Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommutative martingale Hardy space h_p for all 0<p<2 by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of h_p for all 0<p<1, and provide a constructive proof of the atomic decomposition for p=1 which resolves a main problem on the subject left open for the last twelve years. We also study (p,∞)_c-atoms, and show that every (p,2)_c-atom can be decomposed into a sum of (p,∞)_c-atoms; consequently, for every 0<p≤1, the (p,q)_c-atoms lead to the same atomic space for all 2≤q≤∞. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space h_p (0<p<1) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to prove some sharp martingale inequalities.