The splitting of exact sequences of PLS-spaces and smooth dependence of solutions of linear partial differential equations

We investigate the splitting of short exact sequences of the form0→X→Y→E→0,0→X→Y→E→0, where E is the dual of a Fréchet Schwartz space and X, Y   are PLS-spaces, like the spaces of distributions or real analytic functions or their subspaces. In particular, we characterize pairs (E,X)(E,X) as above such that Ext1(E,X)=0Ext1(E,X)=0 in the category of PLS-spaces and apply this characterization to many natural spaces X and E  . In particular, we discover an extension of the (DN)(DN)–(Ω)(Ω) splitting theorem of Vogt and Wagner. These abstract results are applied to parameter dependence of linear partial differential operators and surjectivity of such operators on spaces of vector-valued distributions.