Fourier Integrals in Classical Analysis

✍ Scribed by Christopher D. Sogge

• Publisher: Cambridge University Press
• Year: 1993
• Tongue: English
• Leaves: 248
• Series: Cambridge Tracts in Mathematics
• Category: Library

✦ Synopsis

Fourier Integrals in Classical Analysis is an advanced treatment 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, 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.

✦ Table of Contents

Index of Notation......Page 4
Contents......Page 8
Preface......Page 10
0.1. Fourier Transform......Page 12
0.2. Basic Real Variable Theory......Page 20
0.3. Fractional Integration and Sobolev Embedding Theorems......Page 33
0.4. Wave Front Sets and the Cotangent Bundle......Page 39
0.5. Oscillatory Integrals......Page 47
Notes......Page 50
1.1. Stationary Phase Estimates......Page 51
1.2. Fourier Transform of Surface-carried Measures......Page 58
Notes......Page 65
2 Non-homogeneous Oscillatory Integral Operators......Page 66
2.1. Non-degenerate Oscillatory Integral Operators......Page 67
2.2. Oscillatory Integral Operators Related to the Restricted Theorem......Page 69
2.3. Riesz Means in R^n......Page 76
2.4. Kakeya Maximal Functions and Maximal Riesz Means in R^2......Page 82
Notes......Page 103
3.1. Some Basics......Page 104
3.2. Equivalence of Phase Functions......Page 111
3.3. Self-adjoint Elliptic Pseudo-differential Operators on Compact Manifolds......Page 117
Notes......Page 123
4 The Half-wave Operator and Functions of Pseudo-differential Operators......Page 124
4.1. The Half-wave Operator......Page 125
4.2. The Sharp Weyl Formula......Page 135
4.3. Smooth Functions of Pseudo-differential Operators......Page 142
Notes......Page 144
5 L^P Estimates of Eigenfunctions......Page 146
5.1. The Discrete L2 Restriction Theorem......Page 147
5.2. Estimates for Riesz Means......Page 160
5.3. More General Multiplier Theorems......Page 164
Notes......Page 169
6 Fourier Integral Operators......Page 171
6.1. Lagrangian Distributions......Page 172
6.2. Regularity Properties......Page 179
6.3. Spherical Maximal Theorems: Take 1......Page 197
Notes......Page 204
7 Local Smoothing of Fourier Integral Operators......Page 205
7.1. Local Smoothing in Two Dimensions and Variable Coefficient Kakeya Maximal Theorems......Page 206
7.2. Local Smoothing in Higher Dimensions......Page 225
7.3. Spherical Maximal Theorems Revisited......Page 235
Notes......Page 238
Appendix: Lagrangian Subspaces of T* R^n......Page 239
Bibliography......Page 241
Index......Page 248