Cauchy transformation and mutual dualities between $A^{-\infty}(\Omega)$ and $A^\infty(\complement\Omega)$ for Carathéodory domains

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.