Harmonic functions on finitely sheeted unlimited covering surfaces

Abstract

We denote by $HP\left(R\right)$ and ($HB\left(R\right)$, 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 unlimited covering surface $\tilde{W}$ of $W$ with projection map $\varphi$. We give a necessary and sufficient condition, in terms of Martin boundary, for $HX\left(W\right)\circ$$\varphi$$=HX\left(\tilde{W}\right)(X=P,B)$. We also give some examples illustrating the above result when $W$ is the unit disc.