Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in C1,η open sets

In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in C1,η open sets. The processes are symmetric pure jump Markov processes with jumping intensity κ(x,y)ψ1(|x−y|)−1|x−y|−d−α, where α∈(0,2). Here, ψ1 is an increasing function on [0,∞), with ψ1(r)=1 on 0<r≤1 and c1ec2rβ≤ψ1(r)≤c3ec4rβ on r>1 for β∈[0,∞], and κ(x,y) is a symmetric function confined between two positive constants, with |κ(x,y)−κ(x,x)|≤c5|x−y|ρ for |x−y|<1 and ρ>α/2. We establish two-sided estimates for the transition densities of such processes in C1,η open sets when η∈(α/2,1]. In particular, our result includes (relativistic) symmetric stable processes and finite-range stable processes in C1,η open sets when η∈(α/2,1].