Heat kernel estimates for the fractional Laplacian

Explicit sharp estimates for the Green function of the Laplacian in C domains were completed in 1986 by Zhao [42]. Sharp estimates of the Green function of Lipschitz domains were given in 2000 by Bogdan [6]. Explicit qualitatively sharp estimates for the classical heat kernel in C domains were established in 2002 by Zhang [41]. Qualitatively sharp heat kernel estimates in Lipschitz domains were given in 2003 by Varopulous [40]. The development of the boundary potential theory of the fractional Laplacian follows a parallel path. Green function estimates were obtained in 1997 and 1998 by Kulczycki [29] and Chen and Song [21] for C domains, and in 2002 by Jakubowski for Lipschitz domains [28]. In 2008 Chen, Kim and Song [19] gave sharp explicit estimates for the heat kernel pD(t, x, y) of the fractional Laplacian on C 1,1 domainsD. The main contribution of the present paper is the following result.