Dirichlet heat kernel estimates for rotationally symmetric Ĺevy processes

In this paper, we consider a large class of purely discontinuous rotationally symmetric Ĺevy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a ?-fat open set, the sharp two-sided estimates are given in terms of surviving probabilities and the global transition density of the Ĺevy process. When D is a C1,1 open set and the Ĺevy exponent of the process is given by ?(?) = f(|?|2) with f being a complete Bernstein function satisfying a mild growth condition at infinity, our two-sided estimates are explicit in terms of ?, the distance function to the boundary of D and the Ĺevy density of X. This gives an affirmative answer to the conjecture posted in Chen, Kim and Song [Global heat kernel estimates for relativistic stable processes in half-space-like open sets. Potential Anal. 36 (2012) 235-261]. Our results are the first sharp two-sided Dirichlet heat kernel estimates for a large class of symmetric Ĺevy processes with general Ĺevy exponents. We also derive an explicit lower bound estimate for symmetric Ĺevy processes on Rd in terms of their Ĺevy exponents.