Weak type commutator and Lipschitz estimates: resolution of the Nazarov-Peller conjecture

Let $\mathcal{M}$ be a semi-finite von Neumann algebra and let $f:{\Bbb R}\rightarrow{\Bbb C}$ be a Lipschitz function. If $A,B\in\mathcal{M}$ are self-adjoint operators such that $[A,B]\in L_1(\mathcal{M})$, then$$\big\|[f(A),B]\big\|_{1,\infty}\leq c_{abs}\|f^\prime\|_{\infty}\big\|[A,B]\big\|_1,$$where $c_{abs}$ is an absolute constant independent of $f$, $\mathcal{M}$ and $A,B$ and $\|\cdot\|_{1,\infty}$ denotes the weak $L_1$-norm. If $X,Y\in\mathcal{M}$ are self-adjoint operators such that $X-Y\in L_1(\mathcal{M})$, then$$\big\|f(X)-f(Y)\big\|_{1,\infty}\leq c_{abs}\|f^\prime\|_{\infty}\|X-Y\|_1.$$This result resolves a conjecture raised by F. Nazarov and V. Peller implying a couple of existing results in perturbation theory.