[Espaces de fonctions sur les tores quantiques]
On considère les espaces de Sobolev, Besov et Triebel–Lizorkin sur un tore quantique 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'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 ; descriptions of completely bounded Fourier multipliers on these spaces.
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@article{CRMATH_2015__353_8_729_0,
author = {Xiong, Xiao and Xu, Quanhua and Yin, Zhi},
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},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.06.002/}
}
TY - JOUR AU - Xiong, Xiao AU - Xu, Quanhua AU - Yin, Zhi TI - Function spaces on quantum tori JO - Comptes Rendus. Mathématique PY - 2015 SP - 729 EP - 734 VL - 353 IS - 8 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2015.06.002/ DO - 10.1016/j.crma.2015.06.002 LA - en ID - CRMATH_2015__353_8_729_0 ER -
%0 Journal Article %A Xiong, Xiao %A Xu, Quanhua %A Yin, Zhi %T Function spaces on quantum tori %J Comptes Rendus. Mathématique %D 2015 %P 729-734 %V 353 %N 8 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2015.06.002/ %R 10.1016/j.crma.2015.06.002 %G en %F CRMATH_2015__353_8_729_0
Xiong, Xiao; Xu, Quanhua; Yin, Zhi. Function spaces on quantum tori. Comptes Rendus. Mathématique, Tome 353 (2015) no. 8, pp. 729-734. doi : 10.1016/j.crma.2015.06.002. http://archive.numdam.org/articles/10.1016/j.crma.2015.06.002/
[1] Limiting embedding theorems for when and applications, J. Anal. Math., Volume 87 (2002), pp. 37-75
[2] Harmonic analysis on quantum tori, Commun. Math. Phys., Volume 322 (2013), pp. 755-805
[3] Interpolation of linear operators on Sobolev spaces, Ann. Math., Volume 109 (1979), pp. 583-599
[4] On the equivalence of the K-functional and moduli of continuity and some applications, Lect. Notes Math., Volume 571 (1976), pp. 119-140
[5] Noncommutative Riesz transforms – a probabilistic approach, Amer. J. Math., Volume 132 (2010), pp. 611-681
[6] On the Bourgain, Brézis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., Volume 195 (2002), pp. 230-238
[7] Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group, Can. J. Math., Volume 63 (2011), pp. 1161-1187
[8] Noncommutative vector-valued spaces and completely p-summing maps, Astérisque, Volume 247 (1998) (vi+131 pp.)
[9] Noncommutative -spaces (Johnson, W.B.; Lindenstrauss, J., eds.), Handbook of the Geometry of Banach Spaces, vol. 2, North-Holland, Amsterdam, 2003, pp. 1459-1517
[10] Sobolev theory for noncommutative tori, Rend. Semin. Mat. Univ. Padova, Volume 86 (1992), pp. 143-156
[11] A symplectic approach to Yang–Mills theory for noncommutative tori, Can. J. Math., Volume 44 (1992), pp. 368-387
[12] Theory of Function Spaces, II, Birkhäuser, Basel, 1992
[13] Hardy–Littlewood theory for semigroups, J. Funct. Anal., Volume 63 (1985), pp. 240-260
[14] Lipschitz algebras and derivations of von Neumann algebras, J. Funct. Anal., Volume 139 (1996), pp. 261-300
[15] α-Lipschitz algebras on the noncommutative torus, J. Oper. Theory, Volume 39 (1998), pp. 123-138
[16] X. Xiong, Q. Xu, Z. Yin, Sobolev, Besov and Triebel–Lizorkin spaces on quantum tori, Preprint, 2015.
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