A Riemannian off-diagonal heat kernel bound for uniformly elliptic operators

We find a Gaussian off-diagonal heat kernel estimate for uniformly elliptic operators with measurable coefficients acting on regions Omega subset of o r equal to R-N, where the order 2m of the operator satisfies N < 2m. The es timate is expressed using certain Riemannian-type metrics, and a geometrica l result is established allowing conversion of the estimate into terms of t he usual Riemannian metric on Omega. Work of Barbatis ([1]) is applied to f ind the best constant in this expression.