Cauchy transformation and mutual dualities between \(A^{-\infty}(\Omega)\) and \(A^\infty(\complement\Omega)\) for Carathéodory domains (Q267515)

Let \(\Omega\) be a domain in the complex plane \(\mathbb{C}\) and denote by \(A^{-\infty}(\Omega)\) the space of functions that are holomorphic in \(\Omega\) with polynomial growth near the boundary \(\partial\Omega\). Define \(\Omega^d:=\overline{\mathbb{C}}\setminus\Omega\) and let \(A^{\infty}(\Omega^d)\) be the space of holomorphic functions in the interior of \(\Omega^d\) which are smooth in the whole of \(\Omega^d\) and vanish at infinity. The authors show that the Cauchy transform constitutes mutual dualities between these two types of spaces for some domains \(\Omega\). Specifically, they show that, if \(\Omega\) is a bounded regular domain, then the Cauchy transform is an epimorphism from \((A^{-\infty}(\Omega))'_b\) onto \(A^{\infty}(\Omega^d)\) (Proposition 3.1). If, in addition, this domain \(\Omega\) also has rectifiable boundary, then the Cauchy transform is an isomorphism from \((A^{\infty}(\Omega^d))'_b\) onto \(A^{-\infty}(\Omega)\) (Theorem 4.5). Injectivity of the first-named epimorphism is obtained for more specific domains. Namely, if \(\Omega\) is a Carathéodory domain, then the Cauchy transform is an isomorphism between \((A^{-\infty}(\Omega))'_b\) and \(A^{\infty}(\Omega^d)\) (Theorem 4.3). Corollary 4.6 establishes mutual duality between \(A^{\infty}(\Omega^d)\) and \(A^{-\infty}(\Omega)\) for Carathéodory domains with rectifiable boundary.