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.

List of contents

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.

Product details

• Authors: Peter Hintz
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Release: 11.08.2025
• EAN: 9783031907050
• ISBN: 978-3-0-3190705-0
• No. of pages: 285
• Illustrations: X, 285 p. 35 illus., 22 illus. in color.
• Series: Graduate Texts in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Kybernetik und Systemtheorie, Partial Differential Equations, Differential equations, Dynamical systems, quantization, pseudodifferential operators, Microlocal Analysis, Sobolev spaces, Lorentzian geometry, wave front set, Schwartz functions, elliptic parametrix, radial sets, differential operators, ellipticity, linear PDEs, microlocalization, semiclassical radial point, graduate text microlocal analysis, radial compactification phase space, semiclassical pseudodifferential operators, hyperbolic evolution equations, Schwartz kernel