Kodai Mathematical Journal

The 3G inequality for a uniformly John domain

Hiroaki Aikawa and Torbjörn Lundh

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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.

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Kodai Math. J. Volume 28, Number 2 (2005), 209-219.

First available in Project Euclid: 11 August 2005

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Aikawa, Hiroaki; Lundh, Torbjörn. The 3G inequality for a uniformly John domain. Kodai Math. J. 28 (2005), no. 2, 209--219. doi:10.2996/kmj/1123767003. http://projecteuclid.org/euclid.kmj/1123767003.

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