1. The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes
- Author
-
Karsten Kruse
- Subjects
General Mathematics ,Holomorphic function ,Duality (optimization) ,Omega ,Cauchy-Riemann ,Combinatorics ,Mathematics - Analysis of PDEs ,Fréchet space ,Computer Science::Systems and Control ,FOS: Mathematics ,35A01, 35B30, 32W05, 46A63 (Primary), 46A32, 46E40 (Secondary) ,ddc:510 ,Mathematik [510] ,Mathematics ,Kernel (set theory) ,Applied Mathematics ,Operator (physics) ,Hausdorff space ,Parameter dependence ,Weight ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Solvability ,Vector-valued ,Smooth ,Complex plane ,Analysis of PDEs (math.AP) - 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 ) .
- Published
- 2023