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

Karsten Kruse

• This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator ∂¯¯¯ on spaces EV(Ω,E) of C∞-smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights V. Vector-valued means that these functions have values in a locally convex Hausdorff space E over C. We derive a counterpart of the Grothendieck-Köthe-Silva duality O(C∖K)/O(C)≅A(K) with non-empty compact K⊂R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of ∂¯¯¯:EV(Ω,E)→EV(Ω,E) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on EV(Ω,C).