In this talk, we consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal L'evy processes with weak scaling conditions. We first establish sharp two-sided heat kernel estimates for this processes in $C^{1,rho}$ open sets. As a corollary of our main result, we obtain a sharp two-sided Green function and a scale invariant boundary Harnack inequality with explicit decay rates in $C^{1,rho}$ open sets.

This is a joint work with Tomasz Grzywny and Kyung-Youn Kim.