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# The 3G inequality for a uniformly John domain

Hiroaki Aikawa ; Torbjörn Lundh (Institutionen för matematiska vetenskaper)
Kodai Mathematical Journal Vol. 28 (2005), 2, p. 209-219.

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.

Nyckelord: Green function, 3G inequality, boundary Harnack principle, uniformly Johyan domain, inner uniform domain

CPL Pubid: 23829

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