PUB - Publikationen an der Universität Bielefeld

Let D be a bounded open subset in R-d, d >= 2 and let G denote the Green function for D with respect to (-Delta)(alpha/2), 0 < alpha <= 2, alpha < d. If alpha < 2, assume that D satisfies the interior corkscrew condition-, if alpha=2, i.e., if G is the classical Green function on D, assume-more restrictively-that D is a uniform domain. Let g = G((.), y(0)) Lambda 1 for some y(0) is an element of D. Based on the uniform boundary Harnack principle, it is shown that G has the generalized triangle property which states that G(z, y)/g(z) <= C G(x, y)/g(x) when d(z, x) <= d(z, y). An intermediate step is the approximation G(x, y) approximate to vertical bar x - y vertical bar(alpha-d)g(x)g(y)/g(A)(2), where A is an arbitrary point in a certain set B(x, y). This is discussed in a general setting where D is a dense open subset of a compact metric space satisfying the interior corkscrew condition and G is a quasi-symmetric positive numerical function on D x D which has locally polynomial decay and satisfies Harnack's inequality. Under these assumptions, the uniform boundary Harnack principle, the approximation for G, and the generalized triangle property turn out to be equivalent. (c) 2005 Elsevier Inc. All rights reserved.