New Real-Variable Characterizations of Musielak-Orlicz Hardy Spaces

Let $\varphi: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight. The Musielak-Orlicz Hardy space $H^{\varphi}(\mathbb R^n)$ is defined to be the space of all $f\in{\mathcal S}'({\mathbb R^n})$ such that the grand maximal function $f^*$ belongs to the Musielak-Orlicz space $L^\varphi(\mathbb R^n)$. Luong Dang Ky established its atomic characterization. In this paper, the authors establish some new real-variable characterizations of $H^{\varphi}(\mathbb R^n)$ in terms of the vertical or the non-tangential maximal functions, or the Littlewood-Paley $g$-function or $g_\lambda^\ast$-function, via first establishing a Musielak-Orlicz Fefferman-Stein vector-valued inequality. Moreover, the range of $\lambda$ in the $g_\lambda^\ast$-function characterization of $H^\varphi(\mathbb R^n)$ coincides with the known best results, when $H^\varphi(\mathbb R^n)$ is the classical Hardy space $H^p(\mathbb R^n)$, with $p\in (0,1]$, or its weighted variant.