Sub-Laplacians of holomorphic \(L^p\)-type on rank one AN-groups and related solvable groups
type: Item
instance of: scholarly article
title: Sub-Laplacians of holomorphic \(L^p\)-type on rank one AN-groups and related solvable groups
zbMATH Open document ID: 0957.22013
published in: Journal of Functional Analysis
publication date: 2001-03-25T00:00:00Z
review text: Let \(G=\exp{\mathfrak g}\) be an exponential solvable Lie group endowed with a left-invariant Haar measure and \(L=\int^\infty_0 \lambda dE_\lambda\) the spectral resolution of a sub-Laplacian \(L\) on \(G\). We call a bounded function \(m\) on \([0,\infty[\) an \(L^p\)-multiplier for \(L\) \((1\leq p<\infty)\) if \(m(L)=\int^\infty_0 m(\lambda)d E_\lambda\) extends from \(L^p(G)\cap L^2(G)\) to a bounded operator on \(L^p(G)\). We say that \(L\) is of holomorphic \(L^p\)-type if there exists some non-isolated point \(\lambda_0\in \text{Spec}_{L^2 (G)}L= [0, \infty[\) and an open complex neighborhood \(\Omega\) of \(\lambda_0\) such that every continuous \(L^p\)-multiplier \(m(\lambda)\) vanishing at infinity extends holomorphically to \(\Omega\). The authors prove that, if the Boidol condition holds for some \(l\in{\mathfrak g}^*\) and \({\mathfrak g}={\mathfrak g}(l)+ [{\mathfrak g},{\mathfrak g}]\), \({\mathfrak g}(l)\) is the stabilizer of \(l\), then \(L\) is of holomorphic \(L^1\)-type; and moreover, if the assumption on \(l\) holds on a whole neighborhood of \(l\), then \(G\) is unimodular and \(L\) is of holomorphic \(L^p\)-type for \(1<p< \infty\), \(p\neq 2\). Let \({\mathfrak n}\) be a nilpotent ideal of \({\mathfrak g}\) containing \([{\mathfrak g}, {\mathfrak g}]\) and \({\mathfrak a}\) a linear subspace of \({\mathfrak g}\) such that \({\mathfrak g}={\mathfrak a}\oplus {\mathfrak n}\). For \(l=(\tau, l_0)\in {\mathfrak g}^* ={\mathfrak a}^* \oplus{\mathfrak n}^*\), let \({\mathfrak p}(l)\) denote a Vergne-polarization for \(l\) and put \(P(l)=\exp {\mathfrak p}(l)\). We denote by \(\pi_l=\text{ind}^G_{P (l)} \chi_l\), where \(\chi_l(p) =e^{il(\log p)}\) \((p\in P(l))\), the induced representation of \(G\) realized on \(L^2(\mathbb{R}^\nu)\). Then, by applying continuous perturbation theory, the Boidol condition yields that an eigenvalue \(\lambda (l)\) and an eigenfunction \(\varphi_l\) in \(L^2(\mathbb{R}^\nu)\) to \(\pi_l(L)\) depend continuously on…
reviewed by: Takeshi Kawazoe
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 1419626
zbMATH Keywords: \(AN\)-group
zbMATH Keywords: \(L^p\)-multiplier
zbMATH Keywords: exponential solvable Lie group
zbMATH Keywords: perturbation theory
author: Jean Ludwig
author: Detlef Müller
MaRDI profile type: Publication
full work available at URL: jfan.1999.3517
OpenAlex ID: W1985333761
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DOI: 10.1006/JFAN.1999.3517
description: scientific article; zbMATH DE number 1419626