We denote by and (, resp.) the class of positive (bounded, resp.) harmonic functions on a Riemann surface . Consider an open Riemann surface possessing a Green's function and a -sheeted unlimited covering surface of with projection map . We give a necessary and sufficient condition, in terms of Martin boundary, for . We also give some examples illustrating the above result when is the unit disc.
Hiroaki MASAOKA. Shigeo SEGAWA. "Harmonic functions on finitely sheeted unlimited covering surfaces." J. Math. Soc. Japan 55 (2) 323 - 334, April, 2003. https://doi.org/10.2969/jmsj/1191419119