Weak type operator Lipschitz and commutator estimates for commuting tuples

Soit $f:{ℝ}^d\to ℝ$ une fonction Lipschitzienne. Si $B$ est un opérateur borné auto-adjoint et si ${\left\{A_k\right\}}_{k=1}^d$ sont des opérateurs bornés auto-adjoints qui commutent et tels que $[A_k,B]\in L_1\left(H\right),$ alors

$$\parallel [f(A_1,\cdots ,A_d),B]{\parallel }_{1,\infty }\le c\left(d\right)\parallel \nabla \left(f\right){\parallel }_{\infty }\underset{1\le k\le d}{max}{\parallel [A_k,B]\parallel }_1,$$

où $c\left(d\right)$ est une constante indépendante de $f$, $ℳ$ et $A,B$ et ${\parallel ·\parallel }_{1,\infty }$ désigne la norme $L_1$-faible.

Si ${\left\{X_k\right\}}_{k=1}^d$ (respectivement ${\left\{Y_k\right\}}_{k=1}^d$) sont des opérateurs bornés qui commutent et tels que $X_k-Y_k\in L_1\left(H\right),$ alors

$$\parallel f(X_1,\cdots ,X_d)-f(Y_1,\cdots ,Y_d){\parallel }_{1,\infty }\le c\left(d\right)\parallel \nabla \left(f\right){\parallel }_{\infty }\underset{1\le k\le d}{max}{\parallel X_k-Y_k\parallel }_1.$$

Let $f:{ℝ}^d\to ℝ$ be a Lipschitz function. If $B$ is a bounded self-adjoint operator and if ${\left\{A_k\right\}}_{k=1}^d$ are commuting bounded self-adjoint operators such that $[A_k,B]\in L_1\left(H\right),$ then

$$\parallel [f(A_1,\cdots ,A_d),B]{\parallel }_{1,\infty }\le c\left(d\right)\parallel \nabla \left(f\right){\parallel }_{\infty }\underset{1\le k\le d}{max}{\parallel [A_k,B]\parallel }_1,$$

where $c\left(d\right)$ is a constant independent of $f$, $ℳ$ and $A,B$ and ${\parallel ·\parallel }_{1,\infty }$ denotes the weak $L_1$-norm.

If ${\left\{X_k\right\}}_{k=1}^d$ (respectively, ${\left\{Y_k\right\}}_{k=1}^d$) are commuting bounded self-adjoint operators such that $X_k-Y_k\in L_1\left(H\right),$ then

$$\parallel f(X_1,\cdots ,X_d)-f(Y_1,\cdots ,Y_d){\parallel }_{1,\infty }\le c\left(d\right)\parallel \nabla \left(f\right){\parallel }_{\infty }\underset{1\le k\le d}{max}{\parallel X_k-Y_k\parallel }_1.$$