Harmonic functions for a class of integro-differential operators

We consider the operator LL defined on C2(Rd)C2(Rd) functions by Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇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 RdRd 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.