The 3G inequality for a uniformly John domain
Journal article, 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


Hiroaki Aikawa

Hokkaido University

Torbjörn Lundh

University of Gothenburg

Chalmers, Mathematical Sciences

Kodai Mathematical Journal

Vol. 28 2 209-219

Subject Categories

Mathematical Analysis

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