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Harmonic Functions for a Class of Integro-differential Operators

Authors:

Mohammud Foondun

Affiliation:

(1) Department of Mathematics, The University of Utah, 155 S. 1400 E., Salt Lake City, UT 84112–0090, USA

Abstract:

We consider the operator ${cal L}$

defined on $C^2({Bbb R}^d)$ functions by

$begin{array}{lll} {cal L} f(x)&=&frac{1}{2}sumlimits_{i,j=1}^d a_{ij}(x)frac{partial^2f(x)}{partial x_ipartial x_j}+sumlimits_{i=1}^d b_i(x)frac{partial f(x)}{partial x_i} &&+int_{{Bbb R}^dbackslash{0}}left[f(x+h)-f(x)-1_{(|h|leq1)}hcdot {nabla} f(x)right]n(x,h)dh. end{array}$

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 ${Bbb R}^d$ 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.

Keywords:

Harnack inequality Harmonic functions Jump processes Integro-differential operators

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