The 3G inequality for a uniformly John domain

Abstract

Let G be the Green function for a domain D R^d with d ≥ 3. The Martin boundary of D and the 3G inequality:

for x,y,z D

are studied. We give the 3G inequality for a bounded uniformly John domain D, although the Martin boundary of D need not coincide with the Euclidean boundary. On the other hand, we construct a bounded domain such that the Martin boundary coincides with the Euclidean boundary and yet the 3G inequality does not hold.