$L^2$ 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 $$ T^{\ast}_{b,k}f(x) = \sup_{\varepsilon > 0} \biggl| \int_{|x-y| > \varepsilon} \frac{\Omega(x,x-y)}{|x-y|^n} (b(x)-b(y))^{k} f(y) dy \biggl|. $$ 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 $$ M_{\Omega;b,k}f(x) = \sup_{r > 0} \dfrac{1}{r^{n}} \int_{|x-y| < r} |\Omega(x,x-y)| |b(x)-b(y)|^{k} |f(y)| dy, $$ is also given here. These results obtained in this paper are substantial improvement and extension of some known results.