Harnack Inequality for Non-Local Schrodinger Operators

Let x is an element of R-d, d >= 3, and f : R-d -> R be a twice differentiable function with all second partial derivatives being continuous. For 1 <= i, j <= d, let a(ij) : R-d -> R be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrodinger operator associated to Lf (x) =1/2 Sigma(d)(i=1) Sigma(d)(i=1) partial derivative/partial derivative x(i) (a(ij) (.) partial derivative f/partial derivative x(j)) (x) + integral(Rd\{0}) [f(y) - f (x)] J (x, y) dy where J : R-d x R-d -> R is a symmetric measurable function. Let q : R-d -> R. We specify assumptions on a, q, and J so that non-negative bounded solutions to Lf + qf = 0 satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to Lf = 0.