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

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 (Stud. Math. 68:225–240, 1980), resp. its generalization $(P\Omega )$ introduced by Bonet and Domański (J. Funct. Anal. 230:329–381, 2006).