Classical Microlocal Analysis in the Space of Hyperfunctions

By: Seiichiro Wakabayashi

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Paperback | 1 July 2000

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370 Pages

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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.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Complex Analysis Medicine Medicine in General Differential Calculus & Equations Integral Calculus & Equations Combinatorics & Graph Theory Functional Analysis & Transforms

Hyperfunctions	p. 5
Function spaces?	p. 5
Supports	p. 13
Localization	p. 23
Hyperfunctions	p. 28
Further applications of the Runge approximation theorem	p. 34
Basic calculus of Fourier integral operators and pseudo-diferential operators	p. 41
Preliminary lemmas	p. 41
Symbol classes	p. 52
Definition of Fouier integral operators	p. 57
Product formula of Fourier integral operators I	p. 65
Product formula of Fourier integral operators II	p. 87
Pseudolocal properties	p. 93
Pseudodifferential operators in ß	p. 107
Parametrices of elliptic operators	p. 112
Analytic wave ront sets and micro functions	p. 115
Analytic wave front sets	p. 115
Action of Fourier integral operators on wave front sets	p. 130
The boundary values of analytic functions	p. 155
Operations on hyperfunctions	p. 165
Hyperfunctions supported by a half-space	p. 183
Microfunctions	p. 192
Formal analytic symbols	p. 201
Microlocal uniqueness	p. 205
Preliminary lemmas	p. 205
General results	p. 222
Microhyperbolic operators	p. 231
Canonical transformation	p. 239
Hypoellipticity	p. 244
Local solvability	p. 259
Preliminaries	p. 259
Necessary conditions on local solvability and hypoellipticity 268
Sufficient conditions on local solvability	p. 272
Some examples	p. 285
A Proofs of product formulae	p. 295
Proof of Theorem 2.4.4	p. 295
Proof of Corollary 2.4.5	p. 323
Proof of Theorem 2.4.6	p. 328
Proof of Corollary 2.4.7	p. 336
Proof of Theorem 2.5.3	p. 338
A priori estimates	p. 351
Grusin operators	p. 351
A class of operators with double characteristics	p. 355
Table of Contents provided by Publisher. All Rights Reserved.

Classical Microlocal Analysis in the Space of Hyperfunctions

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