On the extension of separately hyperharmonic functions and \(H^ p\)- functions
[Shortened version of the original review.] Given the definition of an n-hyperharmonic function and an n-small set in \({\mathbb{C}}^ n\) the author proves the following theorems: Theorem 1: Let G be an open set in \({\mathbb{C}}^ n\). Let \(E\subset G\) be closed in G and n-small. Let \(u: G\setminus E\to (-\infty,\infty]\) be n- hyperharmonic. If u is locally bounded below in G, then u has a unique n- hyperharmonic extension \(u^*: G\to (-\infty,\infty].\) Theorem 2. Let G be an open set in \({\mathbb{C}}^ n\). Let \(E\subset G\) be closed in G and n-small. Let \(f: G\setminus E\to {\mathbb{C}}\) be a holomorphic function such that for some \(p>0\), \(| f|^ p\) has an n-hyperharmonic majorant u in \(G\setminus E\). If u is superharmonic, then f has a unique holomorphic extension \(f^*: G\to {\mathbb{C}}.\) Theorem 3. Let \(E\subset U^ n\) be closed in \(U^ n\) and n-small. Let \(f: U^ n\setminus E\to {\mathbb{C}}\) be a holomorphic function such that for some \(p>0\), \(| f|^ p\) has an n-harmonic majorant u in \(U^ n\setminus E\). Then f has a unique holomorphic extension \(f^*: U^ n\to {\mathbb{C}}\) such that \(| f^*|^ p\) has an n-harmonic majorant in \(U^ n.\) Theorem 3 includes the results of \textit{P. Järvi} [Proc. Am. Math. Soc. 86, 596-598 (1982; Zbl 0532.32004)], and the author in J. Lond. Math. Soc., II. Ser. 27, 281-288 (1983; Zbl 0479.30007)]. Several further theorems and corollaries are proved.
scientific article; zbMATH DE number 3937523