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) is surjective on the space of E-valued (ultra-)distributions over an arbitrary convex set if E is a nuclear Fréchet space with property (DN). In particular, this holds if E is isomorphic to the space of tempered distributions S or to the space of germs of holomorphic functions over a one-point set H({0}). This result has an interpretation in terms of solving the scalar equation P(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 Rd is obtained as well. To get the above results we develop a splitting theory for short exact sequences of the form 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) − ( ) splitting theorem of Vogt and Wagner is obtained. © 2005 Elsevier Inc. All rights reserved.