Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in 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 ∈ ( 0 , 2 ) . Here, ψ 1 is an increasing function on [ 0 , ∞ ) , with ψ 1 ( r ) = 1 on 0 < r ≤ 1 and c 1 e c 2 r β ≤ ψ 1 ( r ) ≤ c 3 e c 4 r β 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 ) | ≤ 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 ∈ ( 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 ∈ ( α / 2 , 1 ] .