Classical Microlocal Analysis in the Space of Hyperfunctions / Edition 1 available in Paperback
- Pub. Date:
- Springer Berlin Heidelberg
The book develops "Classical Microlocal Analysis" in the spaces of hyperfunctions and microfunctions, which makes it possible to apply the methods in the distribution category to the studies on partial differential equations in the hyperfunction category. Here "Classical Microlocal Analysis" means that it does not use "Algebraic Analysis." The main tool in the text is, in some sense, integration by parts. The studies on microlocal uniqueness, analytic hypoellipticity and local solvability are reduced to the problems to derive energy estimates (or a priori estimates). The author assumes basic understanding of theory of pseudodifferential operators in the distribution category.
Table of ContentsChapter 1 Hyperfunctions > 1.1 Function spaces > 1.2 Supports > 1.3 Localization > 1.4 Hyperfunctions > 1.5 Further applications of the Runge approximation theorem >Chapter 2 Basic calculus of Fourier integral operators and pseudodifferential operators > 2.1 Preliminary lemmas > 2.2 Symbol classes > 2.3 Definition of Fourier integral operators > 2.4 Product formula of Fourier integral operators I > 2.5 Product formula of Fourier integral operators II > 2.6 Pseudolocal properties > 2.7 Pseudodifferential operators in B > 2.8 Parametrices of elliptic operators >Chapter 3 Analytic wave front sets and microfunctions > 3.1 Analytic wave front sets > 3.2 Action of Fourier integral operators on wave front sets > 3.3 The boundary values of analytic functions > 3.4 Operations on hyperfunctions > 3.5 Hyperfunctions supported by a half-space > 3.6 Microfunctions > 3.7 Formal analytic symbols >Chapter 4 Microlocal uniqueness > 4.1 Preliminary lemmas > 4.2 General results > 4.3 Microhyperbolic operators > 4.4 Canonical transformation > 4.5 Hypoellipticity >Chapter 5 Local solvability > 5.1 Preliminaries > 5.2 Necessary conditions on local solvability and hypoellipticity > 5.3 Sufficient conditions on local solvability > 5.4 Some examples >Chapter A Proofs of product formulae > A.1 Proof of Theorem 2.4.4 > A.2 Proof of Corollary 2.4.5 > A.3 Proof of Theorem 2.4.6 > A.4 Proof of Corollary 2.4.7 > A.5 Proof of Theorem 2.5.3 >Chapter B A priori estimates > B.1 Grusin operators > B.2 A class of operators with double characteristics