One-sided extendability of bounded functions
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}
scientific article; zbMATH DE number 7370983
scientific article