Intrinsic square function characterizations of weak Musielak–Orlicz Hardy spaces

Abstract

Let $\phi :{\mathbb{R}}^n\times [0,\infty )\to [0,\infty )$ satisfy that, for any given $x\in {\mathbb{R}}^n$, $\phi (x,\cdot )$ is an Orlicz function and $\phi (\cdot ,t)$ is a Muckenhoupt $A_{\infty }({\mathbb{R}}^n)$ weight uniformly in $t\in (0,\infty )$. In this article, via using the atomic and Littlewood–Paley function characterizations of the weak Musielak–Orlicz Hardy space ${\mathit{WH}}^{\phi }({\mathbb{R}}^n)$, for any $\alpha \in (0,1]$ and $s\in {\mathbb{Z}}_{+}$, we first establish its $s$-order intrinsic square function characterizations in terms of the intrinsic Lusin area function $S_{\alpha ,s}$, the intrinsic $g$-function $g_{\alpha ,s}$ and the intrinsic $g_{\lambda }^{\ast }$-function $g_{\lambda ,\alpha ,s}^{\ast }$ with the best known range $\lambda \in (2+2(\alpha +s)/n,\infty )$.