Atomic Characterizations of Weak Martingale Musielak--Orlicz Hardy Spaces and Their Applications

TL;DR

This paper develops atomic characterizations for weak martingale Musielak--Orlicz Hardy spaces, enabling new boundedness results for operators and inequalities, thus generalizing and improving existing theories in martingale Hardy spaces.

Contribution

It introduces atomic characterizations for weak martingale Musielak--Orlicz Hardy spaces, leading to new boundedness and inequality results that extend prior work.

Findings

Established atomic characterizations of weak martingale Musielak--Orlicz Hardy spaces.

Proved boundedness of sublinear operators from these Hardy spaces to weak Musielak--Orlicz spaces.

Derived new martingale inequalities and relationships among various weak martingale Musielak--Orlicz Hardy spaces.

Abstract

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