Harnack Inequality for Non-Local Schrödinger Operators
Let x∈ Rd, d ≥ 3, and f: Rd→ R be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let aij: Rd→ R be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to (Formula presented.) where J: Rd× Rd→ R is a symmetric measurable function. Let q: Rd→ 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.
Athreya, Siva and Ramachandran, Koushik, "Harnack Inequality for Non-Local Schrödinger Operators" (2018). Journal Articles. 1412.