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： 2017

E-ISBN: 9781108234498

P-ISBN(Paperback): 9781107120075

Subject： O174.21 orthogonal series (Fourier series)

Keyword： 数值分析

Language： ENG

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Fourier Integrals in Classical Analysis

Description

This advanced monograph is 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. In particular, the author uses microlocal analysis to study problems involving maximal functions and Riesz means using the so-called half-wave operator. To keep the treatment self-contained, the author begins with a rapid review of Fourier analysis and also develops the necessary tools from microlocal analysis. This second edition includes two new chapters. The first presents Hörmander's propagation of singularities theorem and uses this to prove the Duistermaat–Guillemin theorem. The second concerns newer results related to the Kakeya conjecture, including the maximal Kakeya estimates obtained by Bourgain and Wolff.

Chapter

Cover

Series page

Title page

Copyright information

Dedication

Table of contents

Preface to the Second Edition

Preface to the First Edition

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

The One-Dimensional Case

Stationary Phase in Higher Dimensions

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 Nikodym 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

Application: Unique Continuation for the Laplacian

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

Sharpness of Results

6.3 Spherical Maximal Theorems: Take 1

Notes

7 Propagation of Singularities and Refined Estimates

7.1 Wave Front Sets Redux

7.2 Propagation of Singularities

7.3 Improved Sup-norm Estimates of Eigenfunctions

7.4 Improved Spectral Asymptotics

Notes

8 Local Smoothing of Fourier Integral Operators

8.1 Local Smoothing in Two Dimensions and Variable Coefficient Nikodym Maximal Theorems

Orthogonality Arguments in Two Dimensions

Variable Coefficient Nikodym Maximal Functions

8.2 Local Smoothing in Higher Dimensions

Orthogonality Arguments in Higher Dimensions

8.3 Spherical Maximal Theorems Revisited

Notes

9 Kakeya-type Maximal Operators

9.1 The Kakeya Maximal Operator and the Kakeya Problem

9.2 Universal Bounds for Kakeya-type Maximal Operators

9.3 Negative Results in Curved Spaces

Negative Results for Oscillatory Integrals Arising in Curved Spaces

9.4 Wolff's Bounds for Kakeya-type Maximal Operators

Auxiliary Maximal Function Bounds

Wolff’s Bounds for Nikodym Maximal Functions in Rn

9.5 The Fourier Restriction Problem and the Kakeya Problem

Notes

Appendix Lagrangian Subspaces of T∗Rn

References

Index of Notation

Index

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