Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds
type: Item
instance of: scholarly article
title: Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds
zbMATH Open document ID: 1080.58022
author: Nick Dungey
published in: Mathematische Zeitschrift
publication date: 2005-02-11T00:00:00Z
review text: The main results of the paper are some precise estimates for the gradient of the heat kernel on certain non-compact manifolds and boundedness of the Riesz transforms in spaces \(L^p\). Here is a brief summary of the paper. Let \(M\) be a Riemannian manifold, with cocompact, smooth, proper and free action of a discrete, finitely generated nilpotent (or, more generally, of polynomial growth) group \(G\); assume also that \(G\) acts on \(M\) by isometries. Let \(H\) be the positive Laplacian on \(M\), \(K(t,x,y)\) -- the corresponding heat kernel. Let \(\nabla\) be the gradient, \(d(x, y)\) -- the Riemannian distance, and \(V(x, r)\) -- the volume of the ball on \(M\). The main result of the paper is the following estimate: \[ |\nabla_x K(t,x,y)|\leq ct^{-1/2}V(x, t^{1/2})^{-1}\exp(- bd(x,y)^2/t), \] for some constants \(c\) and \(b\). The author also proves that \(\|\nabla f\|_p\leq c_p\| H^{1/2}f\|_p\) for some constant \(c\) and \(1< p<\infty\), thus establishing boundedness of the Riesz transform \(\nabla H^{-1/2}\) in spaces \(L^p\). The main idea of the proof is to consider separately two cases: the case of ``small'' time \((0< t< 1)\) heat propagation, which is not related to the group action and can be established using estimates in a (relatively compact) fundamental domain \(X\subseteq M\), and the case of ``large'' time \((t\geq 1)\) heat propagation, which can be related to the group action, and treated as the case of operators acting in the space \(L^2(G;L^2(X))\) using some elementary relations between ball volumes in \(M\) and the order of polynomial growth of the group \(G\), and also some earlier estimate obtained by the author for the length function on nilpotent groups. For this second case the author establishes some lovely estimates for the discrete oscillation of the gradient of the heat kernel. The proof of the main theorems is obtained by combining estimates for these two cases.
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 2133506
zbMATH Keywords: heat kernel
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Laplace operator
zbMATH Keywords: nilpotent group
zbMATH Keywords: Galois covering
reviewed by: Wiesław Oledzki
MaRDI profile type: Publication
full work available at URL: s00209-003-0646-4
OpenAlex ID: W2078000761
Wikidata QID: Q115388857
DOI: 10.1007/S00209-003-0646-4
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description: scientific article; zbMATH DE number 2133506