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

TL;DR

This paper characterizes the condition $( abla)$ for kernels of differential operators using shifted fundamental solutions, providing new insights into parameter dependence and solution spaces for differential equations.

Contribution

It introduces new characterizations of kernel conditions via shifted fundamental solutions and applies these to parameter dependence and solution space properties.

Findings

Characterization of $( abla)$ condition for smooth kernels.

Characterization of $(P abla)$ and $(Par{ar{ abla}})$ for distributional kernels.

New proof that solution spaces satisfy $( abla)$ for any differential operator.

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 $X \subseteq \mathbb{R}^d$.