Heat kernel estimates for the Dirichlet fractional Laplacian

In this paper, we consider the fractional Laplacian -(-Δ)α/2 on an open subset in ℝ^d with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such Dirichlet fractional Laplacian in C1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric α -stable process killed upon leaving a C1,1 open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.