Fourier Integrals in Classical Analysis

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.

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 refined estimates; 8. Local smoothing of fourier integral operators; 9. Kakeya type maximal operators; Appendix. Lagrangian subspaces of T*Rn; References; Index of Notation; Index.

Christopher D. Sogge is the J. J. Sylvester Professor of Mathematics at The John Hopkins University and the editor-in-chief of the American Journal of Mathematics. His research concerns Fourier analysis and partial differential equations. In 2012, he became one of the Inaugural Fellows of the American Mathematical Society. He is also a fellow of the National Science Foundation, the Alfred P. Sloan Foundation and the Guggenheim Foundation, and he is a recipient of the Presidential Young Investigator Award. In 2007, he received the Diversity Recognition Award from The Johns Hopkins University.