Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces

Abstract:

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 uniformly in $t$. In this article, for any $\alpha \in (0,1]$ and $s\in \mathbb {Z}_+$, the authors establish the $s$-order intrinsic square function characterizations of $H^{\varphi }(\mathbb R^n)$ in terms of the intrinsic Lusin area function $S_{\alpha ,s}$, the intrinsic $g$-function $g_{\alpha ,s}$ and the intrinsic $g_{\lambda }^*$-function $g^\ast _{\lambda , \alpha ,s}$ with the best known range $\lambda \in (2+2(\alpha +s)/n,\infty )$, which are defined via $\mathrm {Lip}_\alpha ({\mathbb R}^n)$ functions supporting in the unit ball. A $\varphi$-Carleson measure characterization of the Musielak-Orlicz Campanato space ${\mathcal L}_{\varphi ,1,s}({\mathbb R}^n)$ is also established via the intrinsic function. To obtain these characterizations, the authors first show that these $s$-order intrinsic square functions are pointwise comparable with those similar-looking $s$-order intrinsic square functions defined via $\mathrm {Lip}_\alpha ({\mathbb R}^n)$ functions without compact supports, which when $s=0$ was obtained by M. Wilson. All these characterizations of $H^{\varphi }(\mathbb R^n)$, even when $s=0$, \[ \varphi (x,t):=w(x)t^p\ \textrm {for\ all}\ t\in [0,\infty )\ \textrm {and}\ x\in {\mathbb R}^n\] with $p\in (n/(n+\alpha ), 1]$ and $w\in A_{p(1+\alpha /n)}(\mathbb R^n)$, also essentially improve the known results.