John-Nirenberg inequality and atomic decomposition for noncommutative martingales

TL;DR

This paper extends classical inequalities and atomic decompositions to noncommutative martingales, providing new inequalities, duality results, and answering a previously open question in the field.

Contribution

It establishes refined John-Nirenberg inequalities and atomic decompositions for noncommutative martingales, improving prior results and addressing open problems.

Findings

Crude and fine versions of John-Nirenberg inequality for noncommutative BMO.

Extension of atomic decomposition for noncommutative Hardy spaces.

Negative answer to a question about BMO posed by Junge and Musat.

Abstract

In this paper, we study the John-Nirenberg inequality for BMO and the atomic decomposition for H1 of noncommutative martingales. We first establish a crude version of the column (resp. row) John-Nirenberg inequality for all 0 < p < \infty. By an extreme point property of Lp -space for 0 < p \leq 1, we then obtain a fine version of this in equality. The latter corresponds exactly to the classical John-Nirenberg inequality and enables us to obtain an exponential integrability inequality like in the classical case. These results extend and improve Junge and Musat's John-Nirenberg inequality. By duality, we obtain the corresponding q-atomic decomposition for different Hardy spaces H1 for all 1<q\leq\infty, which extends the 2-atomic decomposition previously obtained by Bekjan et al. Finally, we give a negative answer to a question posed by Junge and Musat about BMO.