Microlocal analysis for differential operators. An introduction. (English) Zbl 0804.35001

Grigis, Alain; Sjöstrand, Johannes

London Mathematical Society Lecture Note Series. 196. Cambridge: Cambridge University Press. 151 p. £19.95; $ 29.95 (1994).

This is a textbook for a graduate course on microlocal analysis of linear differential operators written on the actual experience of the authors at the universities in Paris. The reader is required to be familiar with the theory of distributions and Fourier transforms. The book consists of 12 chapters with exercises and historical notes. This concise book will be very convenient for professors planning to give a half-year course on this subject. The titles of the chapters are: 1. Symbols and oscillatory integrals, 2. The method of stationary phase, 3. Pseudodifferential operators, 4. Application to elliptic operators and \(L^ 2\) continuity, 5. Local symplectic geometry I (Hamilton-Jacobi theory), 6. The strictly hyperbolic Cauchy problem, 7. The wavefront set of a distribution, 8. Propagation of singularities for operators of real principal type, 9. Local symplectic geometry II, 10. Canonical transformations of pseudodifferential operators, 11. Global theory of Fourier integral operators, 12. Spectral theory for elliptic operators.

Reviewer: A.Kaneko (Komaba)

MSC:

35-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35A27	Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35S05	Pseudodifferential operators as generalizations of partial differential operators

Keywords:

propagation of singularities; oscillatory integrals; symplectic geometry; wavefront set; Fourier integral operators