Linear topological invariants for kernels of convolution and differential operators

We establish the condition $(\mathrm{\Omega })$ for smooth kernels of various types of convolution and differential operators. By the $(DN)$-$(\mathrm{\Omega })$ 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 ${\mathcal{D}}^{\mathrm{\prime }}(Y)$ of distributions over an open set $Y \subseteq $\mathbb{R}$^n$orthespace$\mathscr{S}'($\mathbb{R}$^n)$ of tempered distributions. Most notably, we show that: $(i)$ ${\mathcal{E}}_P(X)=\{f\in \mathcal{E}(X)\mathrm{\mid }P(D)f=0\}$ satisfies $(\mathrm{\Omega })$ 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 $P(D):\mathcal{E}(X)\to \mathcal{E}(X)$ is surjective. Then, ${\mathcal{E}}_P(X)$ satisfies $(\mathrm{\Omega })$. $(iii)$ Let $\mu \in {\mathcal{E}}^{\mathrm{\prime }}({\mathbb{R}}^d)$ be such that $ \mathscr{E}($\mathbb{R}$^d) \rightarrow \mathscr{E}($\mathbb{R}$^d), , f \mapsto \mu \ast f$issurjective\mathrm{.}Then,$ { 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 $(\mathrm{\Omega })$ if and only if the space of distributional zero solutions of the equation satisfies the condition $(P\mathrm{\Omega })$. The above and related results then follow from known results concerning $(P\mathrm{\Omega })$ for distributional kernels of convolution and differential operators.