Analysis of joint spectral multipliers on Lie groups of polynomial growth

Martini, Alessio

doi:10.5802/aif.2721

Analysis of joint spectral multipliers on Lie groups of polynomial growth
[Analyse de multiplicateurs spectraux conjoints sur des groupes de Lie à croissance polynomiale]

Martini, Alessio ¹

¹ Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa (Italy)

Annales de l'Institut Fourier, Tome 62 (2012) no. 4, pp. 1215-1263.

Résumé
Abstract

On étudie la bornitude $L^{p}$ ( $1 < p < \infty$ ) des opérateurs de la forme $m (L_{1}, \dots, L_{n})$ pour un système commutatif $L_{1}, \dots, L_{n}$ d’opérateurs différentiels autoadjoints invariants à gauche sur un groupe de Lie $G$ à croissance polynomiale, qui engendrent une algèbre contenant un opérateur sous-coercif pondéré. En particulier, quand $G$ est un groupe homogène et $L_{1}, \dots, L_{n}$ sont homogènes, on prouve des analogues des theorèmes de multiplicateurs de Mihlin-Hörmander et Marcinkiewicz.

We study the problem of $L^{p}$ -boundedness ( $1 < p < \infty$ ) of operators of the form $m (L_{1}, \dots, L_{n})$ for a commuting system of self-adjoint left-invariant differential operators $L_{1}, \dots, L_{n}$ on a Lie group $G$ of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when $G$ is a homogeneous group and $L_{1}, \dots, L_{n}$ are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.

MR Zbl

DOI : 10.5802/aif.2721

Classification : 43A22, 22E30, 42B15
Keywords: spectral multipliers, joint functional calculus, differential operators, Lie groups, polynomial growth, singular integral operators
Mots-clés : multiplicateurs spectraux, calcul fonctionnel conjoint, opérateurs différentiels, groupes de Lie, croissance polynomiale, opérateurs intégraux singuliers

Affiliations des auteurs :

Martini, Alessio ¹

¹ Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa (Italy)

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Martini, Alessio. Analysis of joint spectral multipliers on Lie groups of polynomial growth. Annales de l'Institut Fourier, Tome 62 (2012) no. 4, pp. 1215-1263. doi : 10.5802/aif.2721. https://aif.centre-mersenne.org/articles/10.5802/aif.2721/

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