Linear topological invariants for kernels of convolution and differential operators

• Published in: arXiv.org 2023-02
• Main Authors: Debrouwere, Andreas, Kalmes, Thomas
• Format: Article
• Language: English
• Subjects: Convolution; Differential equations; Kernels; Operators (mathematics); Splitting
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2204.11733

We establish the condition \((\Omega)\) for smooth kernels of various types of convolution and differential operators. By the \((DN)\)-\((\Omega)\) splitting theorem of Vogt and Wagner, this implies that these operators are surjective on the corresponding spaces of vector-valued smooth functions with values in a product of Montel \((DF)\)-spaces whose strong duals satisfy the condition \((DN)\), e.g., the space \(\mathscr{D}'(Y)\) of distributions over an open set \(Y \subseteq \mathbb{R}^n\) or the space \(\mathscr{S}'(\mathbb{R}^n)\) of tempered distributions. Most notably, we show that: \((i)\) \(\mathscr{E}_P(X) = \{ f \in \mathscr{E}(X) \, | \, P(D)f = 0 \}\) satisfies \((\Omega)\) for any differential operator \(P(D)\) and any open convex set \(X \subseteq \mathbb{R}^d\). \((ii)\) Let \(P\in\mathbb{C}[\xi_1,\xi_2]\) and \(X \subseteq \mathbb{R}^2\) open be such that \(P(D):\mathscr{E}(X)\rightarrow\mathscr{E}(X)\) is surjective. Then, \(\mathscr{E}_P(X)\) satisfies \((\Omega)\). \((iii)\) Let \(\mu \in \mathscr{E}'(\mathbb{R}^d)\) be such that \( \mathscr{E}(\mathbb{R}^d) \rightarrow \mathscr{E}(\mathbb{R}^d), \, f \mapsto \mu \ast f\) is surjective. Then, \( \{ f \in \mathscr{E}(\mathbb{R}^d) \, | \, \mu \ast f = 0 \}\) satisfies \((\Omega)\). The central result in this paper states that the space of smooth zero solutions of a general convolution equation satisfies the condition \((\Omega)\) if and only if the space of distributional zero solutions of the equation satisfies the condition \((P\Omega)\). The above and related results then follow from known results concerning \((P\Omega)\) for distributional kernels of convolution and differential operators.