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

In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in $C^{1,η}$ open sets. The processes are symmetric pure jump Markov processes with jumping intensity $κ(x,y) ψ_1 (|x-y|)^{-1} |x-y|^{-d-α}$, where $α\in (0,2)$. Here, $ψ_1$ is an increasing function on $[ 0, \infty )$, with $ψ_1(r)=1$ on $0<r \le 1$ and $c_1e^{c_2r^β} \le ψ_1(r) \le c_3 e^{c_4r^β}$ on $r>1$ for $β\in [0,\infty]$, and $κ( x, y)$ is a symmetric function confined between two positive constants, with $|κ(x,y)-κ(x,x)|\leq c_5|x-y|^ρ$ for $|x-y|<1$ and $ρ>α/2$. We establish two-sided estimates for the transition densities of such processes in $C^{1,η}$ open sets when $η\in (α/2, 1]$. In particular, our result includes (relativistic) symmetric stable processes and finite-range stable processes in $C^{1,η}$ open sets when $η\in (α/2, 1]$.

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