The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes
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Kruse, Karsten [1]
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[1] Hamburg University of Technology
Hamburg University of Technology
Hamburg, Freie und Hansestadt, Alemania
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- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 74, Fasc. 1, 2023, págs. 81-112
- Idioma: inglés
- DOI: 10.1007/s13348-021-00337-2
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Resumen
This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator \overline{\partial } on spaces {\mathcal {E}}{\mathcal {V}}(\varOmega ,E) of {\mathcal {C}}^{\infty }-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}}. Vector-valued means that these functions have values in a locally convex Hausdorff space E over {\mathbb {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) with non-empty compact K\subset {\mathbb {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) 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}}).
