Completeness of spaces of harmonic functions under restricted supremum norms (Q1880499)

Let \(\Omega\) be a domain in \({\mathbb R}^n, n\geq 2, E\subset \Omega , E\neq\Phi, \) and let \(h_E(\Omega)\) be the collection of all harmonic functions on \(\Omega\) bounded on \(E.\) Define \(\| h\| _E=\sup_E | h| ,\) where \(h\in h_E(\Omega).\) Is the set \(h_E(\Omega)\) endowed by \(\| \cdot \| _E\) a Banach space? Let \(\Omega \cup \{ A \}\) denote the Alexandroff (one-point) compactification of \(\Omega.\) Given an open subset \(\omega \subset \Omega \) the point \(A\) is called accessible from \(\omega\) if there is a continuous function \(\gamma: [0,\infty)\rightarrow \omega \) such that \(\gamma (t)\to A\) as \(t\to\infty.\) Arcozzi and Björn conjectured the following Theorem. The pair \((h_E(\Omega), \| \cdot\| _E)\) is a Banach space if and only if \(A\) is not accessible from \(\Omega \setminus \bar{E}.\) The author proves this theorem.