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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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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Haonan Zhang Lp-Lq Fourier Multipliers on Locally Compact Quantum Groups, Journal of Fourier Analysis and Applications, Volume 29 (2023) no. 4 | DOI:10.1007/s00041-023-10029-z
Vishvesh Kumar; Michael Ruzhansky Hausdorff–Young inequality for Orlicz spaces on compact homogeneous manifolds, Indagationes Mathematicae, Volume 31 (2020) no. 2, p. 266 | DOI:10.1016/j.indag.2020.01.004
Rauan Akylzhanov; Michael Ruzhansky L-L multipliers on locally compact groups, Journal of Functional Analysis, Volume 278 (2020) no. 3, p. 108324 | DOI:10.1016/j.jfa.2019.108324
Rauan Akylzhanov; Shahn Majid; Michael Ruzhansky Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups, Communications in Mathematical Physics, Volume 362 (2018) no. 3, pp. 761-799 | DOI:10.1007/s00220-018-3219-4
Duván Cardona Besov continuity for pseudo-differential operators on compact homogeneous manifolds, Journal of Pseudo-Differential Operators and Applications, Volume 9 (2018) no. 4, p. 861 | DOI:10.1007/s11868-017-0226-8

Cité par 5 documents. Sources : Crossref, NASA ADS

^☆ 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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