Parameter dependence of solutions of differential equations on spaces of distributions and the splitting of short exact sequences

We show that a linear partial differential operator with constant coefficients P(D)P(D) is surjective on the space of E  -valued (ultra-)distributions over an arbitrary convex set if E′E′ is a nuclear Fréchet space with property (DN). In particular, this holds if E   is isomorphic to the space of tempered distributions S′S′ or to the space of germs of holomorphic functions over a one-point set H({0})H({0}). This result has an interpretation in terms of solving the scalar equation P(D)u=fP(D)u=f such that the solution u depends on parameter whenever the right-hand side f   also depends on the parameter in the same way. A suitable analogue for surjective convolution operators over RdRd is obtained as well. To get the above results we develop a splitting theory for short exact sequences of the form0⟶X⟶Y⟶Z⟶0,0⟶X⟶Y⟶Z⟶0,where Z is a Fréchet Schwartz space and X, Y   are PLS-spaces, like the spaces of distributions or real analytic functions or their subspaces. In particular, an extension of the (DN)-(Ω)(DN)-(Ω) splitting theorem of Vogt and Wagner is obtained.