Grafakos Modern Fourier Analysis

2. Auflage 2009
ISBN: 978-0-387-09434-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 507 Seiten

Reihe: Mathematics and Statistics

The great response to the publication of the book Classical and Modern Fourier Analysishasbeenverygratifying.IamdelightedthatSpringerhasofferedtopublish 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 second volume is intended to serve as a text for a seco- semester course in the subject. It is designed to be a continuation of the rst v- ume. Chapters 1-5 in the rst volume contain Lebesgue spaces, Lorentz spaces and interpolation, maximal functions, Fourier transforms and distributions, an introd- tion to Fourier analysis on the n-torus, singular integrals of convolution type, and Littlewood-Paley theory. Armed with the knowledgeof this material, in this volume,the reader encounters more advanced topics in Fourier analysis whose development has led to important theorems. These theorems are proved in great detail and their proofs are organized to present the ow of ideas. The exercises at the end of each section enrich the material of the corresponding section and provide an opportunity to develop ad- tional intuition and deeper comprehension. The historical notes in each chapter are intended to provide an account of past research but also to suggest directions for further investigation. The auxiliary results referred to the appendix can be located in the rst volume.

Autoren/Hrsg.

Grafakos, Loukas

Inhaltsverzeichnis

1;Preface;7
2;Acknowledgements;8
3;Contents;10
4;6 Smoothness and Function Spaces;15
4.1;6.1 Riesz Potentials, Bessel Potentials, and Fractional Integrals;15
4.2;6.2 Sobolev Spaces;26
4.3;6.3 Lipschitz Spaces;38
4.4;6.4 Hardy Spaces;51
4.5;6.5 Besov–Lipschitz and Triebel–Lizorkin Spaces;82
4.6;6.6 Atomic Decomposition;92
4.7;6.7 Singular Integrals on Function Spaces;107
5;7 BMO and Carleson Measures;130
5.1;7.1 Functions of Bounded Mean Oscillation;130
5.2;7.2 Duality between H1 and BMO;143
5.3;7.3 Nontangential Maximal Functions and Carleson Measures;148
5.4;7.4 The Sharp Maximal Function;159
5.5;7.5 Commutators of Singular Integrals with BMO Functions;170
6;8 Singular Integrals of Nonconvolution Type;181
6.1;8.1 General Background and the Role of BMO;181
6.2;8.2 Consequences of L2 Boundedness;194
6.3;8.3 The T(1) Theorem;205
6.4;8.4 Paraproducts;224
6.5;8.5 An Almost Orthogonality Lemma and Applications;235
6.6;8.6 The Cauchy Integral of Calder´on and the T(b) Theorem;250
6.7;8.7 Square Roots of Elliptic Operators;268
7;9 Weighted Inequalities;291
7.1;9.1 The Ap Condition;291
7.2;9.2 Reverse H¨older Inequality for Ap Weights and Consequences;305
7.3;9.3 The A8 Condition;314
7.4;9.4 Weighted Norm Inequalities for Singular Integrals;321
7.5;9.5 Further Properties of Ap Weights;336
8;10 Boundedness and Convergence of Fourier Integrals;351
8.1;10.1 The Multiplier Problem for the Ball;352
8.2;10.2 Bochner–Riesz Means and the Carleson–Sj¨olin Theorem;363
8.3;10.3 Kakeya Maximal Operators;380
8.4;10.4 Fourier Transform Restriction and Bochner–Riesz Means;399
8.5;10.5 Almost Everywhere Convergence of Bochner–Riesz Means;415
9;11 Time–Frequency Analysis and the Carleson–Hunt Theorem;435
9.1;11.1 Almost Everywhere Convergence of Fourier Integrals;435
9.2;11.2 Distributional Estimates for the Carleson Operator;468
9.3;11.3 The Maximal Carleson Operator and Weighted Estimates;487
10;Glossary;494
11;References;498
12;Index;511