Analysis of joint spectral multipliers on Lie groups of polynomial growth

Martini, Alessio

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

Annales de l'Institut Fourier, Tome 62 (2012), p. 1215-1263 / Harvested from Numdam

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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.

Publié le : 2012-01-01

MR 3025742 | Zbl 1255.43003

DOI : https://doi.org/10.5802/aif.2721
Classification: 43A22, 22E30, 42B15
Mots clés: multiplicateurs spectraux, calcul fonctionnel conjoint, opérateurs différentiels, groupes de Lie, croissance polynomiale, opérateurs intégraux singuliers

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     title = {Analysis of joint spectral multipliers on Lie groups of polynomial growth},
     journal = {Annales de l'Institut Fourier},
     volume = {62},
     year = {2012},
     pages = {1215-1263},
     doi = {10.5802/aif.2721},
     zbl = {1255.43003},
     mrnumber = {3025742},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2012__62_4_1215_0}
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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) pp. 1215-1263. doi : 10.5802/aif.2721. http://gdmltest.u-ga.fr/item/AIF_2012__62_4_1215_0/

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