Gaussian upper bounds for the heat kernel on arbitrary Riemannian manifolds
Let \(M\) be an arbitrary smooth connected Riemannian manifold and let us denote by \(p(t,x,y)\) the heat kernel of a Laplace-Beltrami operator. The purpose of the paper is to provide a direct and universal way of obtaining the Gaussian upper bounds of the heat kernel from an on-diagonal one. The following is the main result: Let \(x,y\) be two points on \(M\), and let us have for all \(t\in (0,T)\) (where \(T\) may be equal to \(\infty\) or be a positive number) (1) \(p(t,x,x)\leq 1/f(t)\) and (2) \(p(t,y,y)\leq 1/g(t)\) where \(f,g\) are in some sense regular functions. Then for any \(C>4\), for some \(\delta= \delta(C)>0\), and all \(t\in (0,T)\) \[ p(t,x,y)\leq {{\text{const}}\over {\sqrt{f(\delta t)g(\delta t)}}}\exp \Biggl(- {{d(x,y)^2}\over {Ct}}\Biggr) \] where \(d(x,y)\) is the geodesic distance between \(x,y\). Unlike many other known results of this kind, the on-diagonal bounds (1) and (2) are assumed only at two points \(x,y\) rather than at all points. The regularity conditions which are imposed on the functions \(f\) and \(g\) are rather mild and include, for example, the functions \(t^a\), \(\log^b t\), \(e^{t^c}\) and their combinations \((a,b,c>0)\). The main ingredient of the proof is the method of \textit{V. I. Ushakov} [Math. USSR, Sb. 39, 87-105 (1981); translation from Mat. Sb., Nov. Ser. 111(153), 95-115 (1980; Zbl 0428.35044)] of estimating the integral of \(p^2\) over the exterior of a ball.
scientific article; zbMATH DE number 1000571