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 $$|[f(A_1,\cdots,A_d),B]|{1,\infty}\leq c(d)|\nabla(f)|{\infty}\max{1\leq k\leq d}|[A_k,B]|1,$$ where $c(d)$ is a constant independent of $f$, $\mathcal{M}$ and $A,B$ and $|\cdot|{1,\infty}$ 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 $$|f(X_1,\cdots,X_d)-f(Y_1,\cdots,Y_d)|{1,\infty}\leq c(d)|\nabla(f)|{\infty}\max_{1\leq k\leq d}|X_k-Y_k|_1.$$