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 < ∞. 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.

In this paper, we establish UMD lattice-valued variational inequalities for differential operators, ergodic averages and analytic semigroups. These results generalize, on the one hand some scalar-valued variational inequalities in ergodic theory, on the other hand Xu's very recent result on UMD lattice-valued maximal inequality. As a consequence, we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence.

In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let (M, τ ) be a noncommutative probability space equipped with a weak- * dense filtration of von Neumann subalgebras (Mn) n≥1 . Let En denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for 1 < p < 2 and x ∈ Lp(M, τ ) one can find a, b ∈ Lp(M, τ ) and contractions un, vn ∈ M such thatMoreover, it turns out that aun and vnb converge in the row/column Hardy spaces H r p (M) and H c p (M) respectively. In particular, this solves a problem posed by Defant and Junge in 2004. In the case p = 1, our results establish a noncommutative form of Davis celebrated theorem on the relation between martingale maximal and square functions in L 1 , whose noncommutative form has remained open for quite some time. Given 1 ≤ p ≤ 2, we also provide new weak type maximal estimates, which imply in turn left/right almost uniform convergence of En(x) in row/column Hardy spaces. This improves the bilateral convergence known so far. Our approach is based on new forms of Davis martingale decomposition which are of independent interest, and an algebraic atomic description for the involved Hardy spaces. The latter results are new even for commutative von Neumann algebras.