  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.
