Heat kernel estimates for the Dirichlet fractional Laplacian

Abstract

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 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 two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.