Linear topological invariants for kernels of differential operators by shifted fundamental solutions

We characterize the condition $(\mathrm{\Omega })$ for smooth kernels of partial differential operators in terms of the existence of shifted fundamental solutions satisfying certain properties. The conditions $(P\mathrm{\Omega })$ and $(P\overline{\stackrel{‾}{\mathrm{\Omega }}})$ for distributional kernels are characterized in a similar way. By lifting theorems for Fréchet spaces and (PLS)-spaces, this provides characterizations of the problem of parameter dependence for smooth and distributional solutions of differential equations by shifted fundamental solutions. As an application, we give a new proof of the fact that the space $\{f\in \mathcal{E}(X)\mathrm{\mid }P(D)f=0\}$ satisfies $(\mathrm{\Omega })$ for any differential operator $P(D)$ and any open convex set $X\subseteq {\mathbb{R}}^d$.