arXiv2018
Let $(Ω,\mathcal{F},\mathbb{P})$ be a probability space and $φ:\ Ω\times[0,\infty)\to[0,\infty)$ be a Musielak-Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak-Orlicz space $L^φ(Ω)$. Using this and extrapolation method, the authors then establish a Fefferman-Stein vector-valued Doob maximal inequality on $L^φ(Ω)$. As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for $L^φ(Ω)$, which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak-Orlicz Hardy spaces $H_φ^s(Ω)$, $P_φ(Ω)$, $Q_φ(Ω)$, $H_φ^S(Ω)$ and $H_φ^M(Ω)$. From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak-Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on $H_φ^S(Ω)$ and $H_φ^M(Ω)$, the authors obtain the Burkholder-Davis-Gundy inequality associated with Musielak--Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from $H_φ[0,1)$ to $L^φ[0,1)$, which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.