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

TL;DR

This paper proves a conjecture by Nazarov and Peller, establishing Lipschitz estimates for operator functions and commutators in semi-finite von Neumann algebras, advancing perturbation theory with sharp bounds.

Contribution

It provides the first proof of the Nazarov-Peller conjecture, offering new weak-type Lipschitz estimates for operator functions and commutators in non-commutative analysis.

Findings

Established weak $L_1$-norm bounds for commutators of Lipschitz functions of operators.

Resolved the Nazarov-Peller conjecture in the setting of semi-finite von Neumann algebras.

Connected Lipschitz continuity with perturbation bounds in non-commutative operator theory.

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.