Topological properties of kernels of partial differential operators

Abstract

For a linear partial differential operator with constant coefficients on $\sD'(\Omega)$, we investigate topological properties like barrelledness or bornolo\-gicity (which allow applications of fundamental principles like the Banach-Steinhaus or the open mapping theorem) of its kernel. Using recent functional analytic results inspired by homological algebra we prove that almost all barrelledness type conditions are equivalent in this situation and provide two distinct sufficient conditions which, in particular, are satisfied if the operator is surjective or hypoelliptic. This last case generalizes a classical result of Malgrange and H\"ormander.