The Boundary Harnack Principle for Nonlocal Elliptic Operators in Non-divergence Form

We prove a boundary Harnack inequality for nonlocal elliptic operators L in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if Lu-1 = Lu-2 = 0 in omega boolean AND B-1, u(1) = u(2) = 0 in B-1 set minus omega, and u(1),u(2) >= 0 in Double-struck capital R-n, then u(1) and u(2) are comparable in B-1/2. The result applies to arbitrary open sets omega. When omega is Lipschitz, we show that the quotient u(1)/u(2) is Holder continuous up to the boundary in B-1/2. These results will be used in forthcoming works on obstacle-type problems for nonlocal operators.