## Banach Journal of Mathematical Analysis

### Parametric Marcinkiewicz integrals with rough kernels acting on weak Musielak–Orlicz Hardy spaces

Bo Li

#### Abstract

Let $\varphi:\mathbb{R}^{n}\times[0,\infty)\to[0,\infty)$ satisfy that $\varphi(x,\cdot)$, for any given $x\in\mathbb{R}^{n}$, is an Orlicz function and that $\varphi(\cdot ,t)$ is a Muckenhoupt $A_{\infty}$ weight uniformly in $t\in(0,\infty)$. The weak Musielak–Orlicz Hardy space $\mathit{WH}^{\varphi}(\mathbb{R}^{n})$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak–Orlicz space $\mathit{WL}^{\varphi}(\mathbb{R}^{n})$. 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 $x\in\mathbb{R}^{n}$,

$$\mu^{\rho}_{\Omega}(f)(x):=(\int_{0}^{\infty}\vert\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}f(y){dy}\vert^{2}\frac{dt}{t^{2\rho+1}})^{1/2},$$ 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 $\mathit{WH}^{\varphi}(\mathbb{R}^{n})$ to weak Musielak–Orlicz space $\mathit{WL}^{\varphi}(\mathbb{R}^{n})$. 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.

#### Article information

Source
Banach J. Math. Anal., Volume 13, Number 1 (2019), 47-63.

Dates
Accepted: 7 May 2018
First available in Project Euclid: 30 October 2018

https://projecteuclid.org/euclid.bjma/1540865070

Digital Object Identifier
doi:10.1215/17358787-2018-0015

Mathematical Reviews number (MathSciNet)
MR3892337

Zentralblatt MATH identifier
07002031

#### Citation

Li, Bo. Parametric Marcinkiewicz integrals with rough kernels acting on weak Musielak–Orlicz Hardy spaces. Banach J. Math. Anal. 13 (2019), no. 1, 47--63. doi:10.1215/17358787-2018-0015. https://projecteuclid.org/euclid.bjma/1540865070

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