Harmonic functions on finitely sheeted unlimited covering surfaces

Dedicated to Professor Masayuki Ito on his sixtieth birthday

Abstract

We denote by HP(R) and (HB(R), resp.) the class of positive (bounded, resp.) harmonic functions on a Riemann surface R. Consider an open Riemann surface W possessing a Green's function and a p-sheeted ( 1<p<∞) unlimited covering surface ˜{W} of W with projection map \varphi. We give a necessary and sufficient condition, in terms of Martin boundary, for HX(W)\circ\varphi=HX(˜{W})(X=P, B). We also give some examples illustrating the above result when W is the unit disc.