## 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.

## Citation

Bo Li. "Parametric Marcinkiewicz integrals with rough kernels acting on weak Musielak–Orlicz Hardy spaces." Banach J. Math. Anal. 13 (1) 47 - 63, January 2019. https://doi.org/10.1215/17358787-2018-0015

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