Harmonic functions for a class of integro-differential operators

TL;DR

This paper establishes a Harnack inequality and regularity results for harmonic functions associated with a class of integro-differential operators, under conditions ensuring ellipticity and suitable behavior of the jump kernel.

Contribution

It introduces conditions under which the Harnack inequality holds for nonlocal operators and demonstrates its failure without these conditions, advancing understanding of harmonic functions for integro-differential operators.

Findings

Harnack inequality holds under uniform ellipticity and kernel conditions.

Failure of Harnack inequality without suitable kernel conditions.

Regularity results for nonnegative harmonic functions.

Abstract

We consider the operator $\sL$ defined on $C^2(\bR^d)$ functions by \sL f(x)&=&{1/2}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial^2f(x)}{\partial x_i\partial x_j}+\sum_{i=1}^d b_i(x)\frac{\partial f(x)}{\partial x_i} &+&\int_{\bR^d\backslash\{0\}}[f(x+h)-f(x)-1_{(|h|\leq1)}h\cdot \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