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 $f:\mathbb{R}\to \mathbb{C}$ be a Lipschitz function. If $A,B\in \mathcal{M}$ are self-adjoint operators such that $[A,B]\in L_1(\mathcal{M}),$ then $$\Vert [f(A),B]{\Vert }_{1,\infty }\le c_{abs}\Vert f^{\prime }{\Vert }_{\infty }\Vert [A,B]{\Vert }_1,$$ where $c_{abs}$ is an absolute constant independent of $f$, $\mathcal{M}$ and $A,B$ and $\Vert \cdot {\Vert }_{1,\infty }$ 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 $$\Vert f(X)-f(Y){\Vert }_{1,\infty }\le c_{abs}\Vert f^{\prime }{\Vert }_{\infty }\Vert X-Y{\Vert }_1.$$ This result resolves a conjecture raised by F. Nazarov and V. Peller implying a couple of existing results in perturbation theory.