Function spaces on quantum tori

Xiao Xiong; Quanhua Xu; Zhi Yin

doi:10.1016/j.crma.2015.06.002

Functional analysis

Function spaces on quantum tori
[Espaces de fonctions sur les tores quantiques]

Présenté par : Gilles Pisier

Xiao Xiong ¹ ; Quanhua Xu ^1,² ; Zhi Yin ²

¹ Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon cedex, France
² School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Comptes Rendus. Mathématique, Volume 353 (2015) no. 8, pp. 729-734.

Résumé
Abstract

On considère les espaces de Sobolev, Besov et Triebel–Lizorkin sur un tore quantique $T_{θ}^{d}$ de d générateurs. Les principaux résultats comprennent : le plongement de Besov et Sobolev ; des caractérisations à la Littlewood–Paley pour les espaces de Besov et Triebel–Lizorkin ; une formule explicite de la K-fonctionnelle de $(L_{p} (T_{θ}^{d}), W_{p}^{k} (T_{θ}^{d}))$ ; l'indépendance en θ des multiplicateurs de Fourier complètement bornés sur ces espaces.

We study Sobolev, Besov and Triebel–Lizorkin spaces on quantum tori. These spaces share many properties with their classical counterparts. The results announced include: Besov and Sobolev embedding theorems; Littlewood–Paley-type characterizations of Besov and Triebel–Lizorkin spaces; an explicit description of the K-functional of $(L_{p} (T_{θ}^{d}), W_{p}^{k} (T_{θ}^{d}))$ ; descriptions of completely bounded Fourier multipliers on these spaces.

Reçu le : 2015-05-11
Accepté le : 2015-06-03
Publié le : 2015-06-30

DOI : 10.1016/j.crma.2015.06.002

Affiliations des auteurs :

Xiao Xiong ¹ ; Quanhua Xu ^{1, 2} ; Zhi Yin ²

¹ Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon cedex, France
² School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

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     title = {Function spaces on quantum tori},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {729--734},
     publisher = {Elsevier},
     volume = {353},
     number = {8},
     year = {2015},
     doi = {10.1016/j.crma.2015.06.002},
     language = {en},
}

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Xiao Xiong; Quanhua Xu; Zhi Yin. Function spaces on quantum tori. Comptes Rendus. Mathématique, Volume 353 (2015) no. 8, pp. 729-734. doi : 10.1016/j.crma.2015.06.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.06.002/

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