Markov processes with jump kernels decaying at the boundary

The goal of this work is to develop a general theory for non-local singular operators of the type L^B_f (x) = ₀ ₃, \, |ₘ-ₗ|> (f (y) -f (x) ) B (x, y) |x-y|^-d-\, dy, and L f (x) =L^B_f (x) - (x) f (x), in case D is a C^1, 1 open set in Rᵈ, d 2. The function B (x, y) above may vanish at the boundary of D, and the killing potential may be subcritical or critical. From a probabilistic point of view we study the reflected process on the closure D with infinitesimal generator L^B_, and its part process on D obtained by either killing at the boundary D, or by killing via the killing potential (x). The general theory developed in this work (i) contains subordinate killed stable processes in C^1, 1 open sets as a special case, (ii) covers the case when B (x, y) is bounded between two positive constants and is well approximated by certain H\"older continuous functions, and (iii) extends the main results known for the half-space in Rᵈ. The main results of the work are the boundary Harnack principle and its possible failure, and sharp two-sided Green function estimates. Our results on the boundary Harnack principle completely cover the corresponding earlier results in the case of half-space. Our Green function estimates extend the corresponding earlier estimates in the case of half-space to bounded C^1, 1 open sets.