The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes

Abstract

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator overline{partial } on spaces {mathcal {E}}{mathcal {V}}(varOmega ,E) of {mathcal {C}}^{infty }-smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights {mathcal {V}}. Vector-valued means that these functions have values in a locally convex Hausdorff space E over {mathbb {C}}. We derive a counterpart of the Grothendieck-Köthe-Silva duality {mathcal {O}}({mathbb {C}}setminus K)/{mathcal {O}}({mathbb {C}})cong {mathscr {A}}(K) with non-empty compact Ksubset {mathbb {R}} for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of overline{partial }:{mathcal {E}} {mathcal {V}}(varOmega ,E)rightarrow {mathcal {E}}{mathcal {V}} (varOmega ,E) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on {mathcal {E}}{mathcal {V}}(varOmega ,{mathbb {C}}).