Surjectivity of differential operators and linear topological invariants for spaces of zero solutions

TL;DR

This paper establishes conditions under which linear differential operators with constant coefficients are surjective on spaces of smooth functions and distributions, and explores the topological properties of their zero solution spaces.

Contribution

It provides a characterization of surjectivity for certain differential operators and links this to the topological invariants of their zero solution spaces.

Findings

Sufficient and necessary conditions for surjectivity of differential operators.

Identification of topological invariants in zero solution spaces.

Extension of results to both $C^ abla$ and distribution spaces.

Abstract

We provide a sufficient condition for a linear differential operator with constant coefficients $P(D)$ to be surjective on $C^\infty(X)$ and $\mathscr{D}'(X)$, respectively, where $X\subseteq\mathbb{R}^d$ is open. Moreover, for certain differential operators this sufficient condition is also necessary and thus a characterization of surjectivity for such differential operators on $C^\infty(X)$, resp. on $\mathscr{D}'(X)$, is derived. Additionally, we obtain for certain surjective differential operators $P(D)$ on $C^\infty(X)$, resp. $\mathscr{D}'(X)$, that the spaces of zero solutions $C_P^\infty(X)=\{u\in C^\infty(X);\, P(D)u=0\}$, resp. $\mathscr{D}_P'(X)=\{u\in\mathscr{D}'(X);\,P(D)u=0\}$ possess the linear topological invariant $(\Omega)$ introduced by Vogt and Wagner in [27], resp. its generalization $(P\Omega)$ introduced by Bonet and Doma\'nski in [1].