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 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 \(\{ f \in {\mathscr {E}}(X) \, | \, P(D)f = 0\}\) satisfies \((\Omega )\) for any differential operator P(D) and any open convex set \(X \subseteq {\mathbb {R}}^d\).