A refinement of Peetre's theorem
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 definition, the space of anisotropic distributions (of order \(l\) and \(m\)) is the topological dual space \(\mathscr{D}'_{l,m}(\Omega)\) of \(C_c^{l,m}(\Omega)\). Refinements of the above theorems of Peetre are shown for linear mappings \(A:C_c^{\infty}(\Omega)\rightarrow C_c^{l,m}(\Omega)\), where then \(a_\alpha\in C^{l,m}(\Omega)\) (with the obvious definition of \(C^{l,m}\)), and \(B:C_c^\infty(\Omega)\rightarrow\mathscr{D}'_{m,m}(\Omega)\), where then \(b_\alpha\in\mathscr{D}'_{m,m}(\Omega)\). The proof of the latter statement relies on a version of the Schwartz kernel theorem in the context of anisotropic distributions: for every separately continuous bilinear form \(t\) on \(C_c^\infty(\Omega_1)\times C_c^m(\Omega_2)\) there is a unique \(T\in\mathscr{D}'_{\infty,m}(\Omega_1\times\Omega_2)\) such that \(t(\varphi,\psi)=T(\varphi\otimes\psi)\), \(\varphi\in C_c^\infty(\Omega_1)\), \(\psi\in C_c^m(\Omega_2)\). Additionally, the paper contains some interesting structural results on anisotropic distributions.
scientific article; zbMATH DE number 7138466
scientific article