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

In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in C1,n open sets. The processes are symmetric pure jump Markov processes with jumping intensity K (x, Y)1fri. (Ix Y 1)-1 lx Y d cg, where a E (0, 2). Here, Vri is an increasing function on [0, oo), with *I (r) = 1 on 0 < r < 1 and c1ec2rfi < (r) < c3ec4rfi on r > 1 for,8 E [0, oo], and K (x, y) is a symmetric function confined between two positive constants, with 1K (x, y) tc(x, x) I < C5 lx y IP for Ix y I < 1 and p > a/2. We establish two-sided estimates for the transition densities of such processes in C1'17 open sets when n E (a/2, 1]. In particular, our result includes (relativistic) symmetric stable processes and finite-range stable processes in C101 open sets when ri E (a/2, 1]. (C) 2014 Elsevier B.V. All rights reserved.