Lecture notes for pseudodifferential operators and microlocal analysis

TL;DR

This lecture note introduces pseudodifferential operators and microlocal analysis, covering symbols, oscillatory integrals, stationary phase lemmas, wavefront sets, and singularity propagation, with detailed proofs and exercises.

Contribution

It provides a comprehensive, accessible introduction to pseudodifferential operators and microlocal analysis, including new proofs of stationary phase lemmas for general functions.

Findings

Stationary phase lemmas extended to non-compactly supported functions.

Development of wavefront set and singularity propagation concepts.

Detailed proofs and exercises enhance understanding of microlocal analysis.

Abstract

This is a introductory course focusing some basic notions in pseudodifferential operators ($\Psi$DOs) and microlocal analysis. We start this lecture notes with some notations and necessary preliminaries. Then the notion of symbols and $\Psi$DOs are introduced. In Chapter 3 we define the oscillatory integrals of different types. Chapter 4 is devoted to the stationary phase lemmas. One of the features of the lecture is that the stationary phase lemmas are proved for not only compactly supported functions but also for more general functions with certain order of smoothness and certain order of growth at infinity. We build the results on the stationary phase lemmas. Chapters 5, 6 and 7 covers main results in $\Psi$DOs and the proofs are heavily built on the results in Chapter 4. Some aspects of the semi-classical analysis are similar to that of microlocal analysis. In Chapter 8 we finally introduce the notion of wavefront, and Chapter 9 focuses on the propagation of singularities of solution of partial differential equations. Important results are circulated by black boxes and some key steps are marked in red color. Exercises are provided at the end of each chapter.