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

Abstract

We investigate the splitting of short exact sequences of the form 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 ) as above such that Ext 1 ( 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 ( D N ) – ( 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.