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John-Nirenberg inequality and atomic decomposition for noncommutative martingales

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.