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Publisher DOI: 10.1007/s13348-021-00337-2
Title: The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes
Language: English
Authors: Kruse, Karsten  
Keywords: Cauchy-Riemann; Parameter dependence; Smooth; Solvability; Vector-valued; Weight
Issue Date: 19-Oct-2021
Publisher: Springer
Source: Collectanea Mathematica (in Press) : (2021)
Abstract (english): 
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).
DOI: 10.15480/882.3893
ISSN: 2038-4815
Journal: Collectanea mathematica 
Institute: Mathematik E-10 
Document Type: Article
Project: Projekt DEAL 
More Funding information: Open Access funding enabled and organized by Projekt DEAL.
Peer Reviewed: Yes
License: CC BY 4.0 (Attribution) CC BY 4.0 (Attribution)
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