Boundary Harnack Principle and Gradient Estimates for Fractional Laplacian Perturbed by Non-local Operators

Suppose d ≥ 2 and 0 < β < α < 2. We consider the non-local operator ℒ^b= Δ ^α/2 + S^b, whereSbf(x):=limε→0A(d,−β)∫|z|>ε(f(x+z)−f(x))b(x,z)|z|d+βdy.Here b(x, z) is a bounded measurable function on ℝ^d× ℝ^d that is symmetric in z, and A(d, − β) is a normalizing constant so that when b(x, z)≡1, S^b becomes the fractional Laplacian Δ^β/2:=−(−Δ)^β/2. In other words, ℒ^b f(x) : =limε→0 A(d, - β) ∫ _{|z| > ε} (f(x+z) – f(x)) j^b (x.z) dz where j^b(x, z) : = A(d, − α) |z|^−(d+α)+ A(d, − β) b(x, z) |z| ^−(d+β). It is recently established in Chen and Wang [11] that, when j^b(x, z)≥0 on ℝ^d× ℝ^d, there is a conservative Feller process X^b having ℒ^b as its infinitesimal generator. In this paper we establish, under certain conditions on b, a uniform boundary Harnack principle for harmonic functions of X^b (or equivalently, of ℒ^b) in any κ-fat open set. We further establish uniform gradient estimates for non-negative harmonic functions of X^b in open sets.