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

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 $T_{b,k}^{\ast }$ 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.