Multiplicateurs de Mikhlin pour une classe particulière de groupes non-unimodulaires. (Mikhlin multipliers for a particular class of non-unimodular groups.)
type: Item
instance of: scholarly article
title: Multiplicateurs de Mikhlin pour une classe particulière de groupes non-unimodulaires. (Mikhlin multipliers for a particular class of non-unimodular groups.)
zbMATH Open document ID: 0911.22002
author: Sami Mustapha
publication date: 1998-11-09T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75316
review text: Let \(G\) be a connected real Lie group, and let \(X_1, \dots, X_n\) be left invariant vector fields on \(G\) satisfying a certain Hörmander's condition. Let \(\Delta=-\sum^n_{ j=1} X^2_j\) be the associated sub-Laplacian and \(\Delta= \int^\infty_0 \lambda dE_\lambda\) its spectral decomposition. Then, for any bounded measurable function \(m\) on \([0,\infty)\), the operator \(m(\Delta) =\int^\infty_0 m(\lambda) dE_\lambda\) is a bounded linear operator on \(L^2 (G)\). The question of which conditions on \(m\) guarantee that \(m(\Delta)\) is bounded on \(L^p (G)\) for \(1\leq p\leq\infty\), has long been a matter of interest and has been studied by several authors for various types of groups \(G\). When \(G\) is of polynomial growth, in particular, when \(G\) is a stratified nilpotent Lie group, the results are quite satisfying and generalize the classical theorems for \(\mathbb{R}^d\) due to Hörmander and Mikhlin. However, the above problem is considerably more difficult when \(G\) is of exponential growth. Some special results have recently been obtained, e.g. for extensions of the Heisenberg groups by \(\mathbb{R}\). Let \({\mathfrak n}\) be a stratified real nilpotent Lie algebra of rank \(r\), that is, \({\mathfrak n}\) decomposes into a direct sum \({\mathfrak n}= V_1\oplus \cdots \oplus V_r\) of linear subspaces such that \(V_1\) generates the Lie algebra \({\mathfrak n}\), \([V_1,V_r]=0\) and \([V_1,V_j] \subseteq V_{j+1}\), \(1\leq j\leq r-1\). In the paper under review the author studies the semi-direct product group \(G=\mathbb{R} \ltimes N\), where \(N=\exp{\mathfrak n}\) and \(\mathbb{R}\) acts on \({\mathfrak n}\) by \(t\cdot (v_1 \oplus \cdots \oplus v_r) =e^{t_{v_1}} \oplus \cdots \oplus e^{rt} v_r\). Let \(X_1, \dots, X_m\) denote the system of left invariant vector fields associated to a basis of \(V_1\), and consider the sub-Laplacian \(\Delta= -(X^2_1 +\cdots +X^2_m+ {\partial^2 \over \partial t^2})\) on \(G\). It is shown that if \(m\) has comp…
reviewed by: Eberhard Kaniuth
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 1220348
zbMATH Keywords: multiplier
zbMATH Keywords: real Lie group
zbMATH Keywords: sub-Laplacian
zbMATH Keywords: bounded linear operator
zbMATH Keywords: nilpotent Lie group
zbMATH Keywords: exponential growth
zbMATH Keywords: Heisenberg groups
zbMATH Keywords: heat kernel
MaRDI profile type: Publication
OpenAlex ID: W2072093877
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DOI: 10.5802/AIF.1644
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1220348