Weak type operator Lipschitz and commutator estimates for commuting tuples

Abstract

Let $f:{\mathbb{R}}^d\to \mathbb{R}$ 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 $[A_k,B]\in L_1(H),$ then $$\Vert [f(A_1,\cdots ,A_d),B]{\Vert }_{1,\infty }\le c(d)\Vert \nabla (f){\Vert }_{\infty }\underset{1\le k\le d}{\max }\Vert [A_k,B]{\Vert }_1,$$ where $c(d)$ is a constant independent of $f$, $\mathcal{M}$ and $A,B$ and $\Vert \cdot {\Vert }_{1,\infty }$ 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 $X_k-Y_k\in L_1(H),$ then $$\Vert f(X_1,\cdots ,X_d)-f(Y_1,\cdots ,Y_d){\Vert }_{1,\infty }\le c(d)\Vert \nabla (f){\Vert }_{\infty }\underset{1\le k\le d}{\max }\Vert X_k-Y_k{\Vert }_1.$$