On solvability of linear partial differential equations in local spaces \(B^{loc}_{p,k}(G)\)
type: Item
instance of: scholarly article
title: On solvability of linear partial differential equations in local spaces \(B^{loc}_{p,k}(G)\)
zbMATH Open document ID: 0661.35013
DOI: 10.4171/ZAA/287
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1988-01-01T00:00:00Z
review text: Let \(L(x,D)=\sum_{| \sigma | \leq r}a_{\sigma}(x)D^{\sigma}\) be a linear partial differential operator with \(C^{\infty}(G)\)- coefficients \(a_{\sigma}\), where G is an open subset in \({\mathbb{R}}^ n\). Denote by \(\Lambda^{\sim}_{p,k}(G)\) the minimal closed realization of L(x,D) in the local Hörmander space \({\mathcal B}^{loc}_{p,k}(G)\). The closedness of the range \(R(\Lambda^{\sim}_{p,k}(G))\) and of the range \(R(\Lambda^{\sim +}_{p,k}(G))\) of the dual operator \(\Lambda^{\sim +}_{p,k}(G)\) is considered. Among other things, one shows necessary and sufficient conditions for the closedness of \(R(\Lambda^{\sim}_{p,k}(G))\). The surjectivity of \(\Lambda^{\sim}_{p,k}(G)\) is also characterized. As an application a sufficient condition for the closedness of \(R(\Lambda^{\sim}_{2,1}(B(0,R)))\), where B(0,R) is the open ball in \({\mathbb{R}}^ n\), is established, when certain a priori estimates for the formal transpose L'(x,D) of L(x,D) hold.
Mathematics Subject Classification ID: 35G05
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 4079991
zbMATH Keywords: minimal closed realization
zbMATH Keywords: local Hörmander space
zbMATH Keywords: closedness of the range
zbMATH Keywords: a priori estimates
author: Jouko Tervo
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4079991