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:
\frac{G(x, y)G(y, z)}{G(x, z)}≤A(|x−y|^2−d+|y−z|^2−d)   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.