Weak Type Endpoint Estimates for the Commutators of Rough Singular Integral Operators

Abstract

Let $\Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere $S^{n-1}$, $T_{\Omega }$ be the convolution singular integral operator with kernel $\frac{\Omega (x)}{|x{|}^n}$. For $b\in \mathrm{BMO}({\mathbb{R}}^n)$, let $T_{\Omega ,b}$ be the commutator of $T_{\Omega }$. In this paper, by establishing suitable sparse dominations, the authors establish some weak type endpoint estimates of $L\log L$ type for $T_{\Omega ,b}$ when $\Omega \in L^q(S^{n-1})$ for some $q\in (1,\infty ]$.

15 pages