John-Nirenberg inequalities for noncommutative column BMO and Lipschitz martingales

TL;DR

This paper advances the understanding of noncommutative BMO and Lipschitz spaces by establishing John-Nirenberg inequalities using a novel noncommutative stopping time approach, solving open questions and revealing duality with Hardy spaces.

Contribution

It introduces a new noncommutative stopping time method to prove John-Nirenberg inequalities, addressing open problems and providing new insights into duality and space characterizations.

Findings

Established John-Nirenberg inequalities for noncommutative BMO and Lipschitz spaces.

Solved two open questions from previous literature.

Showed Lipschitz space as dual to noncommutative Hardy space.

Abstract

In this paper, we continue the study of John-Nirenberg theorems for BMO/Lipschitz spaces in the noncommutative martingale setting. As conjectured from the classical case, a desired noncommutative ``stopping time" argument was discovered to obtain the distribution function inequality form of John-Nirenberg theorem. This not only provides another approach without using duality and interpolation to the results for spaces $\mathsf{bmo}^c(\mathcal M)$ and ${{\Lambda}^{{c}}_{\beta}}(\mathcal{M})$, but also allows us to find the desired version of John-Nirenberg inequalities for spaces $\mathcal{BMO}^c(\mathcal M)$ and ${\mathcal L^{{c}}_{\beta}}(\mathcal{M})$. And thus we solve two open questions after \cite{ref5, ref3}. As an application, we show that Lipschitz space is also the dual space of noncommutative Hardy space defined via symmetric atoms. Finally, our results for ${\mathcal L^{{c}}_{\beta}}(\mathcal{M})$ as well as the approach seem new even going back to the classical setting.