Estimates on the Dirichlet heat kernel of domains above the graphs of bounded C1,1 functions

Suppose that D is the domain in , d ≥ 3, above the graph of a bounded C1,1 function : and that pD(t, x, y) is the Dirichlet heat kernel in D. In this paper we show that there exist positive constants C1, C2, C3, C4 such that for all t > 0 and x, y D, C1((x)(y) / t 1) t -d/2 e -C|x-y|^2 / t ≤ pD(t, x, y), pD(t, x, y) ≤ C3((x)(y) / t 1) t -d/2 e -C|x-y|^2 / t, where (x) stands for the distance between x and D