Harnack inequalities for non-local operators of variable order

Abstract

We consider harmonic functions with respect to the operator \[ L u ( x ) = ∫ [ u ( x + h ) − u ( x ) − 1 ( | h | ≤ 1 ) h ⋅ ∇ u ( x ) ] n ( x , h ) d h . \mathcal {L} u(x)=\int [u(x+h)-u(x)-1_{(|h|\leq 1)} h\cdot \nabla u(x)] n(x,h) \, dh. \] Under suitable conditions on n ( x , h ) n(x,h) we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator L \mathcal {L} is allowed to be anisotropic and of variable order.