Boundary Harnack inequality for Markov processes with jumps
  We prove a boundary Harnack inequality for jump-type Markov processes on metric measure state spaces, under comparability estimates of the jump kernel and Urysohn-type property of the domain of the generator of the process. The result holds for positive harmonic functions in arbitrary open sets. It applies, e.g., to many subordinate Brownian motions, L\'evy processes with and without continuous part, stable-like and censored stable processes, jump processes on fractals, and rather general Schr\"odinger, drift and jump perturbations of such processes. 
Krzysztof Bogdan, Takashi Kumagai, Mateusz Kwa\'snicki
math.PR math.AP