Fourier Integrals in Classical Analysis

Christopher D. Sogge

doi:10.1017/CBO9780511530029

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.

"The writing is agile and somewhat colloquial, giving a refreshing informal tone to the presentation of quite arduous topics. The goals of the collection in which this book has been published state that '[works] have to be rigorous, definite, and of lasting value to mathematicians working in the relevant disciplines'. I believe that the book under consideration generously fulfills these goals." Josefina Alvarez, Mathematical Reviews

"...provides an excellent introduction to oscillatory integral operators and a detailed treatment of some of the most recent developments....rewards the reader with a thorough account of some of the last decade's most important developments in Fourier analysis, many of them due to its author. It belongs on the bookshelf of anyone seriously interested in the subject." Allan Greenleaf, Bulletin of the American Mathematical Society

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Fourier Integrals in Classical Analysis

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Contents

Frontmatter
pp i-vi

Contents
pp vii-viii

Preface
pp ix-x

0 - Background
pp 1-39

1 - Stationary Phase
pp 40-54

2 - Non-homogeneous Oscillatory Integral Operators
pp 55-92

3 - Pseudo-differential Operators
pp 93-112

4 - The Half-wave Operator and Functions of Pseudo-differential Operators
pp 113-134

5 - Lp Estimates of Eigenfunctions
pp 135-159

6 - Fourier Integral Operators
pp 160-193

7 - Local Smoothing of Fourier Integral Operators
pp 194-227

Appendix: Lagrangian Subspaces of T*IRn
pp 228-229

Bibliography
pp 230-236

Index
pp 237-238

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