Linear topological invariants for kernels of convolution and differential operators

Abstract

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 ${\mathcal{D}}^{\prime }(Y)$ of distributions over an open set $Y\subseteq {\mathbb{R}}^n$ or the space ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ of tempered distributions. Most notably, we show that: $(i)$ ${\mathcal{E}}_P(X)=\{f\in \mathcal{E}(X)|P(D)f=0\}$ 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 $P(D):\mathcal{E}(X)\to \mathcal{E}(X)$ is surjective. Then, ${\mathcal{E}}_P(X)$ satisfies $(\Omega )$. $(iii)$ Let $\mu \in {\mathcal{E}}^{\prime }({\mathbb{R}}^d)$ be such that $\mathcal{E}({\mathbb{R}}^d)\to \mathcal{E}({\mathbb{R}}^d),f\mapsto \mu \ast f$ is surjective. Then, $\{f\in \mathcal{E}({\mathbb{R}}^d)|\mu \ast f=0\}$ 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.

17 pages; correction of typos; accepted for publication in Journal of Functional Analysis

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