Atomic decompositions for noncommutative martingales
Abstract
We prove an atomic type decomposition for the noncommutative martingale Hardy space 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 p for all 0< p < 1, and provide a constructive proof of the atomic decomposition for p=1. We also study (p,\8)c-atoms, and show that every (p,2)c-atom can be decomposed into a sum of (p,\8)c-atoms; consequently, for every 0<p 1, the (p,q)c-atoms lead to the same atomic space for all 2 q\8. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space 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 proving some sharp martingale inequalities.