An Introduction to Microlocal Analysis

Microlocal analysis provides a powerful, versatile, and modular perspective on the analysis of linear partial differential equations. This text, developed from a first-year graduate course, provides an accessible introduction and develops, from first principles, the core notions and results including pseudodifferential operators, wave front sets, and propagation phenomena. The reader is assumed to have some exposure to functional analysis and the theory of smooth manifolds. With detailed proofs, a wealth of exercises of varying levels of difficulty, and connections to contemporary research in general relativity, the book serves as both a comprehensive textbook for graduate students and a useful reference for researchers.

Preface.- 1. Introduction.- 2. Schwartz functions and tempered distributions.- 3. Symbols.- 4. Pseudodifferential operators.- 5. Pseudodifferential operators on manifolds.- 6. Microlocalization.- 7. Hyperbolic evolution equations and Egorov's theorem.- 8. Real principal type propagation of singularities.-  9. Solving wave-type equations.- 10. Propagation of singularities at radial sets.- 11. Late time asymptotics of linear waves on de Sitter space.- Bibliography.- Index.

Peter Hintz is a Professor of mathematics at ETH, Zurich. His research focuses on partial differential equations arising in general relativity. Much of his work is concerned with stability problems for solutions of the Einstein field equations and with the global asymptotic control (regularity, decay) of solutions to related linear and nonlinear wave equations. Methods and ideas from microlocal analysis and spectral/scattering theory feature prominently in his research.