# BMO-estimates for non-commutative vector valued Lipschitz functions

## TL;DR

This paper develops BMO-space estimates for non-commutative Lipschitz functions using Markov semigroups on finite von Neumann algebras, extending classical commutator bounds to the non-commutative setting.

## Contribution

It introduces a framework for BMO estimates for non-commutative Lipschitz functions via Markov semigroups, including multivariate cases with Hörmander-Mikhlin conditions.

## Key findings

- Established BMO bounds for commutators with Lipschitz functions
- Extended results to multivariate functions under Hörmander-Mikhlin assumptions
- Proved automatic continuity of Markov dilations in this context

## Abstract

We construct Markov semi-groups $\mathcal{T}$ and associated BMO-spaces on a finite von Neumann algebra $(\mathcal{M}, \tau)$ and obtain results for perturbations of commutators and non-commutative Lipschitz estimates. In particular, we prove that for any $A \in \mathcal{M}$ self-adjoint and $f: \mathbb{R} \rightarrow \mathbb{R}$ Lipschitz there is a Markov semi-group $\mathcal{T}$ such that for $x \in \mathcal{M}$, \[ \Vert [f(A), x] \Vert_{{\rm BMO}(\mathcal{M}, \mathcal{T})} \leq c_{abs} \Vert f' \Vert_\infty \Vert [A, x] \Vert_\infty. \] We obtain an analogue of this result for more general von Neumann valued-functions $f: \mathbb{R}^n \rightarrow \mathcal{N}$ by imposing H\"ormander-Mikhlin type assumptions on $f$. In establishing these result we show that Markov dilations of Markov semi-groups have certain automatic continuity properties. We also show that Markov semi-groups of double operator integrals admit (standard and reversed) Markov dilations.