Opérateurs uniformément sous-elliptiques sur les groupes de Lie. (Uniformly sub-elliptic operators on Lie groups)
type: Item
instance of: scholarly article
title: Opérateurs uniformément sous-elliptiques sur les groupes de Lie. (Uniformly sub-elliptic operators on Lie groups)
zbMATH Open document ID: 0734.58041
DOI: 10.1016/0022-1236(91)90092-J
published in: Journal of Functional Analysis
publication date: 1991-01-01T00:00:00Z
review text: Let G be a connected unimodular Lie group G, and let \(X_ 1,...,X_ k\) be left invariant vector fields which generate the Lie algebra of G. Let \(a: G\to {\mathbb{R}}^ k\otimes {\mathbb{R}}^ k\) be a smooth symmetric matrix valued function which satisfies \[ \alpha | \xi |^ 2\leq \sum a_{ij}(x)\xi_ i\xi_ j \leq \alpha^{-1}| \xi |^ 2,\quad x\in G,\quad \xi \in {\mathbb{R}}^ k,\text{ for some } \alpha \in]0,1]. \] Put \({\mathcal L}_{\alpha}=\sum^{k}_{i,j=1}X_ ia_{ij}X_ j\). If G has polynomial volume growth, the authors prove Gaussian estimates for the heat kernel \(e^{t{\mathcal L}\alpha}\). As a consequence they obtain regularity results and Harnack inequalities. Their results extend earlier results of De Giorgi-Nash-Moser-Aronson (for parabolic operators in \({\mathbb{R}}^ n)\) and of Varopoulos (for evolution operators associated to the Kohn Laplacian).
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 4215002
zbMATH Keywords: subelliptic operators
zbMATH Keywords: Lie groups
zbMATH Keywords: Gaussian estimates
author: Laurent Saloff-Coste
author: Daniel W. Stroock
reviewed by: Paul Godin
MaRDI profile type: Publication
cites work: Bounds for the fundamental solution of a parabolic equation
cites work: Upper bounds for symmetric Markov transition functions
cites work: Explicit Constants for Gaussian Upper Bounds on Heat Kernels
cites work: Q3252874
cites work: A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash
cites work: Q3969295
cites work: Fundamental solutions for second order subelliptic operators
cites work: Croissance polynomiale et périodes des fonctions harmoniques
cites work: The Poincaré inequality for vector fields satisfying Hörmander's condition
cites work: Q3780156
cites work: Q3772595
cites work: Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator
cites work: Q4720495
cites work: Q3684932
cites work: Q3771355
cites work: Correetion to “A Harnack Inequality for Parabolic Differential Equations”
cites work: Balls and metrics defined by vector fields. I: Basic properties
cites work: Continuity of Solutions of Parabolic and Elliptic Equations
cites work: Hypoelliptic differential operators and nilpotent groups
cites work: Q4731650
cites work: Analyse sur les groupes de Lie à croissance polynômiale. (Analysis on Lie groups of polynomial growth)
cites work: Harnack's inequality for elliptic equations and the Hölder property of their solutions
cites work: Fundamental solutions and geometry of the sum of squares of vector fields
cites work: Hardy-Littlewood theory for semigroups
cites work: Analysis on Lie groups
cites work: Q4722267
description: scientific article; zbMATH DE number 4215002