Estimates for Parametric Marcinkiewicz Integrals on Musielak-Orlicz Hardy Spaces

Abstract

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

30 pages, accepted by JMI