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 $\mathrm{\Omega }\subseteq {\mathrm{\Re }}^N$, where the order $2m$ of the operator satisfies $N<2m$. The estimate is expressed using certain Riemannian-type metrics, and a geometrical result is established allowing conversion of the estimate into terms of the usual Riemannian metric on $\mathrm{\Omega }$. Work of Barbatis is applied to find the best constant in this expression.