Quantitative weighted bounds for Calder\'{o}n commutator with rough kernel

We consider weighted $L^p(w)$ boundedness ($1<p<\mathrm{\infty }$ and $w$ a Muckenhoupt $A_p$ weight) of the Calder'{o}n commutator ${\mathcal{C}}_{\mathrm{\Omega }}$ associated with rough homogeneous kernel, under the condition $\Omega\in L^q(\mathbb S^{n-1})$for$q_0<q\leq\infty$with$q_0$ a fixed constant depending on $w$. Comparing to the previous related known results (assuming $\mathrm{\Omega }\in L^{\mathrm{\infty }}({\mathbb{S}}^{n-1})$), our result for $\Omega\in L^q(\mathbb S^{n-1})$with$q$intherange$(q_0,\infty)$ is new. We also obtain a quantitative weighted bound for this ${\mathcal{C}}_{\mathrm{\Omega }}$ on $L^p(w)$, which is the best known quantitative result for this class of operators.