On some martingale inequalities for mean oscillations in weak spaces

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Given a Banach function space \(X\) over a nonatomic probability space, we define, in a natural way, a quasi-Banach function space which is denoted by \(\mathrm {w}\text {-}{X}\). In this paper, we consider some inequalities in \(\mathrm {w}\text {-}{X}\) involving the mean oscillation and the maximal function of a martingale. The main results give necessary and sufficient conditions for the inequalities to be valid.