# The 3G inequality for a uniformly John domain Artikel i vetenskaplig tidskrift, 2005

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

uniformly Johyan domain

boundary Harnack principle

inner uniform domain

3G inequality

Green function

## Författare

#### Hiroaki Aikawa

Hokkaido University

#### Torbjörn Lundh

Göteborgs universitet

Chalmers, Matematiska vetenskaper

#### Kodai Mathematical Journal

Vol. 28 2 209-219

#### Ämneskategorier

Matematisk analys

2018-09-22