Atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces and their applications

Abstract

Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a probability space, and let $\phi :\Omega \times [0,\infty )\to [0,\infty )$ be a Musielak–Orlicz function. In this article, we establish the atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces ${\mathit{WH}}_{\phi }^s(\Omega )$, ${\mathit{WH}}_{\phi }^M(\Omega )$, ${\mathit{WH}}_{\phi }^S(\Omega )$, $W{P}_{\phi }(\Omega )$, and $W{Q}_{\phi }(\Omega )$. We then use these atomic characterizations to obtain the boundedness of $\sigma$-sublinear operators from weak martingale Musielak–Orlicz Hardy spaces to weak Musielak–Orlicz spaces, as well as some martingale inequalities which further clarify the relationships among these weak martingale Musielak–Orlicz Hardy spaces. All these results improve and generalize the corresponding results on weak martingale Orlicz–Hardy spaces. Moreover, we improve all the known results on weak martingale Musielak–Orlicz Hardy spaces. In particular, both the boundedness of $\sigma$-sublinear operators and the martingale inequalities, for weak weighted martingale Hardy spaces as well as for weak weighted martingale Orlicz–Hardy spaces, are new.