# $L^2$ boundedness for maximal commutators with rough variable kernels

### Yanping Chen

University of Sciences and Technology, Beijing, China### Yong Ding

Beijing Normal University, China### Ran Li

Beijing Normal University, China

## Abstract

For $b\in BMO(\mathbb{R}^n)$ and $k\in\mathbb{N}$, the $k$-th order maximal commutator of the singular integral operator $T$ with rough variable kernels is defined by

In this paper the authors prove that the $k$-th order maximal commutator $T^{\ast}_{b,k}$ is a bounded operator on $L^2(\mathbb{R}^n)$ if $\Omega$ satisfies the same conditions given by Calderón and Zygmund. Moreover, the $L^2$-boundedness of the $k$-th order commutator of the rough maximal operator $M_\Omega$ with variable kernel, which is defined by

is also given here. These results obtained in this paper are substantial improvement and extension of some known results.

## Cite this article

Yanping Chen, Yong Ding, Ran Li, $L^2$ boundedness for maximal commutators with rough variable kernels. Rev. Mat. Iberoam. 27 (2011), no. 2, pp. 361–391

DOI 10.4171/RMI/640