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

Abstract: 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.