Estimates for Parametric Marcinkiewicz Integrals on Musielak-Orlicz Hardy Spaces

Let $\varphi:\mathbb{R}^n\times[0,\,\infty) \rightarrow [0,\,\infty)$ satisfy that $\varphi(x,\,\cdot)$, for any given $x\in\mathbb{R}^n$, is an Orlicz function and $\varphi(\cdot\,,t)$ is a Muckenhoupt $A_\infty$ weight uniformly in $t\in(0,\,\infty)$. The Musielak-Orlicz Hardy space $H^\varphi(\mathbb{R}^n)$ generalizes both of the weighted Hardy space and the Orlicz Hardy space and hence has a wide generality. In this paper, the authors first prove the completeness of both of the Musielak-Orlicz space $L^\varphi(\mathbb{R}^n)$ and the weak Musielak-Orlicz space $WL^\varphi(\mathbb{R}^n)$. Then the authors obtain two boundedness criterions of operators on Musielak-Orlicz spaces. As applications, the authors establish the boundedness of parametric Marcinkiewicz integral $\mu^\rho_\Omega$ from $H^\varphi(\mathbb{R}^n)$ to $L^\varphi(\mathbb{R}^n)$ (resp. $WL^\varphi(\mathbb{R}^n)$) under weaker smoothness condition (resp. some Lipschitz condition) assumed on $\Omega$. These results are also new even when $\varphi(x,\,t):=\phi(t)$ for all $(x,\,t)\in\mathbb{R}^n\times[0,\,\infty)$, where $\phi$ is an Orlicz function.