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Suppose that D is the domain in Rd, d ≥ 3, above the graph of a bounded C1,1 function Γ : Rd−1 → R 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 and C4 such that for all t > 0 and x, y ∈ D, C1( ρ(x)ρ(y) t ∧ 1)te C2|x−y| t ≤ p(t, x, y), p(t, x, y) ≤ C3( ρ(x)ρ(y) t ∧ 1)te C4|x−y| t , where ρ(x) stands for the distance between x and ∂D.

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