Boundary Harnack principle for diffusion with jumps

Consider the operator L^b=L⁰+b₁⋅∇+S^b₂ on R^d, where L⁰ is a second order differential operator of non-divergence form, the drift function b₁ belongs to some Kato class and S^b₂f(x)≔∫_R^df(x+z)−f(x)−∇f(x)⋅z1_{|z|≤1}b₂(x,z)j₀(z)dz,f∈C_b²(R^d).Here j₀(z) is a nonnegative locally bounded function on R^d∖{0} satisfying that ∫_R^d(1∧|z|²)j₀(z)dz<∞ and that there are constants β∈(1,2) and c₀>0 so that [Formula presented] and b₂(x,z) is a real-valued bounded function on R^d×R^d. There is conservative Feller process X^b associated with the non-local operator L^b. We derive sharp two-sided Green function estimates of L^b on bounded C^1,1 domains, identify the Martin and minimal Martin boundary, and establish the Martin integral representation of L^b-harmonic functions on these domains. The latter in particular reveals how the process X^b exits a bounded C^1,1 domain D, or equivalently, the structure of the harmonic measure of L^b on D, which consists of the continuously exiting term and the jump-off term. These results are then used to establish, under some mild conditions, Harnack principle and the boundary Harnack principle with explicit boundary decay rate for the operator L^b on C^1,1 open sets.