Dirichlet heat kernel estimates of subordinate diffusions with continuous components in C^1, open sets

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels for a class of subordinate diffusions with continuous components when the scaling order of purely discontinuous part of the subordinator is between 0 and 1 including 1 in C^1, ( (0, 1]) open sets. As a corollary, we obtain the sharp two-sided estimates for Green functions of these processes in bounded C^1, open sets.