Atomic decompositions of vector-valued weak Musielak-Orlicz martingale spaces and applications

In this paper, some vector-valued weak martingale Hardy spaces of Musielak-Orlicz type are introduced, and several atomic decompositions are established for them. As applications of the atomic decompositions, a sufficient condition for sublinear operators defined on vector-valued weak Musielak-Orlicz martingale spaces to be bounded is given. Using the sufficient condition, some martingale inequalities are deduced. These results closely depend on the geometrical properties of the Banach space in which the martingales take values. The results obtained here generalize the corresponding known results of scalar-valued and vector-valued weak martingale Hardy spaces.