Fourier Integrals in Classical Analysis ( Cambridge Tracts in Mathematics )

Publication series ：Cambridge Tracts in Mathematics

Author： Christopher D. Sogge  

Publisher： Cambridge University Press‎

Publication year： 1993

E-ISBN: 9780511833885

P-ISBN(Paperback): 9780521434645

Subject： O174.2 classical harmonic analysis (Fourier analysis)

Keyword： 泛函分析

Language： ENG

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Fourier Integrals in Classical Analysis

Description

Fourier Integrals in Classical Analysis is an advanced monograph concerned with modern treatments of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. Using microlocal analysis, the author, in particular, studies problems involving maximal functions and Riesz means using the so-called half-wave operator. This self-contained book starts with a rapid review of important topics in Fourier analysis. The author then presents the necessary tools from microlocal analysis, and goes on to give a proof of the sharp Weyl formula which he then modifies to give sharp estimates for the size of eigenfunctions on compact manifolds. Finally, at the end, the tools that have been developed are used to study the regularity properties of Fourier integral operators, culminating in the proof of local smoothing estimates and their applications to singular maximal theorems in two and more dimensions.

Chapter

Cover

Half-title

Title

Copyright

Dedication

Contents

Preface

0. Background

0.1. Fourier Transform

0.2. Basic Real Variable Theory

0.3. Fractional Integration and Sobolev Embedding Theorems

0.4. Wave Front Sets and the Cotangent Bundle

0.5. Oscillatory Integrals

Notes

1. Stationary Phase

1.1. Stationary Phase Estimates

1.2. Fourier Transform of Surface-carried Measures

Notes

2. Non-homogeneous Oscillatory Integral Operators

2.1. Non-degenerate Oscillatory Integral Operators

2.2. Oscillatory Integral Operators Related to the Restriction Theorem

2.3. Riesz Means in Rn

2.4. Kakeya Maximal Functions and Maximal Riesz Means in R2

Notes

3. Pseudo-differential Operators

3.1. Some Basics

3.2. Equivalence of Phase Functions

3.3. Self-adjoint Elliptic Pseudo-differential Operators on Compact Manifolds

Notes

4. The Half-wave Operator and Functions of Pseudo-differential Operators

4.1. The Half-wave Operator

4.2. The Sharp Weyl Formula

4.3. Smooth Functions of Pseudo-differential Operators

Notes

5. Lp Estimates of Eigenfunctions

5.1. The Discrete L2 Restriction Theorem

5.2. Estimates for Riesz Means

5.3. More General Multiplier Theorems

Notes

6. Fourier Integral Operators

6.1. Lagrangian Distributions

6.2. Regularity Properties

6.3. Spherical Maximal Theorems: Take 1

Notes

7. Local Smoothing of Fourier Integral Operators

7.1. Local Smoothing in Two Dimensions and Variable Coefficient Kakeya Maximal Theorems

7.2. Local Smoothing in Higher Dimensions

7.3. Spherical Maximal Theorems Revisited

Notes

Appendix: Lagrangian Subspaces of T* Rn

Bibliography

Index

Index of Notation

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