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

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