arXiv2023
We establish the condition $(Ω)$ for smooth kernels of various types of convolution and differential operators. By the $(DN)$-$(Ω)$ 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 $(Ω)$ for any differential operator $P(D)$ and any open convex set $X \subseteq \mathbb{R}^d$. $(ii)$ Let $P\in\mathbb{C}[ξ_1,ξ_2]$ 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 $(Ω)$. $(iii)$ Let $μ\in \mathscr{E}'(\mathbb{R}^d)$ be such that $ \mathscr{E}(\mathbb{R}^d) \rightarrow \mathscr{E}(\mathbb{R}^d), \, f \mapsto μ\ast f$ is surjective. Then, $ \{ f \in \mathscr{E}(\mathbb{R}^d) \, | \, μ\ast f = 0 \}$ satisfies $(Ω)$. The central result in this paper states that the space of smooth zero solutions of a general convolution equation satisfies the condition $(Ω)$ if and only if the space of distributional zero solutions of the equation satisfies the condition $(PΩ)$. The above and related results then follow from known results concerning $(PΩ)$ for distributional kernels of convolution and differential operators.