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