  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.
