Theorems of Plessner and Riesz types for finely harmonic morphims (Q805805)

Two boundary behavior theorems for finely harmonic morphisms between Riemann surfaces will be proved. The first one is of Plessner type: Let R be a hyperbolic Riemann surface, \(R'\) a Riemann surface, \(U\subset R\) a fine domain and \(\phi: U\to R'\) a finely harmonic morphism. If \(R'\) is hyperbolic, then \(\phi\) has a fine limit at \(\omega_ x\)-almost every point of the minimal Martin boundary part \(\Delta_ 1(U)\), \(x\in R\). If \(R'\) is parabolic and \(R'\setminus \phi(U)\) polar, then either \(\phi\) has a fine limit at \(\zeta\) or the fine cluster set \(\phi^\wedge(\zeta)\) is the whole Martin compactification \(R_ M^{'*}\) at \(\omega_ x\)-almost every point \(\zeta \in \Delta_ 1(U)\), \(x\in R\). The second theorem, of Riesz type, considers the same general situation as above. If then a polar set \(N\subset R'\) exists such that the harmonic measure of \(\{\zeta \in \Delta_ 1(U)|\) \(\phi^\wedge(\zeta)\subset N\}\) is strictly positive, then \(\phi\) reduces to a constant mapping. These theorems will be proved by probabilistic arguments.