Weak type operator Lipschitz and commutator estimates for commuting tuples
Annales de l'Institut Fourier, Volume 68 (2018) no. 4, p. 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 : 2017-03-09
Revised : 2017-09-21
Accepted : 2017-11-07
Published online : 2018-11-23
DOI : https://doi.org/10.5802/aif.3195
Classification:  47B10,  47L20,  47A30
Keywords: Non-commutative L p -spaces, commutator estimates, Calderón–Zygmund theory 
@article{AIF_2018__68_4_1643_0,
     author = {Caspers, Martijn and Sukochev, Fedor and Zanin, Dmitriy},
     title = {Weak type operator Lipschitz and commutator estimates for commuting tuples},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {4},
     year = {2018},
     pages = {1643-1669},
     doi = {10.5802/aif.3195},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_4_1643_0}
}

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/item/AIF_2018__68_4_1643_0/

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