Fourier Integrals in Classical Analysis [Hardcover]

This advanced monograph is concerned with modern treatments of central problems in harmonic analysis.This advanced monograph, concerned with modern treatments of central problems in harmonic analysis, explores the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. New chapters discuss the DuistermaatGuillemin theorem and results related to the Kakeya conjecture.This advanced monograph, concerned with modern treatments of central problems in harmonic analysis, explores the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. New chapters discuss the DuistermaatGuillemin theorem and results related to the Kakeya conjecture.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 H?rmander's propagation of singularities theorem and uses this to prove the DuistermaatGuillemin theorem. The second concerns newer results related to the Kakeya conjecture, including the maximal Kakeya estimates obtained by Bourgain and Wolff.Background; 1. Stationary phase; 2. Non-homogeneous oscillatory integral operators; 3. Pseudo-differential operators; 4. The half-wave operator and functions of pseudo-differential operators; 5. Lp estimates of Eigenfunctions; 6. Fourier integral operators; 7. Propagation of singularities and relóq