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