Kuramochi boundary and harmonic functions with finite Dirichlet integrals on unlimited covering surfaces (Q1885242)

Let \(W\) be an open Riemann surface with Green function and \(\widetilde W\) an \(m\)-sheeted \((1< m< \infty)\) unlimited covering surface of \(W\). Denote by \(\pi= \pi_{\widetilde W}\) the projection of \(\widetilde W\) onto \(W\)- and \(HD(W)\) (resp. \(H\widetilde D(W)\) the set of harmonic functions with finite integrals on \(W\) (resp. \(\widetilde W\)). In this paper the authors obtain necessary and sufficient conditions in terms of the Kuramochi compactification for the property that \(HD(\widetilde W)= HD(W)\circ \pi\) is true.