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

We characterize the condition $$(\Omega )$$ ( Ω ) for smooth kernels of partial differential operators in terms of the existence of shifted fundamental solutions satisfying certain properties. The conditions $$(P\Omega )$$ ( P Ω ) and $$(P\overline{\overline{\Omega }})$$ ( P Ω ¯ ¯ ) 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 $$\{ f \in {\mathscr {E}}(X) \, | \, P(D)f = 0\}$$ { f ∈ E ( X ) | P ( D ) f = 0 } satisfies $$(\Omega )$$ ( Ω ) for any differential operator P ( D ) and any open convex set $$X \subseteq {\mathbb {R}}^d$$ X ⊆ R d .