Harmonic functions on finitely sheeted unlimited covering surfaces. (Q1396417)

It is considered the following objects: the classes \(HP(R)\) and \(HB(R)\) of positive and bounded harmonic functions on a Riemann surface \(R\), open Riemann surface \(W\) possessing Green's functions, p-sheeted unlimited covering surface \(\widetilde{W}\) of \(W\) with projection map \(\varphi\). Let \(X\) be one of the symbols \(P\) or \(B\). Evidently, that \[ HX(W)\circ \varphi :=\{h\circ\varphi: h\in HX(W)\subset HX(\widetilde{W})\} \] The author give a necessary and suffisient condition, in terms of Martin boundary, for \(HX(W)\circ\varphi=HX(\widetilde{W})\). As example they give the following proposition. Let \(A=(1-2^{-n-1}\exp{\frac{2\pi i k}{2^{n+2}}}\), \(n=1,2,\dots\), \(k=1,2,\dots,2^{n+2})\) and let \(\widetilde{D}\) be p-sheets unlimited covering surface of the unit disk \(D\) with branch points in \(\widetilde{D}\) of multiplicity \(p\) for every \(z\in A\). Then \(HP(D)\circ \varphi=HP(\widetilde{D})\).