Comparison of the classes of Wiener functions

For a harmonic space satisfying the axioms of M. Brelot [ΊΓ|, one can define the notion of Wiener functions as a generalization of that for a Riemann surface or a Green space (see [2]). The class of Wiener functions may be used to see global properties of the harmonic space in particular, in order to show that a compactification of the base space be resolutive with respect to the Dirichlet problem, it is enough to verify that every continuous function on the compactification is a Wiener function (see Theorem 4.4 in [2]). Thus, given two harmonic structures ξ>i and ξ>2 on the same base space Ω, it may be useful to know when the inclusion BWCBW holds, where BW(i = l, 2) is the class of bounded Wiener functions with respect to φ, (ϊ = l, 2). In this paper, we shall give a sufficient condition for the above inclusion, which includes the conditions given in [4] and [5] for special cases.