Notes on Beurling's theorem (Q793191)

The paper offers important improvements to the description of exceptional sets regarding to existence of fine limits at the boundary of a harmonic function u on an open Riemann surface R with Kuramochi compactification \(R^*\) and Kuramochi metric d. For any \(p\in \Delta_ 1\), the set of minimal points of \(\Delta =R^*\backslash R\), let \(u^ N(P)\) denote the N-fine cluster set of u at p. Then, as a consequence of the main theorem, the author proves the following: If u is quasi Dirichlet finite, then \(S=\{p\in \Delta_ 1 | diam u^ N(P)>0\}\) is a set of capacity zero. The paper contains an example showing that in some sense, the result is near of being the best possible. In fact in \(R=\{z\in {\mathbb{C}} | | Z|<1\},\) there exists a quasi Dirichlet finite harmonic function such that \(\{p\in \Delta_ 1 | u^ N(P)=\{\infty \}\}\) is a set of positive capacity. Further, the author gives in his context a Riezs-type theorem.