Harmonic Analysis (PMS-43), Volume 43: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. (PMS-43)

✍ Scribed by Elias M. Stein

• Publisher: Princeton University Press
• Year: 2016
• Tongue: English
• Leaves: 710
• Series: Princeton Mathematical Series; 24
• Category: Library

✦ Synopsis

This book contains an exposition of some of the main developments of the last twenty years in the following areas of harmonic analysis: singular integral and pseudo-differential operators, the theory of Hardy spaces, L\sup\ estimates involving oscillatory integrals and Fourier integral operators, relations of curvature to maximal inequalities, and connections with analysis on the Heisenberg group.

✦ Table of Contents

Contents
Preface
Guide to the Reader
Prologue
I. Real-Variable Theory
1. Basic assumptions
2. Examples
3. Covering lemmas and the maximal function
4. Generalization of the Calderón-Zygmund decomposition
5. Singular integrals
6. Examples of the general theory
7. Appendix: Truncation of singular integrals
8. Further results
II.
More about Maximal Functions
1. Vector-valued maximal functions
2. Nontangential behavior and Carleson measures
3. Two applications
4. Singular approximations of the identity
5. Further results
III. H
ardy Spaces
1. Maximal characterization of Hp
2. Atomic decomposition for Hp
3. Singular integrals
4. Appendix: Relation with harmonic functions
5. Further results
IV. H 1 and
BMO
1. The space of functions of bounded mean oscillation
2. The sharp function
3. An elementary approach and a dyadic version
4. Further properties of BMO
5. An interpolation theorem
6. Further results
V. Weighted Inequalities
1. The class Ap
2. Two further characterizations of Ap
3. The main theorem about Ap
4. Weighted inequalities for singular integrals
5. Further properties of Ap weights
6. Further results
VI. Pseudo-Differential and Singular Integral
Operators: Fourier Transform
1. Pseudo-differential operators
2. An L
2 theorem
3. The symbolic calculus
4. Singular integral realization of pseudo-differential operators
5. Estimates in Lp
, Sobolev, and Lipschitz spaces
6. Appendix: Compound symbols
7. Further results
VII. Pseudo-Differential and Singular Integral
Operators: Almost Orthogonality
1. Exotic and forbidden symbols
2. Almost orthogonality
3. L^2 theory of operators with Calderón-Zygmund kernels
4. Appendix: The Cauchy integral
5. Further results
VIII.
Oscillatory Integrals of the First Kind
1. Oscillatory integrals of the first kind, one variable
2. Oscillatory integrals of the first kind, several variables
3. Fourier transforms of measures supported on surfaces
4. Restriction of the Fourier transform
5. Further results
IX. Oscillatory Integrals of the Second Kind
1. Oscillatory integrals related to the Fourier transform
2. Restriction theorems and Bochner-Riesz summability
3. Fourier integral operators: L^2 estimates
4. Fourier integral operators: L^P estimates
5. Appendix: Restriction theorems in two dimensions
6. Further results
X.
Maximal Operators: Some Examples
1. The Besicovitch set
2. Maximal functions and counterexamples
3. Further results
XI. Maximal Averages and Oscillatory Integrals
1. Maximal averages and square functions
2. Averages over a k-dimensional submanifold of finite type
3. Averages on variable hypersurfaces
4. Further results
XII. Introduction to the Heisenberg Group
1. Geometry of the complex ball and the Heisenberg group
2. The Cauchy-Szegö integral
3. Formalism of quantum mechanics and the Heisenberg group
4. Weyl correspondence and pseudo-differential operators
5. Twisted convolution and singular integrals on H^n
6. Appendix: Representations of the Heisenberg group
7. Further results
XIII. More about the Heisenberg Group
1. The Cauchy-Riemann complex and its boundary analogue
2. The operators ∂̄
b and ⌞b on the Heisenberg group
3. Applications of the fundamental solution
4. The Lewy operator
5. Homogeneous groups
6. Appendix: The ∂-Neumann problem
7. Further results
Bibliography
Author Index
Subject Index