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

The abstract Cauchy problem in locally convex spaces

“…Let E be a sequentially complete C-lcHs, K := [0, ∞] or K := {∞} and equip bv K (E) with its usual quotient topology, which is Hausdorff locally convex by [37,Remark 14,p 22]. By [38,Theorem 7.2 (ii), p. 106] the Laplace transform…”

The abstract Cauchy problem in locally convex spaces

“…Let E be a sequentially complete C-lcHs, K ∶= [0, ∞] or K ∶= {∞} and equip bv K (E) with its usual quotient topology, which is Hausdorff locally convex by [37,Remark 14,p 22]. By [38,Theorem 7.2 (ii), p. 41] the Laplace transform…”

Parameter dependence of solutions of the Cauchy-Riemann equation on spaces of weighted smooth functions

“…Ω n ∶= {z ∈ C Im(z) < n}. The case that this sequence has holes along the real axis is treated in [47].…”