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We characterize the condition $(\Omega)$ for smooth kernels of partialdifferential operators in terms of the existence of shifted fundamentalsolutions satisfying certain properties. The conditions $(P\Omega)$ and$(P\overline{\overline{\Omega}})$ for distributional kernels are characterizedin a similar way. By lifting theorems for Fr\'echet spaces and (PLS)-spaces,this provides characterizations of the problem of parameter dependence forsmooth and distributional solutions of differential equations by shiftedfundamental solutions. As an application, we give a new proof of the fact thatthe space $\{ f \in \mathscr{E}(X) \, | \, P(D)f = 0\}$ satisfies $(\Omega)$for any differential operator $P(D)$ and any open convex set $X \subseteq\mathbb{R}^d$.

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