Gaussian upper bounds for the heat kernel on arbitrary Riemannian manifolds
type: Item
instance of: scholarly article
title: Gaussian upper bounds for the heat kernel on arbitrary Riemannian manifolds
zbMATH Open document ID: 0865.58042
author: A. A. Grigor'yan
published in: Journal of Differential Geometry
publication date: 1997-06-11T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1000571
zbMATH Keywords: Gaussian estimates
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: heat kernel
zbMATH Keywords: Laplace-Beltrami operator
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214459753
OpenAlex ID: W1595988316
Wikidata QID: Q115175981
DOI: 10.4310/JDG/1214459753
description: scientific article; zbMATH DE number 1000571