A refinement of Peetre's theorem
type: Item
instance of: scholarly article
title: A refinement of Peetre's theorem
zbMATH Open document ID: 1475.46041
author: Marin Mišur
published in: Results in Mathematics
publication date: 2019-11-28T00:00:00Z
review text: Two classical theorems of \textit{J. Peetre} in [Math. Scand. 7, 211--218 (1959; Zbl 0089.32502); Math. Scand. 8, 116--120 (1960; Zbl 0097.10402)], respectively, characterize linear partial differential operators in an open subset \(\Omega\) of \(\mathbb{R}^N\) in the following coordinate-free manner: \begin{itemize} \item[(i)] A linear mapping \(A:C_c^\infty(\Omega)\rightarrow C_c^\infty(\Omega)\) satisfies \(\mbox{supp}(A\varphi)\subseteq\mbox{supp}\,\varphi\) for all \(\varphi\in C_c^\infty(\Omega)\) if and only if \(A\) is a linear partial differential operator with smooth coefficients, i.e., there is a locally finite family of smooth functions \((a_\alpha)_{\alpha\in\mathbb{N}_0^N}\) on \(\Omega\) such that \(A\varphi=\sum_{\alpha}a_\alpha \partial^\alpha \varphi\), \(\varphi\in C_c^\infty(\Omega)\). \item[(ii)] A linear mapping \(B:C_c^\infty(\Omega)\rightarrow \mathscr{D}'(\Omega)\) satisfies \(\mbox{supp}(B\varphi)\subseteq\mbox{supp}\,\varphi\) for all \(\varphi\in C_c^\infty(\Omega)\) if and only if there is a locally finite family of distributions \((b_\alpha)_{\alpha\in\mathbb{N}_0^N}\) on \(\Omega\) such that \(\mbox{supp}(B\varphi-\sum_{\alpha}b_\alpha \partial^\alpha \varphi)\subseteq \Lambda\), \(\varphi\in C_c^\infty(\Omega)\), where \(\Lambda\subset\Omega\) denotes the so-called (necessarily discrete) set of discontuity points of \(B\). \end{itemize} In this paper, the author generalizes both characterizations of linear partial differential operators to the setting of anisotropic distributions: For \(\Omega\subseteq\mathbb{R}^d\times\mathbb{R}^r\) and \(m,l\in\mathbb{N}_0\cup\{\infty\}\), let \(C_c^{l,m}(\Omega)\) be the space of compactly supported functions on \(\Omega\) which are continuously differentiable up to order \(l\) with respect to \(x\) and up to order \(m\) with respect to \(y\), where \((x,y)\in\mathbb{R}^d\times\mathbb{R}^r\). Then \(C_c^{l,m}(\Omega)\) can be equipped in a natural way with an \textrm{LF}-space topology. By definiti…
reviewed by: Thomas Kalmes
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 7138466
zbMATH Keywords: partial differential operator
zbMATH Keywords: Peetre's theorem
zbMATH Keywords: Schwartz' kernel theorem
zbMATH Keywords: distributions
zbMATH Keywords: anisotropic spaces
MaRDI profile type: Publication
full work available at URL: s00025-019-1125-5
OpenAlex ID: W2989736467
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Wikidata QID: Q126653110
DOI: 10.1007/S00025-019-1125-5
description: scientific article; zbMATH DE number 7138466