数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (5): 1981-1996.doi: 10.1007/s10473-023-0503-y

• • 上一篇    下一篇

THE FRACTIONAL TYPE MARCINKIEWICZ INTEGRALS AND COMMUTATORS ON WEIGHTED HARDY SPACES*

Yanyan han, Huoxiong wu   

  1. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
  • 收稿日期:2022-04-27 修回日期:2023-04-26 发布日期:2023-10-25

THE FRACTIONAL TYPE MARCINKIEWICZ INTEGRALS AND COMMUTATORS ON WEIGHTED HARDY SPACES*

Yanyan han, Huoxiong wu   

  1. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
  • Received:2022-04-27 Revised:2023-04-26 Published:2023-10-25
  • Contact: †Huoxiong wu, E-mail: huoxwu@xmu.edu.cn
  • About author:Yanyan han, E-mail: hanyanyan_bj@163.com

摘要: This paper is devoted to studying the behaviors of the fractional type Marcinkiewicz integrals $\mu_{\Omega, \beta}$ and the commutators $\mu_{\Omega, \beta}^b$ generated by $\mu_{\Omega, \beta}$ with $b\in L_{\rm loc}(\mathbb{R}^n)$ on weighted Hardy spaces. Under the assumption of that the homogeneous kernel $\Omega$ satisfies certain regularities, the authors obtain the boundedness of $\mu_{\Omega, \beta}$ from the weighted Hardy spaces $H^p_{\omega^p}(\mathbb{R}^n)$ to the weighted Lebesgue spaces $L^q_{\omega^q}(\mathbb{R}^n)$ for $n/(n+\beta)\le p\le 1$ with $1/q=1/p-\beta/n$, as well as the same $(H^p_{\omega^p}, L^q_{\omega^q})$-boudedness of $\mu_{\Omega, \beta}^b$ when $b$ belongs to $\mathcal{BMO}_{\omega^p, p}(\mathbb{R}^n)$, which is a non-trivial subspace of ${\rm BMO}(\mathbb{R}^n)$.

关键词: fractional type Marcinkiewicz integrals, commutators, Muckenhoupt weights, ${\rm BMO}$ spaces, Hardy spaces

Abstract: This paper is devoted to studying the behaviors of the fractional type Marcinkiewicz integrals $\mu_{\Omega, \beta}$ and the commutators $\mu_{\Omega, \beta}^b$ generated by $\mu_{\Omega, \beta}$ with $b\in L_{\rm loc}(\mathbb{R}^n)$ on weighted Hardy spaces. Under the assumption of that the homogeneous kernel $\Omega$ satisfies certain regularities, the authors obtain the boundedness of $\mu_{\Omega, \beta}$ from the weighted Hardy spaces $H^p_{\omega^p}(\mathbb{R}^n)$ to the weighted Lebesgue spaces $L^q_{\omega^q}(\mathbb{R}^n)$ for $n/(n+\beta)\le p\le 1$ with $1/q=1/p-\beta/n$, as well as the same $(H^p_{\omega^p}, L^q_{\omega^q})$-boudedness of $\mu_{\Omega, \beta}^b$ when $b$ belongs to $\mathcal{BMO}_{\omega^p, p}(\mathbb{R}^n)$, which is a non-trivial subspace of ${\rm BMO}(\mathbb{R}^n)$.

Key words: fractional type Marcinkiewicz integrals, commutators, Muckenhoupt weights, ${\rm BMO}$ spaces, Hardy spaces

中图分类号: 

  • 47B47