general setting. Many results in the text require working with general measures instead of Lebesgue measure.
1.1 L p and Weak L p
Let X be a measure space and let µ be a positive, not necessarily finite, measure
on X. For 0 < p < ∞, L p (X, µ) denotes the set of all complex-valued µ-measurable
functions on X whose modulus to the pth power is integrable. L∞ (X, µ) is the set
of all complex-valued µ-measurable functions f on X such that for some B > 0, the
set {x : | f (x)| > B} has µ-measure zero. Two functions in L p (X, µ) are considered
equal if they are equal µ-almost everywhere. The notation L p (Rn ) is reserved for
the space L p (Rn , |·|), where |·| denotes n-dimensional Lebesgue measure. Lebesgue
measure on Rn is also denoted by dx. Within context and in the absence of ambi-
L. Grafakos, Classical Fourier Analysis, Second Edition,
DOI: 10.1007/978-0-387-09432-8_1, © Springer Science+Business Media, LLC 2008
1
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1 L p Spaces and Interpolation
2
guity, L p (X, µ) is simply written as L p . The space L p (Z) equipped with counting
measure is denoted by ℓ p (Z) or simply ℓ p .
For 0 < p < ∞, we define the L p quasinorm of a function f by
f
L p (X,µ)
1
p
p
=
X
| f (x)| dµ(x)
(1.1.1)
and for p = ∞ by
f
L∞ (X,µ)
= ess.sup | f | = inf B > 0 : µ({x : | f (x)| > B}) = 0 .
(1.1.2)
It is well known that Minkowski’s (or the triangle) inequality
f +g
L p (X,µ)
≤ f
L p (X,µ)
+ g
(1.1.3)
L p (X,µ)
holds for all f , g in L p = L p (X, µ), whenever 1 ≤ p ≤ ∞. Since in addition
f L p (X,µ) = 0 implies that f = 0 (µ-a.e.), the L p spaces are normed linear spaces
for 1 ≤ p ≤ ∞. For 0 < p < 1, inequality (1.1.3) is reversed when f , g ≥ 0. However,
the following substitute of (1.1.3) holds:
f +g
L p (X,µ)
≤ 2(1−p)/p
f
L p (X,µ)
+ g
L p (X,µ)
,
(1.1.4)
and thus L p (X, µ) is a quasinormed linear space. See also Exercise 1.1.5. For all
0 < p ≤ ∞, it can be shown that every Cauchy sequence in L p (X, µ) is convergent,
and hence the spaces L p (X, µ) are complete. For the case 0 < p < 1 we refer to
Exercise 1.1.8. Therefore, the L p spaces are Banach spaces for 1 ≤ p ≤ ∞ and quasip
.
Banach spaces for 0 < p < 1. For any p ∈ (0, ∞) \ {1} we use the notation p′ = p−1
Moreover, we set 1′ = ∞ and ∞′ = 1, so that p′′ = p for all p ∈ (0, ]. Hăolders
inequality says that for all p [1, ∞] and all measurable functions f , g on (X, µ) we
have
f g L1 ≤ f L p g L p′ .
′
It is a well-known fact that the dual (L p )∗ of L p is isometric to L p for all 1 ≤ p < ∞.
Furthermore, the L p norm of a function can be obtained via duality when 1 ≤ p ≤ ∞
as follows:
f
Lp
=
f g dµ .
sup
g
′ =1
Lp
X
For the endpoint cases p = 1, p = ∞, see Exercise 1.4.12(a), (b).
1.1.1 The Distribution Function
Definition 1.1.1. For f a measurable function on X, the distribution function of f is
the function d f defined on [0, ∞) as follows:
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1.1 L p and Weak L p
3
d f (α) = µ({x ∈ X : | f (x)| > α}) .
(1.1.5)
The distribution function d f provides information about the size of f but not
about the behavior of f itself near any given point. For instance, a function on Rn and
each of its translates have the same distribution function. It follows from Definition
1.1.1 that d f is a decreasing function of α (not necessarily strictly).
df (α)
f(x)
a1
B3
a2
.
B2
.
a3
B1
0
E3
E1
E2
0
x
. .
a3
a2
a1
α
Fig. 1.1 The graph of a simple function f = ∑3k=1 ak χEk and its distribution function d f (α). Here
j
µ(Ek ).
B j = ∑k=1
Example 1.1.2. Recall that simple functions are finite linear combinations of characteristic functions of sets of finite measure. For pedagogical reasons we compute
the distribution function d f of a nonnegative simple function
N
f (x) =
∑ a j χE j (x) ,
j=1
where the sets E j are pairwise disjoint and a1 > · · · > aN > 0. If α ≥ a1 , then clearly
d f (α) = 0. However, if a2 ≤ α < a1 then | f (x)| > α precisely when x ∈ E1 , and in
general, if a j+1 ≤ α < a j , then | f (x)| > α precisely when x ∈ E1 ∪ · · · ∪ E j . Setting
j
Bj =
∑ µ(Ek ) ,
k=1
we have
N
d f (α) =
∑ B j χ[a j+1 ,a j ) (α) ,
j=0
where a0 = ∞ and B0 = aN+1 = 0. Figure 1.1 illustrates this example when N = 3.
We now state a few simple facts about the distribution function d f .
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1 L p Spaces and Interpolation
4
Proposition 1.1.3. Let f and g be measurable functions on (X, µ). Then for all
α, β > 0 we have
(1) |g| ≤ | f | µ-a.e. implies that dg ≤ d f ;
(2) dc f (α) = d f (α/|c|), for all c ∈ C \ {0};
(3) d f +g (α + β ) ≤ d f (α) + dg (β );
(4) d f g (αβ ) ≤ d f (α) + dg (β ).
Proof. The simple proofs are left to the reader.
Knowledge of the distribution function d f provides sufficient information to evaluate the L p norm of a function f precisely. We state and prove the following important description of the L p norm in terms of the distribution function.
Proposition 1.1.4. For f in L p (X, µ), 0 < p < ∞, we have
f
p
Lp
∞
=p
0
α p−1 d f (α) dα .
(1.1.6)
Proof. Indeed, we have
∞
p
0
∞
α p−1 d f (α) dα = p
α p−1
0
X
| f (x)|
=
χ{x: | f (x)|>α} dµ(x) dα
pα p−1 dα dµ(x)
X 0
=
X
= f
| f (x)| p dµ(x)
p
,
Lp
where we used Fubini’s theorem in the second equality. This proves (1.1.6).
Notice that the same argument yields the more general fact that for any increasing
continuously differentiable function ϕ on [0, ∞) with ϕ(0) = 0 we have
∞
X
ϕ(| f |) dµ =
0
ϕ ′ (α)d f (α) dα .
(1.1.7)
Definition 1.1.5. For 0 < p < ∞, the space weak L p (X, µ) is defined as the set of
all µ-measurable functions f such that
f
L p,∞
Cp
αp
1/p
= sup γ d f (γ) : γ > 0
= inf C > 0 : d f (α) ≤
for all α > 0
(1.1.8)
(1.1.9)
is finite. The space weak-L∞ (X, µ) is by definition L∞ (X, µ).
The reader should check that (1.1.9) and (1.1.8) are in fact equal. The weak L p
spaces are denoted by L p,∞ (X, µ). Two functions in L p,∞ (X, µ) are considered equal
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1.1 L p and Weak L p
5
if they are equal µ-a.e. The notation L p,∞ (Rn ) is reserved for L p,∞ (Rn , | · |). Using
Proposition 1.1.3 (2), we can easily show that
kf
L p,∞
= |k| f
L p,∞
,
(1.1.10)
for any complex nonzero constant k. The analogue of (1.1.3) is
f +g
L p,∞
≤ cp
f
L p,∞
+ g
L p,∞
,
(1.1.11)
where c p = max(2, 21/p ), a fact that follows from Proposition 1.1.3 (3), taking both
α and β equal to α/2. We also have that
f
L p,∞ (X,µ)
=0⇒ f =0
µ-a.e.
(1.1.12)
In view of (1.1.10), (1.1.11), and (1.1.12), L p,∞ is a quasinormed linear space for
0 < p < ∞.
The weak L p spaces are larger than the usual L p spaces. We have the following:
Proposition 1.1.6. For any 0 < p < ∞ and any f in L p (X, µ) we have
f L p ; hence L p (X, µ) ⊆ L p,∞ (X, µ).
f
L p,∞
≤
Proof. This is just a trivial consequence of Chebyshev’s inequality:
α p d f (α) ≤
{x: | f (x)|>α}
The integral in (1.1.13) is at most f
f Lp .
p
Lp
| f (x)| p dµ(x) .
(1.1.13)
and using (1.1.9) we obtain that f
L p,∞
≤
The inclusion L p ⊆ L p,∞ is strict. For example, on Rn with the usual Lebesgue
n
measure, let h(x) = |x|− p . Obviously, h is not in L p (Rn ) but h is in L p,∞ (Rn ) with
h L p,∞ (Rn ) = vn , where vn is the measure of the unit ball of Rn .
It is not immediate from their definition that the weak L p spaces are complete
with respect to the quasinorm · L p,∞ . The completeness of these spaces is proved
in Theorem 1.4.11, but it is also a consequence of Theorem 1.1.13, proved in this
section.
1.1.2 Convergence in Measure
Next we discuss some convergence notions. The following notion is important in
probability theory.
Definition 1.1.7. Let f , fn , n = 1, 2, . . . , be measurable functions on the measure
space (X, µ). The sequence fn is said to converge in measure to f if for all ε > 0
there exists an n0 ∈ Z+ such that
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1 L p Spaces and Interpolation
6
n > n0 =⇒ µ({x ∈ X : | fn (x) − f (x)| > ε}) < ε .
(1.1.14)
Remark 1.1.8. The preceding definition is equivalent to the following statement:
For all ε > 0
lim µ({x ∈ X : | fn (x) − f (x)| > ε}) = 0 .
(1.1.15)
n→∞
Clearly (1.1.15) implies (1.1.14). To see the converse given ε > 0, pick 0 < δ < ε
and apply (1.1.14) for this δ . There exists an n0 ∈ Z+ such that
µ({x ∈ X : | fn (x) − f (x)| > δ }) < δ
holds for n > n0 . Since
µ({x ∈ X : | fn (x) − f (x)| > ε}) ≤ µ({x ∈ X : | fn (x) − f (x)| > δ }) ,
we conclude that
µ({x ∈ X : | fn (x) − f (x)| > ε}) < δ
for all n > n0 . Let n → ∞ to deduce that
lim sup µ({x ∈ X : | fn (x) − f (x)| > ε}) ≤ δ .
(1.1.16)
n→∞
Since (1.1.16) holds for all 0 < δ < ε, (1.1.15) follows by letting δ → 0.
Convergence in measure is a weaker notion than convergence in either L p or L p,∞ ,
0 < p ≤ ∞, as the following proposition indicates:
Proposition 1.1.9. Let 0 < p ≤ ∞ and fn , f be in L p,∞ (X, µ).
(1) If fn , f are in L p and fn → f in L p , then fn → f in L p,∞ .
(2) If fn → f in L p,∞ , then fn converges to f in measure.
Proof. Fix 0 < p < ∞. Proposition 1.1.6 gives that for all ε > 0 we have
µ({x ∈ X : | fn (x) − f (x)| > ε}) ≤
1
εp
X
| fn − f | p dµ .
This shows that convergence in L p implies convergence in weak L p . The case p = ∞
is tautological.
Given ε > 0 find an n0 such that for n > n0 , we have
fn − f
1
L p,∞
1
= sup α µ({x ∈ X : | fn (x) − f (x)| > α}) p < ε p +1 .
α>0
Taking α = ε, we conclude that convergence in L p,∞ implies convergence in measure.
Example 1.1.10. Fix 0 < p < ∞. On [0, 1] define the functions
fk, j = k1/p χ( j−1 , j ) ,
k
k
k ≥ 1, 1 ≤ j ≤ k.
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1.1 L p and Weak L p
7
Consider the sequence { f1,1 , f2,1 , f2,2 , f3,1 , f3,2 , f3,3 , . . .}. Observe that
|{x : fk, j (x) > 0}| = 1/k .
Therefore, fk, j converges to 0 in measure. Likewise, observe that
fk, j
L p,∞
(k − 1/k)1/p
= 1,
k1/p
k≥1
= sup α|{x : fk, j (x) > α}|1/p ≥ sup
α>0
which implies that fk, j does not converge to 0 in L p,∞ .
It turns out that every sequence convergent in L p (X, µ) or in L p,∞ (X, µ) has a
subsequence that converges a.e. to the same limit.
Theorem 1.1.11. Let fn and f be complex-valued measurable functions on a measure space (X, µ) and suppose that fn converges to f in measure. Then some subsequence of fn converges to f µ-a.e.
Proof. For all k = 1, 2, . . . choose inductively nk such that
µ({x ∈ X : | fnk (x) − f (x)| > 2−k }) < 2−k
(1.1.17)
and such that n1 < n2 < · · · < nk < · · · . Define the sets
Ak = {x ∈ X : | fnk (x) − f (x)| > 2−k } .
Equation (1.1.17) implies that
∞
µ
∞
∞
≤
Ak
k=m
∑ µ(Ak ) ≤ ∑ 2−k = 21−m
(1.1.18)
k=m
k=m
for all m = 1, 2, 3, . . . . It follows from (1.1.18) that
∞
µ
Ak
k=1
≤ 1 < ∞.
(1.1.19)
Using (1.1.18) and (1.1.19), we conclude that the sequence of the measures of the
∞
sets { ∞
k=m Ak }m=1 converges as m → ∞ to
∞
∞
µ
Ak
= 0.
(1.1.20)
m=1 k=m
To finish the proof, observe that the null set in (1.1.20) contains the set of all x ∈ X
for which fnk (x) does not converge to f (x).
In many situations we are given a sequence of functions and we would like to
extract a convergent subsequence. One way to achieve this is via the next theorem,
which is a useful variant of Theorem 1.1.11. We first give a relevant definition.
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1 L p Spaces and Interpolation
8
Definition 1.1.12. We say that a sequence of measurable functions { fn } on the measure space (X, µ) is Cauchy in measure if for every ε > 0, there exists an n0 ∈ Z+
such that for n, m > n0 we have
µ({x ∈ X : | fm (x) − fn (x)| > ε}) < ε.
Theorem 1.1.13. Let (X, µ) be a measure space and let { fn } be a complex-valued
sequence on X that is Cauchy in measure. Then some subsequence of fn converges
µ-a.e.
Proof. The proof is very similar to that of Theorem 1.1.11. For all k = 1, 2, . . .
choose nk inductively such that
µ({x ∈ X : | fnk (x) − fnk+1 (x)| > 2−k }) < 2−k
(1.1.21)
and such that n1 < n2 < · · · < nk < nk+1 < · · · . Define
Ak = {x ∈ X : | fnk (x) − fnk+1 (x)| > 2−k } .
As shown in the proof of Theorem 1.1.11, (1.1.21) implies that
∞
∞
µ
Ak
= 0.
(1.1.22)
m=1 k=m
For x ∈
/
∞
k=m Ak
and i ≥ j ≥ j0 ≥ m (and j0 large enough) we have
i−1
i−1
l= j
l= j
| fni (x) − fn j (x)| ≤ ∑ | fnl (x) − fnl+1 (x)| ≤ ∑ 2−l ≤ 21− j ≤ 21− j0 .
This implies that the sequence { fni (x)}i is Cauchy for every x in the set (
and therefore converges for all such x. We define a function
∞
lim fn (x) when x ∈
/ ∞
m=1 k=m Ak ,
j
j→∞
f (x) =
∞
0
when x ∈ ∞
m=1 k=m Ak .
∞
c
k=m Ak )
Then fn j → f almost everywhere.
1.1.3 A First Glimpse at Interpolation
It is a useful fact that if a function f is in L p (X, µ) and in Lq (X, µ), then it also lies
in Lr (X, µ) for all p < r < q. The usefulness of the spaces L p,∞ can be seen from
the following sharpening of this statement:
Proposition 1.1.14. Let 0 < p < q ≤ ∞ and let f in L p,∞ (X, µ) ∩ Lq,∞ (X, µ). Then
f is in Lr (X, µ) for all p < r < q and
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1.1 L p and Weak L p
9
f
Lr
≤
1−1
r q
1−1
p q
L p,∞
1
r
r
r
+
r− p q−r
f
f
1−1
p r
1−1
p q
Lq,∞
,
(1.1.23)
with the suitable interpretation when q = ∞.
Proof. Let us take first q < ∞. We know that
q
p
f L p,∞ f Lq,∞
,
αp
αq
d f (α) ≤ min
Set
q
Lq,∞
p
L p,∞
f
B=
f
.
(1.1.24)
1
q−p
.
(1.1.25)
We now estimate the Lr norm of f . By (1.1.24), (1.1.25), and Proposition 1.1.4 we
have
f
r
Lr (X,µ)
∞
=r
0
α r−1 d f (α) dα
≤r
α
r−1
min
0
B
=r
q
p
∞
α r−1−p f
0
f L p,∞ f Lq,∞
,
dα
αp
αq
p
dα + r
L p,∞
∞
α r−1−q f
B
r
r
p
q
f L p,∞ Br−p +
f Lq,∞ Br−q
r− p
q−r
q−r
r
r
p
q
=
+
f L p,∞ q−p f Lq,∞
r− p q−r
q
dα
Lq,∞
(1.1.26)
=
r−p
q−p
.
Observe that the integrals converge, since r − p > 0 and r − q < 0.
The case q = ∞ is easier. Since d f (α) = 0 for α > f L∞ we need to use only
p
the inequality d f (α) ≤ α −p f L p,∞ for α ≤ f L∞ in estimating the first integral in
(1.1.26). We obtain
r
r−p
p
r
f L p,∞ f L∞ ,
f Lr ≤
r− p
which is nothing other than (1.1.23) when q = ∞. This completes the proof.
Note that (1.1.23) holds with constant 1 if L p,∞ and Lq,∞ are replaced by L p and
respectively. It is often convenient to work with functions that are only locally
in some L p space. This leads to the following definition.
Lq ,
p
p
Definition 1.1.15. For 0 < p < ∞, the space Lloc
(Rn , | · |) or simply Lloc
(Rn ) is the
n
set of all Lebesgue-measurable functions f on R that satisfy
K
| f (x)| p dx < ∞
(1.1.27)
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1 L p Spaces and Interpolation
10
for any compact subset K of Rn . Functions that satisfy (1.1.27) with p = 1 are called
locally integrable functions on Rn .
1 (Rn ). More
The union of all L p (Rn ) spaces for 1 ≤ p ≤ ∞ is contained in Lloc
generally, for 0 < p < q < ∞ we have the following:
q
p
(Rn ) ⊆ Lloc
(Rn ) .
Lq (Rn ) ⊆ Lloc
Functions in L p (Rn ) for 0 < p < 1 may not be locally integrable. For example, take
f (x) = |x|−n−α χ|x|≤1 , which is in L p (Rn ) when p < n/(n + α), and observe that f
is not integrable over any open set in Rn containing the origin.
Exercises
1.1.1. Suppose f and fn are measurable functions on (X, µ). Prove that
(a) d f is right continuous on [0, ∞).
(b) If | f | ≤ lim infn→∞ | fn | µ-a.e., then d f ≤ lim infn→∞ d fn .
(c) If | fn | ↑ | f |, then d fn ↑ d f .
Hint: Part (a): Let tn be a decreasing sequence of positive numbers that tends to
zero. Show that d f (α0 + tn ) ↑ d f (α0 ) using a convergence theorem. Part (b): Let
E = {x ∈ X : | f (x)| > α} and En = {x ∈ X : | fn (x)| > α}. Use that µ ∞
n=m En ≤
∞
lim inf à(En ) and E
m=1 n=m En à-a.e.
n
1.1.2. (Hăolders inequality ) Let 0 < p, p1 , . . . , pk ≤ ∞, where k ≥ 2, and let f j be in
L p j = L p j (X, µ). Assume that
1
1
1
=
+···+ .
p
p1
pk
(a) Show that the product f1 · · · fk is in L p and that
f1 · · · fk
Lp
≤ f1
L p1
· · · fk
L pk
.
(b) When no p j is infinite, show that if equality holds in part (a), then it must be the
case that c1 | f1 | p1 = · · · = ck | fk | pk a.e. for some c j ≥ 0.
−1
(c) Let 0 < q < 1. For r < 0 and g > 0 almost everywhere, let g Lr = g−1 L|r| .
Show that for f ≥ 0, g > 0 a.e. we have
fg
L1
≥ f
Lq
g
′
Lq
1.1.3. Let (X, µ) be a measure space.
(a) If f is in L p0 (X, µ) for some p0 < ∞, prove that
lim f
p→∞
Lp
= f
L∞
.
.
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1.1 L p and Weak L p
11
(b) (Jensen’s inequality ) Suppose that µ(X) = 1. Show that
f
Lp
≥ exp
X
log | f (x)| dµ(x)
for all 0 < p < ∞.
(c) If µ(X) = 1 and f is in some L p0 (X, µ) for some p0 > 0, then
lim f
p→0
Lp
= exp
X
log | f (x)| dµ(x)
with the interpretation e−∞ = 0.
Hint: Part (a): Given 0 < ε < f L∞ , find a measurable set E ⊆ X of positive measure such that | f (x)| ≥ f L∞ − ε for all x ∈ E. Then f L p ≥ ( f L∞ − ε)µ(E)1/p
and thus lim inf p→∞ f L p ≥ f L∞ − ε. Part (b) is a direct consequence of Jensen’s
inequality X log |h| dµ ≤ log X |h| dµ . Part (c): Fix a sequence 0 < pn < p0 such
that pn ↓ 0 and define
hn (x) =
1
1
(| f (x)| p0 − 1) − (| f (x)| pn − 1).
p0
pn
Use that 1p (t p − 1) ↓ logt as p ↓ 0 for all t > 0. The Lebesgue monotone convergence
theorem yields X hn dµ ↑ X h dµ, hence X p1n (| f | pn − 1) dµ ↓ X log | f | dµ, where
the latter could be −∞. Use
exp
X
log | f | dµ
≤
1
pn
pn
X
| f | dµ
≤ exp
X
1
(| f | pn − 1) dµ
pn
to complete the proof.
1.1.4. Let a j be a sequence of positive reals. Show that
θ
(a) ∑∞j=1 a j ≤ ∑∞j=1 aθj , for any 0 ≤ θ ≤ 1.
(b) ∑∞j=1 aθj ≤ ∑∞j=1 a j
(c)
(d)
θ
, for any 1 ≤ θ < ∞.
θ
∑Nj=1 a j ≤ N θ −1 ∑Nj=1 aθj , when 1 ≤ θ
θ
∑Nj=1 aθj ≤ N 1−θ ∑Nj=1 a j , when 0 ≤ θ
< ∞.
≤ 1.
1.1.5. Let { f j }Nj=1 be a sequence of L p (X, µ) functions.
(a) (Minkowski’s inequality ) For 1 ≤ p ≤ ∞ show that
N
N
∑ fj
j=1
Lp
≤
∑
j=1
fj
Lp
.
(b) (Minkowski’s inequality ) For 0 < p < 1 and f j ≥ 0 prove that
