Classical microlocal analysis in the space of hyperfunctions
type: Item
instance of: scholarly article
title: Classical microlocal analysis in the space of hyperfunctions
zbMATH Open document ID: 0949.35003
author: Seiichiro Wakabayashi
published in: Lecture Notes in Mathematics
publication date: 2000-07-19T00:00:00Z
review text: The author gives an important contribution to the study of partial differential equations (P.D.E.) with analytic coefficients in the space of hyperfunctions by methods of ``Classical Microlocal Analysis'', i.e. integration by parts and cut-off functions (versus the methods of ``Algebraic Analysis''). He continues directions started by Hörmander, Treves, Boutet de Monvel and Kree (among others). The main difficulty is, of course, the lack of ``cut-off'' techniques in the framework of analytic functions. The starting point is the formula \(p(x,D)u:=(2\pi)^{-n} \int e^{i\langle x,\xi\rangle}p(x,\xi)\widehat u(\xi) d\xi,\) where \(u\in{\mathcal A}'\) is an analytic functional, and \(\widehat u(\xi):=u(e^{-i \langle z,\xi\rangle})\) denotes its Fourier transform. Since, fortunately or unfortunately, \(p(x,D)u\) is not in general an element of \({\mathcal A}'\), when \(p(x,\xi)\) is not a polynomial, the author defines the spaces, corresponding to the Schwartz spaces \({\mathcal S}\) and \({\mathcal S}'\), that play a key role in the calculus: with \(\varepsilon\geq 0,\) and upon denoting by \(F\) the Fourier transform, he defines \[ \begin{aligned} {\mathcal S}_{-\varepsilon}&:= ^t F^{-1}\bigl( \{v\in C^\infty({\mathbb R}^n);\;e^{-\varepsilon\sqrt{1+|\xi|^2}}v(\xi)\in{\mathcal S}\}\bigr), \\ {\mathcal S}_{\varepsilon}&:=F^{-1}\bigl( \{v\in C^\infty({\mathbb R}^n);\;e^{\varepsilon\sqrt{1+|\xi|^2}}v(\xi)\in{\mathcal S}\}\bigr)\end{aligned} \] and \({\mathcal S}'_\varepsilon\) as the dual space of \({\mathcal S}_\varepsilon.\) It turns out that \({\mathcal S}_\varepsilon\) is dense in \({\mathcal S}\) for \(\varepsilon\geq 0,\) and, upon writing \({\mathcal E}_0:= \bigcap_{\varepsilon>0}{\mathcal S}_{-\varepsilon}\) and \({\mathcal F}_0:= \bigcap_{\varepsilon>0}{\mathcal S}'_{\varepsilon},\) that \({\mathcal E}'\subset{\mathcal A}'\subset{\mathcal E}_0\subset{\mathcal F}_0.\) Now, for \(\varepsilon\geq 0\) and \(K\subset{\mathbb R}^n\) a closed set, he defines \({\mathcal A…
reviewed by: Alberto Parmeggiani
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35A27
Mathematics Subject Classification ID: 46F15
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 35S30
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35H10
zbMATH DE Number: 1478720
zbMATH Keywords: propagation of analytic singularities
zbMATH Keywords: cut-off functions
zbMATH Keywords: Fourier transform
zbMATH Keywords: Schwartz spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0103767
OpenAlex ID: W591833795
DOI: 10.1007/BFB0103767
description: scientific article; zbMATH DE number 1478720