Except for minor modifications, this monograph represents the lecture notes of a course I gave at UCLA during the winter and spring quarters of 1991. My purpose in the course was to present the necessary background material and to show how ideas from the theory of Fourier integral operators can be useful for studying basic topics in classical analysis, such as oscillatory integrals and maximal functions. The link between the theory of Fourier integral operators and classical analysis is of course not new, since one of the early goals of microlocal analysis was to provide variable coefficient versions of the Fourier transform. However, the primary goal of this subject was to develop tools for the study of partial differential equations and, to some extent, only recently have many classical analysts realized its utility in their subject.

本书为英文版。

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 R

　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*IRn

Bibliography

Index

Index of Notation