Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommuta-tive 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 = which resolves a main problem on the subject left open for the last twelve years. We also study (p, oo)c-atoms, and show that every (p, 2)c-atom can be decomposed into a sum of (p, oo)c- atoms; consequently, for every 0 < p < 1, the (p, q)c-atoms lead to the same atomic space for all 2 < q < oo. As appli-cations, 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.(c) 2023 Elsevier Inc. All rights reserved.