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march 2016 Cauchy transformation and mutual dualities between $A^{-\infty}(\Omega)$ and $A^\infty(\complement\Omega)$ for Carathéodory domains
A.V. Abanin, Le Hai Khoi
Bull. Belg. Math. Soc. Simon Stevin 23(1): 87-102 (march 2016). DOI: 10.36045/bbms/1457560856

Abstract

Let $\Omega$ be a Carathéodory domain in the complex plane $\mathbb C$, $A^{-\infty}(\Omega)$ the space of functions that are holomorphic in $\Omega$ with polynomial growth near the boundary $\partial\Omega$, and $A^\infty(\complement\Omega)$ the space of holomorphic functions in the interior of $\complement\Omega:=\overline{\mathbb C}\setminus\Omega$, vanishing at infinity and being in $C^\infty(\complement\Omega)$. We prove that the Cauchy transformation of analytic functionals establishes a mutual duality between spaces $A^{-\infty}(\Omega)$ and $A^\infty(\complement\Omega)$. This result, together with those of [3], gives a solution to duality problem for the space $A^{-\infty}(\Omega)$ in both one and several complex variables.

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A.V. Abanin. Le Hai Khoi. "Cauchy transformation and mutual dualities between $A^{-\infty}(\Omega)$ and $A^\infty(\complement\Omega)$ for Carathéodory domains." Bull. Belg. Math. Soc. Simon Stevin 23 (1) 87 - 102, march 2016. https://doi.org/10.36045/bbms/1457560856

Information

Published: march 2016
First available in Project Euclid: 9 March 2016

zbMATH: 1357.46021
MathSciNet: MR3471981
Digital Object Identifier: 10.36045/bbms/1457560856

Subjects:
Primary: 32A10‎, 46F15

Rights: Copyright © 2016 The Belgian Mathematical Society

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Vol.23 • No. 1 • march 2016
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