New Real-Variable Characterizations of Musielak-Orlicz Hardy Spaces

Abstract

Let $\phi :{\mathbb{R}}^n\times [0,\infty )\to [0,\infty )$ be such that $\phi (x,\cdot )$ is an Orlicz function and $\phi (\cdot ,t)$ is a Muckenhoupt $A_{\infty }({\mathbb{R}}^n)$ weight. The Musielak-Orlicz Hardy space $H^{\phi }({\mathbb{R}}^n)$ is defined to be the space of all $f\in {\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ such that the grand maximal function $f^{\ast }$ belongs to the Musielak-Orlicz space $L^{\phi }({\mathbb{R}}^n)$. Luong Dang Ky established its atomic characterization. In this paper, the authors establish some new real-variable characterizations of $H^{\phi }({\mathbb{R}}^n)$ 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 $H^{\phi }({\mathbb{R}}^n)$ coincides with the known best results, when $H^{\phi }({\mathbb{R}}^n)$ is the classical Hardy space $H^p({\mathbb{R}}^n)$, with $p\in (0,1]$, or its weighted variant.

arXiv commentJ. Math. Anal. Appl. (to appear)