John–Nirenberg Inequalities for Noncommutative Column BMO and Lipschitz Martingales

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 $\textsf{bmo}^c(\mathcal M)$ 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 (Chen et al. in Atomic decompositions for noncommutative martingales. arXiv: 2001.08775v1, math. OA, 2020; Hong and Mei in J Funct Anal 263:1064–1097, 2012). 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.