Global heat kernel estimates for fractional Laplacians in unbounded open sets

In this paper, we derive global sharp heat kernel estimates for symmetric ??-stable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded C ^1,1 open sets in ${\mathbb R^d}$ : half-space-like open sets and exterior open sets. These open sets can be disconnected. We focus in particular on explicit estimates for p _D (t, x, y) for all t?>?0 and ${x, y\,{\in}\,D}$ . Our approach is based on the idea that for x and y in D far from the boundary and t sufficiently large, we can compare p _D (t, x, y) to the heat kernel in a well understood open set: either a half-space or ${\mathbb R^d}$ ; while for the general case we can reduce them to the above case by pushing x and y inside away from the boundary. As a consequence, sharp Green functions estimates are obtained for the Dirichlet fractional Laplacian in these two types of open sets. Global sharp heat kernel estimates and Green function estimates are also obtained for censored stable processes (or equivalently, for regional fractional Laplacian) in exterior open sets.