Subcoercive and subelliptic operators on Lie groups: Variable coefficients
type: Item
instance of: scholarly article
title: Subcoercive and subelliptic operators on Lie groups: Variable coefficients
zbMATH Open document ID: 0816.43002
author: A. F. M. ter Elst
author: Derek W. Robinson
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1995-07-13T00:00:00Z
review text: This paper extends the theory of second-order subelliptic differential operators with real coefficients acting on functions over a Lie group in two different directions: a generalization to higher-order operators characterized by a condition of coercivity and the consideration of complex-valued coefficients. Let \(a_ 1,\dots,a_ d\) be an algebraic basis of rank \(r\) in a Lie algebra \(\mathfrak g\) of a connected Lie group \(G\) and let \(A_ i\) be the left differential operator in the direction \(a_ i\) on the \(L_ p\)- spaces \(L_ p(G,d g)\) or \(L_ p(G,d \widehat{g})\) with respect to the left, or right, Haar measure, where \(p \in [1,\infty]\). An \(m\)-th order operator \(H = \sum C_ \alpha A^ \alpha\) with complex variable bounded coefficients \(C_ \alpha : G \to \mathbb{C}\) and multiindex products \(A^ \alpha\) of the operators \(A_ i\) is called subcoercive of step \(r\) if for all \(g \in G\) the form obtained by fixing the \(C_ \alpha\) at \(g \in G\) is subcoercive of step \(r\) and the ellipticity constant is bounded from below uniformly by a positive constant. The main result: the closure of \(H\) generates a continuous heat semigroup \(S\) which interpolates consistently between the spaces \(L_ p(G,d g)\) and \(L_ p(G,d \widehat{g})\), is determined by an integral kernel and which is holomorphic on a non-empty \(p\)-independent sector. In particular, if the matrix of principal coefficients of \(H\) is hermitian then \(S\) is holomorphic in the open right half-plane. With the help of regularization it is proved that these results are valid if only the principal coefficients are \(m\)-times differentiable in \(L_ \infty\). In this connection Gaussian bounds and smoothness properties of the corresponding integral kernel are studied.
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35J30
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 548928
zbMATH Keywords: second-order subelliptic differential operators
zbMATH Keywords: higher-order operators
zbMATH Keywords: connected Lie group
zbMATH Keywords: heat semigroup
zbMATH Keywords: regularization
reviewed by: Yuri I. Karlovich
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195166574
OpenAlex ID: W2152398650
Wikidata QID: Q115224836
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DOI: 10.2977/PRIMS/1195166574
description: scientific article; zbMATH DE number 548928