Harnack inequality and interior regularity for Markov processes with degenerate jump kernels

In this paper we study interior potential-theoretic properties of purely discontinuous Markov processes in proper open subsets $D\subset {\mathbb{R}}^d$. The jump kernels of the processes may be degenerate at the boundary in the sense that they may vanish or blow up at the boundary. Under certain natural conditions on the jump kernel, we show that the scale invariant Harnack inequality holds for any proper open subset $D\subset {\mathbb{R}}^d$ and prove some interior regularity of harmonic functions. We also prove a Dynkin-type formula and several other interior results.