Subelliptic operators on Lie groups: Variable coefficients
type: Item
instance of: scholarly article
title: Subelliptic operators on Lie groups: Variable coefficients
zbMATH Open document ID: 0842.43005
DOI: 10.1007/BF00049421
author: Ola Bratteli
author: Derek W. Robinson
published in: Acta Applicandae Mathematicae
publication date: 1996-08-04T00:00:00Z
review text: Let \(G\) be a Lie group with Lie algebra \(\mathfrak g\), let \(a_1, \dots, a_n\) be an algebraic basis of \(\mathfrak g\) and \(dg\) a right Haar measure. The authors consider the second-order subelliptic differential operator \(H = \sum_{\alpha : |\alpha|\leq 2} c_\alpha A^\alpha\) with real bounded coefficients \(c_\alpha : G \to \mathbb{R}\), where \(A_i = dL(a_i)\) are the corresponding generators of the left translation by \(G\) on one of the Banach spaces \(L_p(G)\), \(p \in [1,\infty)\). The authors prove that if the principal coefficients \(\{C_\alpha : |\alpha |= 2\}\) are left differentiable in the directions \(a_1,\dots, a_n\) with bounded derivates, then the operator \(H\) has a family of semigroup generator extensions on the \(L_p\)-spaces, and the corresponding holomorphic semigroups \(\{S(t)\), \(t \geq 0\}\) are given by the positive integral kernel \[ (S(t) \varphi) (g) = \int_G K_t (g;h) \varphi(h) dh. \]
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 841766
zbMATH Keywords: Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: Haar measure
zbMATH Keywords: subelliptic differential operator
zbMATH Keywords: generators
zbMATH Keywords: translation
zbMATH Keywords: Banach spaces
zbMATH Keywords: principal coefficients
zbMATH Keywords: holomorphic semigroups
Wikidata QID: Q115395284
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00049421
OpenAlex ID: W1994435896
description: scientific article; zbMATH DE number 841766