no. 4
Weak type operator Lipschitz and commutator estimates for commuting tuples
Caspers, Martijn1; Sukochev, Fedor2; Zanin, Dmitriy2
1 Mathematisch Instituut Budapestlaan 6, 3584 CD, Utrecht (The Netherlands)
2 School of Mathematics and Statistics UNSW, Kensington 2052, NSW (Australia)
Annales de l'Institut Fourier, Volume 68 (2018) no. 4, pp. 1643-1669.

Let f:ℝ d →ℝ 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)∥∞max1≤k≤d∥[Ak,B]∥1,

where c(d) is a constant independent of f, ℳ 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)∥∞max1≤k≤d∥Xk-Yk∥1.

Soit f:ℝ d →ℝ 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)∥∞max1≤k≤d∥[Ak,B]∥1,

où c(d) est une constante indépendante de f, ℳ 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)∥∞max1≤k≤d∥Xk-Yk∥1.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3195
Classification: 47B10, 47L20, 47A30
Keywords: Non-commutative $L_p$-spaces, commutator estimates, Calderón–Zygmund theory
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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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, Volume 68 (2018) no. 4, pp. 1643-1669. doi : 10.5802/aif.3195. https://aif.centre-mersenne.org/articles/10.5802/aif.3195/

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