Atomic decompositions for noncommutative martingales

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We prove an atomic type decomposition for the noncommutative martingale Hardy space hp 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 hp 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 hp (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.