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We characterize the condition (Ω) for smooth kernels of partial differential operators in terms of the existence of shifted fundamental solutions satisfying certain properties. The conditions (P Ω) and (P Ω ¯ ¯) 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 ∈ E (X) | P (D) f = 0 } satisfies (Ω) for any differential operator P(D) and any open convex set X ⊆ R d . [ABSTRACT FROM AUTHOR]