The 3G inequality for a uniformly John domain

Abstract

Let G be the Green function for a domain D $\subset$ R^d with d ≥ 3. The Martin boundary of D and the 3G inequality: $$\frac{G(x,y)G(y,z)}{G(x,z)} \le A(|x-y|^{2-d}+|y-z|^{2-d})\quad \text{for}\quad x,y,z \in D$$ 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.