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Annales scientifiques de l'ENS - Titles - série 4, 50 (2017)

Titles < série 4, 50

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 50, fascicule 4 (2017)

Adrián González-Pérez, Marius Junge, Javier Parcet
Smooth Fourier multipliers in group algebras via Sobolev dimension
Annales scientifiques de l'ENS 50, fascicule 4 (2017), 879-925

Télécharger cet article : Fichier PDF

Résumé :
Multiplicateurs de Fourier réguliers dans des algèbres de groupe par la dimension de Sobolev
Nous étudions des multiplicateurs de Fourier à symboles réguliers sur des groupes localement compacts. De nouveaux critères de Hörmander-Mikhlin pour des multiplicateurs spectraux et non spectraux sont établis. Notre approche se base sur trois nouveaux résultats clés. Premièrement, nous utilisons certains opérateurs maximaux dans des espaces L_p non commutatifs pour obtenir un contrôle sur de larges classes de multiplicateurs. Ce principe général — exploité en analyse harmonique euclidienne ces 40 dernières années — présente un intérêt indépendant et pourrait admettre de nouvelles applications. Deuxièmement, en établissant une version non commutative de la théorie de plongement de Sobolev pour les semigroupes de Markov initiée par Varopoulos, la dimension de cocycle utilisée auparavant est remplacée par la dimension de Sobolev. Ceci permet plus de flexibilité sur la régularité du symbole. Enfin, nous introduisons une notion duale de la croissance polynomiale pour exploiter davantage notre principe du maximum sur des multiplicateurs de Fourier non spectraux. La combinaison de ces ingrédients produit de nouvelles estimations L_p pour des multiplicateurs de Fourier réguliers dans des algèbres de groupe.

Abstract:
We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new Hörmander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative L_p spaces. This general principle—exploited in Euclidean harmonic analysis during the last 40 years—is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimension by the Sobolev dimension. This is based on a noncommutative form of the Sobolev embedding theory for Markov semigroups initiated by Varopoulos, and yields more flexibility to measure the smoothness of the symbol. Third, we introduce a dual notion of polynomial growth to further exploit our maximal principle for non-spectral Fourier multipliers. The combination of these ingredients yields new L_p estimates for smooth Fourier multipliers in group algebras.

Keywords: Fourier multiplier, group von Neumann algebra, Sobolev dimension ; Multiplicateur de Fourier, algèbre de von Neumann associée à un groupe, dimension de Sobolev.

Class. math. : 42B15, 46L52, 47L25.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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