ISBN: 3540676031
TITLE: Classical Microlocal Analysis in the Space of Hyperfunctions
AUTHOR: Wakabayashi, Seiichiro
TOC:

1 Hyperfunctions 5
1.1 Function spaces 5
1.2 Supports 13
1.3 Localization 23
1.4 Hyperfunctions 28
1.5 Further applications of the Runge approximation theorem 34
2 Basic calculus of Fourier integral operators and pseudo-differential operators 41
2.1 Preliminary lemmas 41
2.2 Symbol classes 52
2.3 Definition of Fourier integral operators 57
2.4 Product formula of Fourier integral operators I 65
2.5 Product formula of Fourier integral operators II 87
2.6 Pseudolocal properties 93
2.7 Pseudodifferential operators in B 107
2.8 Parametrices of elliptic operators 112
3 Analytic wave front sets and microfunctions 115
3.1 Analytic wave front sets 115
3.2 Action of Fourier integral operators on wave front sets 130
3.3 The boundary values of analytic functions 155
3.4 Operations on hyperfunctions 165
3.5 Hyperfunctions supported by a half-space 183
3.6 Microfunctions 192
3.7 Formal analytic symbols 201
4 Microlocal uniqueness 205
4.1 Preliminary lemmas 205
4.2 General results 222
4.3 Microhyperbolic operators 231
4.4 Canonical transformation 239
4.5 Hypoellipticity 244
5 Local solvability 259
5.1 Preliminaries 259
5.2 Necessary conditions on local solvability and hypoellipticity 268
5.3 Sufficient conditions on local solvability 272
5.4 Some examples 285
A Proofs of product formulae 295
A.1 Proof of Theorem 2.4.4 295
A.2 Proof of Corollary 2.4.5 323
A.3 Proof of Theorem 2.4.6 328
A.4 Proof of Corollary 2.4.7 336
A.5 Proof of Theorem 2.5.3 338
B A priori estimates 351
B.1 Gruin operators 351
B.2 A class of operators with double characteristics 355
END
