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The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes
Citation Link: https://doi.org/10.15480/882.3893
Publikationstyp
Journal Article
Publikationsdatum
2023-01
Sprache
English
Author
Kruse, Karsten
Institut
Enthalten in
Volume
74
Issue
1
Start Page
81
End Page
112
Citation
Collectanea Mathematica 74 (1): 81-112 (2023-01)
Publisher DOI
Scopus ID
2-s2.0-85117313444
ArXiv ID
Publisher
Springer
Peer Reviewed
true
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).
Schlagworte
Cauchy-Riemann
Parameter dependence
Smooth
Solvability
Vector-valued
Weight
DDC Class
510: Mathematik