Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in $C^{1,\eta }$ open sets

In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in $C^{1,\eta }$ open sets. The processes are symmetric pure jump Markov processes with jumping intensity $\kappa(x,y) \psi_1 (|x-y|)^{-1} |x-y|^{-d-\alpha}$,where$\alpha \in (0,2)$\mathrm{.}Here,$\psi_1$isanincreasingfunctionon$[ 0, \infty )$,with$\psi_1®=1$on$0<r \le 1$and$c_1e^{c_2r^{\beta}} \le \psi_1® \le c_3 e^{c_4r^{\beta}}$on$r>1$for$\beta \in [0,\infty]$,and$ \kappa( x, y)$ is a symmetric function confined between two positive constants, with $|\kappa(x,y)-\kappa(x,x)|\leq c_5|x-y|^{\rho}$for$|x-y|<1$and$\rho>\alpha/2$. We establish two-sided estimates for the transition densities of such processes in $C^{1,\eta }$ open sets when $\eta \in (\alpha \mathrm{/}2,1]$. In particular, our result includes (relativistic) symmetric stable processes and finite-range stable processes in $C^{1,\eta }$ open sets when $\eta \in (\alpha \mathrm{/}2,1]$.