Parametric Marcinkiewicz integrals with rough kernels acting on weak 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 that $\phi (\cdot ,t)$ is a Muckenhoupt $A_{\infty }$ weight uniformly in $t\in (0,\infty )$. The weak Musielak–Orlicz Hardy space ${\mathit{WH}}^{\phi }\left({\mathbb{R}}^n\right)$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak–Orlicz space ${\mathit{WL}}^{\phi }\left({\mathbb{R}}^n\right)$. For parameter $\rho \in (0,\infty )$ and measurable function $f$ on ${\mathbb{R}}^n$, the parametric Marcinkiewicz integral ${\mu }_{\Omega }^{\rho }$ related to the Littlewood–Paley $g$-function is defined by setting, for all $x\in {\mathbb{R}}^n$,

$${\mu }_{\Omega }^{\rho }\left(f\right)\left(x\right):=\left({\int }_0^{\infty }\right|{\int }_{|x-y|\le t}\frac{\Omega (x-y)}{|x-y{|}^{n-\rho }}f\left(y\right)dy{|}^2\frac{dt}{t^{2\rho +1}}{)}^{1/2},$$ where $\Omega$ is homogeneous of degree zero satisfying the cancellation condition.

In this article, we discuss the boundedness of the parametric Marcinkiewicz integral ${\mu }_{\Omega }^{\rho }$ with rough kernel from weak Musielak–Orlicz Hardy space ${\mathit{WH}}^{\phi }\left({\mathbb{R}}^n\right)$ to weak Musielak–Orlicz space ${\mathit{WL}}^{\phi }\left({\mathbb{R}}^n\right)$. These results are new even for the classical weighted weak Hardy space of Quek and Yang, and probably new for the classical weak Hardy space of Fefferman and Soria.