On a theorem of I.I.Privalov
Given a simply connected bounded domain G, put \(\rho_ G(z,\zeta):=\inf \{\) length of \(\gamma_{z\zeta}\}\), where inf is taken over all Jordan arcs \(\gamma_{z\zeta}\) in \(\bar G\) connecting z with \(\zeta\). If \(0<\alpha \leq 1\) and E is a subset of \(\bar G,\) let \(C^{\alpha}(E)\) denote the set of all functions f defined on E such that \[ | f(z)- f(\zeta)| \leq c\rho^{\alpha}_ G(z,\zeta),\quad z,\zeta \in E, \] where \(c=c(f,E,G).\) The aim of the paper is to find conditions imposed on G and \(\alpha\) such that for every function f continuous on \(\bar G\) and holomorphic in G the implication is true \[ text{Re}f|_{\partial G}\in C^{\alpha}(\partial G)\Rightarrow f\in C^{\alpha}(\bar G). \] In particular it is shown that if \[ \sup \{\rho_{{\mathbb{C}}\setminus G}(z,\zeta)/| z-\zeta |;\;z,\zeta \in \partial G\} \] is finite and \(0<\alpha <1/2\) then the implication is true.
scientific article; zbMATH DE number 4177485
scientific article