Lower estimates of the heat kernel on conic manifolds and Riesz transform
type: Item
instance of: scholarly article
title: Lower estimates of the heat kernel on conic manifolds and Riesz transform
zbMATH Open document ID: 1076.58017
author: Thierry Coulhon
author: Hong-Quan Li
published in: Archiv der Mathematik
publication date: 2004-11-05T00:00:00Z
review text: In [Bull. Sci. Math. 124, 365--384 (2000; Zbl 0977.58024)], the second author established Gaussian upper estimates for the heat kernel \(p_{t}\) associated to the Laplace-Beltrami operator \(\Delta\) on \(C(N)\), where \(N\) is a compact connected Riemannian manifold without boundary and with dimension \(n\). Namely, \(p_{t}(x,y)\leq C_{1}t^{-n/2}e^{-d^{2}(x,y)/C_{2}t}\). In the present paper, the authors prove the corresponding lower bound, namely \(p_{t}(x,y)\geq c_{1}t^{-n/2}e^{-d^{2}(x,y)/c_{2}t}\). Thus, on such manifolds, Gaussian upper and lower bounds for the heat kernel are valid, and the volume of balls has polynomial growth with exponent \(n\), although the so-called Riesz transforms (namely, the operator \(\nabla \Delta^{-1/2}\)) are not \(L^p\)-bounded for all \(p>2\) (this was proven by the second author in [J. Funct. Anal. 168, 145--238 (1999; Zbl 0937.43004)]). The proof of the lower bound for \(p_{t}\) relies on a formula for \(p_{t}\) due to \textit{J. Cheeger} [J. Differ. Geom. 18, 575--657 (1983; Zbl 0529.58034)], and does not go through any Poincaré inequality.
reviewed by: Emmanuel Russ
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 2113842
zbMATH Keywords: conical manifolds
zbMATH Keywords: heat kernel
zbMATH Keywords: Riesz transforms
MaRDI profile type: Publication
full work available at URL: s00013-004-1029-8
OpenAlex ID: W2044422250
DOI: 10.1007/S00013-004-1029-8
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description: scientific article; zbMATH DE number 2113842