Dirichlet heat kernel estimates of subordinate diffusions with continuous components in C1, α 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 C1, α(α∈ (0, 1]) open sets. As a corollary, we obtain the sharp two-sided estimates for Green functions of these processes in bounded C1, α open sets.