Weak type operator Lipschitz and commutator estimates for commuting tuples
[Estimées de type faible pour les fonctions operateur-Lipschitz et les commutateurs de plusieurs variables]
Annales de l'Institut Fourier, Tome 68 (2018) no. 4, pp. 1643-1669.

Soit f: d une fonction Lipschitzienne. Si B est un opérateur borné auto-adjoint et si {A k } k=1 d sont des opérateurs bornés auto-adjoints qui commutent et tels que [A k ,B]L 1 (H), alors

[f(A1,,Ad),B]1,c(d)(f)max1kd[Ak,B]1,

c(d) est une constante indépendante de f, et A,B et · 1, désigne la norme L 1 -faible.

Si {X k } k=1 d (respectivement {Y k } k=1 d ) sont des opérateurs bornés qui commutent et tels que X k -Y k L 1 (H), alors

f(X1,,Xd)-f(Y1,,Yd)1,c(d)(f)max1kdXk-Yk1.

Let f: d be a Lipschitz function. If B is a bounded self-adjoint operator and if {A k } k=1 d are commuting bounded self-adjoint operators such that [A k ,B]L 1 (H), then

[f(A1,,Ad),B]1,c(d)(f)max1kd[Ak,B]1,

where c(d) is a constant independent of f, and A,B and · 1, denotes the weak L 1 -norm.

If {X k } k=1 d (respectively, {Y k } k=1 d ) are commuting bounded self-adjoint operators such that X k -Y k L 1 (H), then

f(X1,,Xd)-f(Y1,,Yd)1,c(d)(f)max1kdXk-Yk1.

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DOI : 10.5802/aif.3195
Classification : 47B10, 47L20, 47A30
Keywords: Non-commutative $L_p$-spaces, commutator estimates, Calderón–Zygmund theory
Mot clés : Espaces $L_p$ non commutatifs, estimées de commutateurs, théorie de Calderón–Zygmund
Caspers, Martijn 1 ; Sukochev, Fedor 2 ; Zanin, Dmitriy 2

1 Mathematisch Instituut Budapestlaan 6, 3584 CD, Utrecht (The Netherlands)
2 School of Mathematics and Statistics UNSW, Kensington 2052, NSW (Australia)
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Caspers, Martijn; Sukochev, Fedor; Zanin, Dmitriy. Weak type operator Lipschitz and commutator estimates for commuting tuples. Annales de l'Institut Fourier, Tome 68 (2018) no. 4, pp. 1643-1669. doi : 10.5802/aif.3195. http://archive.numdam.org/articles/10.5802/aif.3195/

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