Harmonic functions for a class of integro-differential operators

2008/12/11 by Foondun, Mohammud
#60J75 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0812.2082

Abstract

We consider the operator \sL defined on C²(\bR^d) functions by \sL f(x)&=&1/2∑_i,j=1^d a_ij(x)(∂²f(x))/(∂ x_i∂ x_j)+∑_i=1^d b_i(x)(∂ f(x))/(∂ x_i) &+&∫_\bR^d\backslash\0\[f(x+h)-f(x)-1_(|h|≤1)h⋅ \grad f(x)]n(x,h)dh. Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on n(x,h), we establish a Harnack inequality for functions that are nonnegative in \bR^d and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on n(x,h). A regularity theorem for those nonnegative harmonic functions is also proved

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