Heat kernel estimates for the Dirichlet fractional Laplacian

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 a Dirichlet fractional Laplacian in C ^1,1, open sets. This heat kernel is also the transition density of a rotationally symmetric α-stable process killed upon leaving a C ^1,1, open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets.