Distinguished normal operators on open Riemann surfaces

for any V\Z> Vi and any continuous function/ on dV\. Consider the Kerekjartό-Stoilow compactification W* of W and denote the boundary by β(W)= W*— W. Partition β(W) into two disjoint sets a and γ where a is non-empty closed. The purpose of this paper is to investigate the following boundary value problems: Suppose that the closure of Wo e V in W* contains a and that / is continuously differentiate in WQ and has DWo(f)< °° Then is there uniquely a function Hf satisfying the following conditions? (I) Hf is harmonic in W and has Dw(Hf)< oo, (II) Hf = Lv(Hf) for any Ve V such that the intersection of β(ΐP) with the closure of V is contained in r, (III) lim Hf(z) = lim/(.2r) for almost all curves r where each r is a locally