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

Abstract

We characterize the condition $(\Omega)$ for smooth kernels of partial differential operators in terms of the existence of shifted fundamental solutions satisfying certain properties. The conditions $(P\Omega)$ and $(P\overline{\overline{\Omega}})$ for distributional kernels are characterized in a similar way. By lifting theorems for Fr\'echet 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 $\{ f \in \mathscr{E}(X) \, | \, P(D)f = 0\}$ satisfies $(\Omega)$ for any differential operator $P(D)$ and any open convex set $X \subseteq \mathbb{R}^d$.