On Fatou's and Beurling's Theorems
Zenjiro Kuramochi · Hokkaido Mathematical Journal · 1982
The purpose of the present paper is to give simple proofs for well known Fatou and Beurling's theorems for harmonic functions and to amelio \cdot rate the Beurling's theorem for analytic functions given in a previous paperO.Let R be a Riemann surface ot\in 0_{g} and \{R_{n}\} : n=0,1 , \cdots be its exhaustion.We suppose \alpha -Martin's topology is defined on \overline{R}=R+\Delta^{a} , where \alpha=K or N.Let U(z) : z\in R be a harmonic function, i .e .U(z) is a mapping from R into a real axis.Let \Delta_{1}^{\alpha} be the set of \alpha -minimal points^{2)} of \Delta^{\alpha} .The fine cluster set A(U(p))\alpha at p\in\Delta_{1}^{\alpha} is defined aswhere G_{\tau} is a fine neighbourhood of p with respect to \alpha -Martin's topology.If A^{\alpha}(U(p)) is a single point, we say U(z) has a fine limit denoted by U^{\alpha}(p) .Then the following Lemma is well known.Lemma 1.1) Let G be an open set in R and v(p) be a neighbourhood of p relative to \alpha -Martin's topology.Then \alpha 1) Let p\in\Delta_{1}^{\alpha} .Then a) v(p) i p .b) There exists only one component G' of G such that G' i pa and G^{\alpha} i p implies (CG)^{0}\exists^{\alpha} i p .If G_{i}^{\alpha} i p(i=1,2, \cdots , i_{0}) , (\begin{array}{l}\cap G_{i}i_{0}i=1\end{array}) i p\alpha .Hence A^{a}(U(p)) is a point or continuum.2) a) Let G'\subset G\subset RG' and G be open sets and let F be a closed set in \Delta_{1}^{K} .If the H. M. {harmonic measure) of F\cap\overline{G}' relative to G>0 : w(F\cap G', z, G)>0,(2) then there exists at least a point p\in F\cap\Delta_{1}^{K} such that G^{K} i p .b) Let G'\subset G and let F be a closed set in \Delta_{1}^{N} .If the C. P. [capacity) F\cap G' relative to