## Taiwanese Journal of Mathematics

### ON THE COMPACTNESS OF COMMUTATORS FOR ROUGH MARCINKIEWICZ INTEGRAL OPERATORS

#### Abstract

Let $b \in \operatorname{BMO}(\mathbb{R}^n)$ and $\mathscr{M}_\Omega$ be the Marcinkiewicz integral operatorwith kernel $\frac{\Omega(x)}{|x|^{n-1}}$, where $\Omega$ is homogeneous of degree zero, integrable and has mean value zero on the unit sphere $S^{n-1}$. In this paper, by means of Fourier transform estimates and approximationto the operator $\mathscr{M}_\Omega$ with integral operators having smooth kernelswe show that if $b \in \operatorname{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain weak size condition, then the commutator $\mathscr{M}_{\Omega,b} = [b, \mathscr{M}_\Omega]$ generated by $b$ and $\mathscr{M}_\Omega$ is a compact operator on $L^p(\mathbb{R}^n)$ for some $1\lt p\lt \infty$.

#### Article information

Source
Taiwanese J. Math., Volume 19, Number 6 (2015), 1777-1793.

Dates
First available in Project Euclid: 4 July 2017

https://projecteuclid.org/euclid.twjm/1499133739

Digital Object Identifier
doi:10.11650/tjm.19.2015.5656

Mathematical Reviews number (MathSciNet)
MR3434277

Zentralblatt MATH identifier
1357.42010

#### Citation

Mao, Suzhen; Sawano, Yoshihiro; Wu, Huoxiong. ON THE COMPACTNESS OF COMMUTATORS FOR ROUGH MARCINKIEWICZ INTEGRAL OPERATORS. Taiwanese J. Math. 19 (2015), no. 6, 1777--1793. doi:10.11650/tjm.19.2015.5656. https://projecteuclid.org/euclid.twjm/1499133739

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