Short-time asymptotics of heat kernels of hypoelliptic Laplacians on unimodular Lie groups
type: Item
instance of: scholarly article
title: Short-time asymptotics of heat kernels of hypoelliptic Laplacians on unimodular Lie groups
zbMATH Open document ID: 1253.43006
published in: Journal of Functional Analysis
publication date: 2012-05-30T00:00:00Z
reviewed by: Jean Ludwig
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 6039128
zbMATH Keywords: SubRiemannian geometry
zbMATH Keywords: Hypoelliptic heat kernel. Non-commutative harmonic analysis
zbMATH Keywords: Trotter-Kato product formula
MaRDI profile type: Publication
full work available at URL: j.jfa.2012.02.004
OpenAlex ID: W1971911267
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DOI: 10.1016/J.JFA.2012.02.004
author: C. Séguin
author: A. Mansouri
review text: This paper uses the observation that the non-commutative harmonic analysis approach proposed by \textit{A. Agrachev}, \textit{U. Boscain}, \textit{J.-P. Gauthier} and \textit{F. Rossi} in [J. Funct. Anal. 256, No. 8, 2621--2655 (2009; Zbl 1165.58012)] may be used to compute hypoelliptic heat kernel short-time asymtotics even in situations where the exact expression of the heat kernel is impossible to compute due to intractable operator equations. The main result of the paper gives such short-time asymptotics for a unimodular type I Lie group \( G \) and a left invariant hypoelliptic sub-Laplacian \( \Delta \) satisfying the following conditions. For every irreducible unitary representation \( (\lambda,\mathcal H^\lambda) \) of \( G \) let \( \hat \Delta^\lambda:=d\lambda (\Delta) \), which is a selfadjoint operator on \(\mathcal H_\lambda \). Then it is assumed that \( \mathcal H^\lambda =L^2(\Omega_\lambda) \) for some measure space \( \Omega_\lambda \), where \( \Omega_\lambda \) is a smooth variety, and \( \hat \Delta^\lambda \) can be written as a sum \( A^\lambda + B^\lambda\), both generating strongly semigroups \( (T^\lambda (t))_{t\geq 0} \) and \( (S^\lambda (t))_{t\geq 0} \). It is supposed furthermore that for every compact subset \( K\subset \Omega_\lambda \) there exists a constant \( \alpha_{K,\lambda} \), such that for every \( f\in C_c^\infty (K) \) and for every operator \( E \) which is a composition of the operators \( B^\lambda,T^\lambda(t) \) and \( S^\lambda (t) \) one has that \( \| E(f)\|_1\leq \alpha^{k}_{K,\lambda}\| f\|_1 \), where \( k \) is the number of times the operator \(B\) appears in \( E \). For each \( t\geq 0, \) \(T^\lambda(t) \) is an integral operator with kernel \( k^\lambda_t(s,r) \) and there exists an integrable function \( K^\lambda_{t/n}(s,r) \) uniformly bounded in \( s \) by an integrable function \( G^\lambda_t(s,r) \) for all \( n\geq 1 \) such that NEWLINE\[NEWLINE\begin{multlined} (t/n)[T^\lambda(t/n)^nB^\lambda+T…
description: scientific article; zbMATH DE number 6039128