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 \(X \subseteq {\mathbb {R}}^d\) .
