Fourier multipliers and group von Neumann algebras

Akylzhanov, Rauan; Ruzhansky, Michael

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

Dans cette note, nous établissons des Lp–Lq bornitudes de multiplicateurs de Fourier sur les groupes unimodulaires localement compacts pour 1<p≤2≤q<∞. 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 Rn et des résultats déjà connus associés aux multiplicateurs de Fourier sur les groupes de Lie compacts.

In this paper we establish the Lp–Lq boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices 1<p≤2≤q<∞. 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 Rn and the corresponding known results for Fourier multipliers on compact Lie groups.

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