Boundary Harnack principle for non-local operators on metric measure spaces

In this paper, a necessary and sufficient condition is obtained for the scale invariant boundary Harnack inequality (BHP in abbreviation) for a large class of Hunt processes on metric measure spaces that are in weak duality with another Hunt process. We next consider a discontinuous subordinate Brownian motion with Gaussian component $X_t=W_{S_t}$ in ${\bf R}^d$ for which the L\'evy density of the subordinator $S$ satisfies some mild comparability condition. We show that the scale invariant BHP holds for the subordinate Brownian motion $X$ in any Lipschitz domain satisfying the interior cone condition with common angle $\theta\in (\cos^{-1}(1/\sqrt d), \pi)$, but fails in any truncated circular cone with angle $\theta \leq \cos^{-1}(1/\sqrt d)$, a Lipschitz domain whose Lipschitz constant is larger than or equal to $1/\sqrt{d-1}.$