Hypoellipticity and local solvability of pseudolocal continuous linear operators in Gevrey classes
type: Item
instance of: scholarly article
title: Hypoellipticity and local solvability of pseudolocal continuous linear operators in Gevrey classes
zbMATH Open document ID: 1065.35099
author: Alessandro Morando
published in: Tsukuba Journal of Mathematics
publication date: 2005-03-30T00:00:00Z
review text: The author considers a linear map on the Gevrey functions of order \(s> 1\) and the corresponding ultradistributions, \(P: G^s_0(\Omega)\to G^s(\Omega)\), \(E_s'(\Omega)\to D_s'(\Omega)\), \(\Omega\subset\mathbb{R}^n\). Assume that \(P\) is \(s\)-pseudo-local in the sense that the following inclusions hold for the Gevrey \(s\)-singular supports: \[ s\text{-sing supp}(Pu)\subset s\text{-sing supp}\,u,\quad u\in E_s'(\Omega). \] The first result of the paper states that if \(P\) is \(s\)-hypoelliptic in \(\Omega\), then the transposed operator \({^tP}\) is \(s\)-locally solvable in \(\Omega\). This generalizes a result of \textit{A. A. Albanese}, \textit{A. Corli} and \textit{L. Rodino} [Math. Nachr. 242, 5--16 (2002; Zbl 1056.35049)] concerning the case when \(P\) is a linear partial differential operator. New interesting applications are given here to \(s\)-hypoelliptic pseudo-differential operators. Further results concern the existence of a fundamental kernel \(K(x, y)\in D_s'(\Omega\times\Omega)\) for \({^tP}\), under the \(s\)-hypoellipticity assumption for \(P\).
reviewed by: Luigi Rodino
Mathematics Subject Classification ID: 35H10
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35S10
zbMATH DE Number: 2150094
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496164718
OpenAlex ID: W2157648972
DOI: 10.21099/TKBJM/1496164718
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description: scientific article; zbMATH DE number 2150094