One-sided extendability of bounded functions
type: Item
instance of: scholarly article
title: One-sided extendability of bounded functions
zbMATH Open document ID: 1473.30030
author: Myrto Manolaki
author: Konstantinos Maronikolakis
author: V. Nestoridis
published in: Complex Analysis and its Synergies
publication date: 2021-07-12T00:00:00Z
review text: Let \(\gamma: I\longrightarrow\mathbb C\) be a rectifiable Jordan arc, where \(I\subset\mathbb R\) is a closed interval. Set \(J:=\gamma(I)\). We say that a function \(f: J\longrightarrow\mathbb C\) admits a one-sided bounded holomorphic extension if there exist a Jordan domain \(\Omega\), a non-trivial arc \(A\subset J\), and a function \(F\in\mathcal H^\infty(\Omega)\) such that \(A\subset\partial\Omega\) and \(\lim_{z\to\zeta}F(z)=f(\zeta)\) for a.e. \(\zeta\in A\). The main results of the paper are the following theorems: \begin{itemize} \item[\(\bullet\)] The set \(\{f\in L^\infty(J): f\) admits a one-sided bounded holomorphic extension\(\}\) is an \(\mathcal F_\sigma\) subset of \(L^\infty(J)\) of the first Baire category. \item[\(\bullet\)] Let \(p>1\) and let \(J\) be a subarc of the unit circle \(\mathbb T=\partial\mathbb D\). Then for any \(M>0\) and any compact \(K\subset J\) of positive measure, the set \[\{f\in L^p(J): \exists_{F\in\mathcal H^p(\mathbb D)}: \|F\|_{\mathcal H^p}\leq M,\lim_{\substack{z\to\zeta\\\text{non-tangential}}}F(z)=f(\zeta)\] for a.e. \(\zeta\in K\}\) is closed and has empty interior in \(L^p(J)\). \end{itemize}
reviewed by: Marek Jarnicki
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30B30
zbMATH DE Number: 7370983
zbMATH Keywords: \(L^\infty\) spaces
zbMATH Keywords: holomorphic extensions
zbMATH Keywords: non-tangential limits
MaRDI profile type: Publication
full work available at URL: s40627-021-00076-x
OpenAlex ID: W3154121200
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DOI: 10.1007/S40627-021-00076-X
description: scientific article; zbMATH DE number 7370983