Linear topological invariants for kernels of convolution and differential operators

TL;DR

This paper establishes topological invariants for kernels of convolution and differential operators, showing these kernels satisfy the condition $( ext{}oldsymbol{ ext{Ω}} ext{)}$ and related properties, with implications for surjectivity on spaces of smooth functions.

Contribution

It proves that kernels of certain convolution and differential operators satisfy the condition $( ext{}oldsymbol{ ext{Ω}})$, linking smooth and distributional solutions and extending known results.

Findings

Kernels of differential operators satisfy $( ext{}oldsymbol{ ext{Ω}} ext{)}$.

Spaces of solutions to convolution equations satisfy $( ext{}oldsymbol{ ext{Ω}})$ if and only if their distributional counterparts satisfy $( ext{P}oldsymbol{ ext{Ω}})$.

Surjectivity of convolution operators on smooth functions implies their kernels satisfy $( ext{}oldsymbol{ ext{Ω}} ext{)}$.

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 $\mathscr{D}'(Y)$ 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.