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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2015-11-28
Accepted: 2016-05-17
Published online: 2016-05-26

DOI: 10.1016/j.crma.2016.05.010

Author's affiliations:

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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     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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^☆ 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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