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, Lévy processes with and without continuous part, stable-like and censored stable processes, jump processes on fractals, and rather general Schrödinger, drift and jump perturbations of such processes.