John–Nirenberg inequality and atomic decomposition for noncommutative martingales

In this paper, we study the John–Nirenberg inequality forBMO and the atomic decomposition forH1 of noncommutative martingales. We first establish a crude version of the column (resp. row) John–Nirenberg inequality for all 0 < p <∞. By an extreme point property of Lp-space for 0 < p 1, we then obtain a fine version of this inequality. 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 ∞, 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. © 2012 Elsevier Inc. All rights reserved.