Fourier multipliers and group von Neumann algebras

Rauan Akylzhanov; Michael Ruzhansky

doi:10.1016/j.crma.2016.05.010

Mathematical analysis/Harmonic analysis

Fourier multipliers and group von Neumann algebras
[Multiplicateurs de Fourier et algèbres de von Neumann]

Présenté par : the Editorial Board

Rauan Akylzhanov ¹ ; Michael Ruzhansky ¹

¹ Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

Comptes Rendus. Mathématique, Volume 354 (2016) no. 8, pp. 766-770.

Résumé
Abstract

Dans cette note, nous établissons des $L^{p}$ – $L^{q}$ bornitudes de multiplicateurs de Fourier sur les groupes unimodulaires localement compacts pour $1 < p \leq 2 \leq q < \infty$ . Notre approche est basée sur la technique des algèbres des opérateurs. Pour cela, nous prouvons une version de l'inégalité de Hausdorff–Young sur les groupes unimodulaires localement compacts. En particulier, le résultat obtenu implique le théorème de Hörmander sur les multiplicateurs de Fourier dans $R^{n}$ et des résultats déjà connus associés aux multiplicateurs de Fourier sur les groupes de Lie compacts.

In this paper we establish the $L^{p}$ – $L^{q}$ boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices $1 < p \leq 2 \leq q < \infty$ . Our approach is based on the operator algebras techniques. The result depends on a version of the Hausdorff–Young–Paley inequality that we establish on general locally compact separable unimodular groups. In particular, the obtained result implies the corresponding Hörmander's Fourier multiplier theorem on $R^{n}$ and the corresponding known results for Fourier multipliers on compact Lie groups.

Reçu le : 2015-11-28
Accepté le : 2016-05-17
Publié le : 2016-05-26

DOI : 10.1016/j.crma.2016.05.010

Affiliations des auteurs :

Rauan Akylzhanov ¹ ; Michael Ruzhansky ¹

¹ Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

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     title = {Fourier multipliers and group von {Neumann} algebras},
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     pages = {766--770},
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     number = {8},
     year = {2016},
     doi = {10.1016/j.crma.2016.05.010},
     language = {en},
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Rauan Akylzhanov; Michael Ruzhansky. Fourier multipliers and group von Neumann algebras. Comptes Rendus. Mathématique, Volume 354 (2016) no. 8, pp. 766-770. doi : 10.1016/j.crma.2016.05.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.05.010/

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Cité par Sources :

^☆ The second author was supported by the Leverhulme Research Grant RPG-2014-02 and by the EPSRC Grant EP/K039407/1. No new data was collected or generated during the course of the research.

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