On a class of holomorphic functions representable by Carleman formulas in some class of bounded, simply connected domains from their values on an analytic arc
type: Item
instance of: scholarly article
title: On a class of holomorphic functions representable by Carleman formulas in some class of bounded, simply connected domains from their values on an analytic arc
zbMATH Open document ID: 1110.30020
author: George Chailos
author: Alekos Vidras
published in: Monatshefte für Mathematik
publication date: 2007-01-26T00:00:00Z
review text: Let \({\mathcal U}\) be a bounded simply connected domain with Jordan rectifiable boundary containing an open analytic arc \(M\) (\(0 < m(M) < m({\mathcal U})\)). The quenching function is defined as an exponent of the solution to the Schwarz boundary value problem with piece-wise constant data (equal to 1 on \(M\) and to 0 on \(\partial {\mathcal U}\setminus M\)). It is proved that if \(f\) is a holomorphic function in \({\mathcal U}\) having angular boundary values almost everywhere on \(M\), Lebesgue-summable on \(M\), and \(\partial {\mathcal U}\) is regular with respect to \(M\), then 1) if \(f\) belongs to Hardy class \({\mathcal H}^1\) near the base \(M\) then it can be represented via Carleman formula \[ f(z) = \lim_{n\rightarrow \infty} \frac{1}{2\pi i} \int_{M} \frac{\omega^n(\zeta)}{\omega^n(z)} \frac{f(\zeta) \,d \zeta}{\zeta - z}, \] where convergence is uniform with respect to any compact subset of \({\mathcal U}\); 2) if the Carleman formula holds for \(f\) point-wise and in addition \(f\) has boundary values at the end-points of \(M\), then \(f\) belongs to Smirnov class \(E^1({\mathcal U}_{\tau})\) for certain Ahlfors-regular exhaustion of \({\mathcal U}\).
Mathematics Subject Classification ID: 30E20
Mathematics Subject Classification ID: 30E25
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 31A20
zbMATH DE Number: 5118930
zbMATH Keywords: holomorphic function
zbMATH Keywords: Carleman formulas
zbMATH Keywords: analytic arc
zbMATH Keywords: quenching function
zbMATH Keywords: angular boundary values
reviewed by: Sergei V. Rogosin
MaRDI profile type: Publication
full work available at URL: s00605-006-0401-0
OpenAlex ID: W1990616604
cites work: Q4532411
cites work: On a class of holomorphic functions representable by Carleman formulas in the disk from their values on the arc of the circle
cites work: La solution des conjectures de Calderon
cites work: L'intégrale de Cauchy définit un opératuer borne sur \(L^ 2 \)pour les courbes lipschitziennes
cites work: Q5574105
DOI: 10.1007/S00605-006-0401-0
description: scientific article; zbMATH DE number 5118930