Two-sided heat kernel estimates for censored stable-like processes

In this paper, we study the precise behavior of the transition density functions of censored (resurrected) α-stable-like processes in C^1,1 open sets in ℝ^d, where d ≥ 1 and α ε (1, 2). We first show that the semigroup of the censored α-stable-like process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp two-sided estimates for the transition density functions of a large class of censored α-stable-like processes in C^1,1 open sets. We further obtain sharp two-sided estimates for the Green functions of these censored α-stable-like processes in bounded C^1,1 open sets.