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

We provide a sufficient condition for a linear differential operator with constant coefficients $P(D)$ to be surjective on $C^{\mathrm{\infty }}(X)$ and ${\mathcal{D}}^{\mathrm{\prime }}(X)$, 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 $C^{\mathrm{\infty }}(X)$, resp. on ${\mathcal{D}}^{\mathrm{\prime }}(X)$, is derived. Additionally, we obtain for certain surjective differential operators $P(D)$ on $C^{\mathrm{\infty }}(X)$, resp. ${\mathcal{D}}^{\mathrm{\prime }}(X)$, that the spaces of zero solutions $C_P^{\mathrm{\infty }}(X)=\{u\in C^{\mathrm{\infty }}(X);P(D)u=0\}$, resp. ${\mathcal{D}}_P^{\mathrm{\prime }}(X)=\{u\in {\mathcal{D}}^{\mathrm{\prime }}(X);P(D)u=0\}$ possess the linear topological invariant $(\mathrm{\Omega })$ introduced by Vogt and Wagner in [27], resp. its generalization $(P\mathrm{\Omega })$ introduced by Bonet and Doma'nski in [1].