Fourier Integrals in Classical Analysis

This advanced monograph is concerned with modern treatments of central problems in harmonic analysis.

Christopher D. Sogge (Author)

9781107120075, Cambridge University Press

Hardback, published 27 April 2017

348 pages, 2 b/w illus.
23.6 x 16 x 2.8 cm, 0.68 kg

'Fourier Integrals and Classical Analysis is an excellent book on a beautiful subject seeing a lot of high-level activity. Sogge notes that the book evolved out of his 1991 UCLA lecture notes, and this indicates the level of preparation expected from the reader: that of a serious advanced graduate student in analysis, or even a beginning licensed analyst, looking to do work in this area. But a lot of advantage can be gained even by fellow travelers, all modulo enough mathematical maturity, training, and Sitzfleisch.' Michael Berg, MAA Reviews

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.

Subject Areas: Wave mechanics [vibration & acoustics PHDS], Differential calculus & equations [PBKJ], Calculus & mathematical analysis [PBK]