On heat kernels on Lie groups
type: Item
instance of: scholarly article
title: On heat kernels on Lie groups
zbMATH Open document ID: 0792.22007
DOI: 10.1007/BF02571816
author: Waldemar Hebisch
published in: Mathematische Zeitschrift
publication date: 1993-02-02T00:00:00Z
full work available at URL: https://eudml.org/doc/174421
review text: Let \(p_ t\) be the heat kernel of a convolution semigroup generated by a sub-Laplacian \(L = \sum U^ 2_ j\), where \(U_ j\) are left invariant vector fields on a unimodular Lie group \(G\) which generate the Lie algebra of \(G\). The paper is concerned with proving upper and lower pointwise estimates of the heat kernel \(p_ t\) in the case when \(G\) is a Lie group of polynomial growth or \(G\) is a unimodular compact extension of a solvable Lie group of exponential growth. Some of the results were obtained previously by \textit{N. Varopoulos} [J. Funct. Anal. 76, 346-410 (1988; Zbl 0634.22008)] and \textit{L. Saloff-Coste} [Ark. Mat. 28, 315-331 (1990; Zbl 0715.43009)] and in those cases new proofs of the results were supplied, while some others are new. The results obtained provide estimates for both \(p_ t(e)\) and \(p_ t(x)\), \(x \neq e\) with \(t\) small and large.
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 108648
zbMATH Keywords: heat kernel
zbMATH Keywords: convolution semigroup
zbMATH Keywords: sub-Laplacian
zbMATH Keywords: left invariant vector fields
zbMATH Keywords: unimodular Lie group
zbMATH Keywords: Lie group of polynomial growth
zbMATH Keywords: solvable Lie group of exponential growth
reviewed by: A. Strasburger
MaRDI profile type: Publication
Wikidata QID: Q115391980
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description: scientific article; zbMATH DE number 108648