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 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 characterise the weights V which give 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 to prove that the kernel ker ∂ of the Cauchy-Riemann operator ∂ in EV(Ω) ∶= EV(Ω, C) has the property (Ω) of Vogt. Then an application of the splitting theory of Vogt for Fréchet spaces and of Bonet and Domański for (PLS)-spaces in combination with some previous results on the surjectivity of the Cauchy-Riemann operator ∂∶ EV(Ω) → EV(Ω) yields the surjectivity of the Cauchy-Riemann operator on EV(Ω, E) if E ∶= F ′ b with some Fréchet space F satisfying the condition (DN ) or if E is an ultrabornological (PLS)-space having the property (P A). This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on EV(Ω).