On quantitative weighted bounds for commutators of rough singular integral operators

Let $\mathrm{\Omega }$ be a homogeneous function of degree zero, integrable, and have a mean value of zero on the unit sphere ${\mathbb{S}}^{n-1}$, $n\ge 2$. Let $T_{\mathrm{\Omega }}$ be the homogeneous convolution singular integral operator with kernel $\frac{\mathrm{\Omega }(x)}{|x{|}^n}$. In this paper, we proved some quantitative two-weight estimates for the commutator, $[b,T_{\mathrm{\Omega }}]$, with a $BMO$ symbol $b$ and the singular integral operator $T_{\mathrm{\Omega }}$ under the condition $\mathrm{\Omega }\in L^q({\mathbb{S}}^{n-1})$ for $q\in (2,\mathrm{\infty })$.