Title: 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
$$\|f(X_1,\cdots,X_d)-f(Y_1,\cdots,Y_d)\|_{1,\infty}\leq
c(d)\|\nabla(f)\|_{\infty}\max_{1\leq k\leq d}\|X_k-Y_k\|_1.$$