On Zahariuta's extremal function for harmonic functions
The author studies the relationship between the \(Lh_0\)-regularity and \(Lh\)-regularity introduced by \textit{V. P. Zahariuta} [Functional analysis, Proc. Conf. Essen 1991, Lect. Notes Pure Appl. Math. 150, 497-522 (1993; Zbl 0792.31001)] and \(H\)-regularity [see \textit{J. Siciak}, Ann. Pol. Math. 20, 267-278 (1968; Zbl 0182.14702)]. A compact subet \(K\) of \(\mathbb{R}^n\) is called \(Lh\)-regular if and only if its complement \(K^*\) is \(Lh\)-regular. If \(E\subseteq \mathbb{R}^n\), then \({\mathcal H}(E)\) denotes the space of functions \(h\) that are harmonic on some open set \(\omega_h\) containing \(E\). Let \(B\) denote the open unit ball in \(\mathbb{R}^n\), let \(K\) be a compact set in \(\mathbb{R}^n\) such that \(K^*\) is connected, and let \(\Omega\) be a connected open set in \(\mathbb{R}^n\). It is shown that (i) \({\mathcal H}(\Omega)\) is isomorphic to \({\mathcal H}(B)\) if and only if \(\Omega\) is \(Lh\)-regular, (ii) if \(K\subset \Omega\) and \(K,\Omega\) are \(Lh\)-regular, then \({\mathcal H}(K)\) and \({\mathcal H}(\Omega)\) possess a common basis. Also, \(K\) is \(Lh_0\)-regular if and only if \(K\) is \(H\)-regular, provided that \(K\) is non-polar. The paper contains some results related to the three-balls theorem of \textit{J. Korevaar} and \textit{J. L. H. Meyers} [Bull. Lond. Math. Soc. 26, 353-362 (1994; Zbl 0819.31001)].
scientific article; zbMATH DE number 1369982