On subharmonic extension and extension in the Hardy-Orlicz classes (Q1339647)

Let \(E\) be a polar subset of a domain \(D\) in \(\mathbb{R}^ n\) \((n \geq 2)\). In the case where \(E\) is closed, a classical result of Brelot asserts that a function \(u\) that is bounded above and subharmonic on \(D \backslash E\) has a subharmonic extension to \(D\). Generalizations have been obtained by several authors, and the results of the present paper contain many of these generalizations. The authors define, in terms of Blaschke-Privalov operators, a class of functions \(A(D \backslash E)\). In the case where \(E\) is closed, \(A(D \backslash E)\) coincides with the class of subharmonic functions on \(D \backslash E\). The following theorem, which includes several known results, is proved: if \(u \in A(D \backslash E)\) and there exist functions \(v,w \in A(D \backslash E)\) with \(w \leq 0\) such that \[ \limsup_{x \to y} \biggl \{ \bigl( u(x) + \varepsilon v(x) \bigr)/ \bigl( -w(x) \bigr) \biggr\} \leq 0 \] for all \(\varepsilon > 0\) and all \(y \in E\), then \(u\) has a subharmonic extension to \(D\). Extension results are proved for the Hardy-Orlicz classes and the Smirnov class. In particular, it is shown that for certain Hardy-Orlicz classes polar sets (not necessarily closed) are removable.