Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold
type: Item
instance of: scholarly article
title: Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold
zbMATH Open document ID: 0842.58070
author: A. A. Grigor'yan
published in: Journal of Functional Analysis
publication date: 1995-06-14T00:00:00Z
review text: Let \(M\) be a smooth connected complete non-compact Riemannian manifold, with corresponding Laplace operator \(\Delta\). Let \(p(x, y, t)\) be the heat kernel, i.e. the smallest positive fundamental solution to the heat equation \(\partial_t u - \Delta u = 0\). If \(D > 2\) is a real number, put \(E_n(x,t) = \int_M |\nabla^n p |^2 (x,y,t) \text{exp} \biggl({r^2 \over Dt}\biggr) dy\). The main result of the paper is an upper bound on \(E_n\), \(n \geq 1\), if some upper bounds on \(E_0\) hold. Pointwise upper bounds of \({\partial^n p\over \partial t^n}\) are also obtained, if some pointwise upper bounds on \(p(x,y,t)\) hold.
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 763994
zbMATH Keywords: complete manifolds
zbMATH Keywords: heat kernel
reviewed by: Paul Godin
MaRDI profile type: Publication
OpenAlex ID: W2048773718
DOI: 10.1006/JFAN.1995.1016
description: scientific article; zbMATH DE number 763994