Improved $A_1-A_{\mathrm{\infty }}$ and related estimates for commutators of rough singular integrals

An $A_1-A_{\mathrm{\infty }}$ estimate improving a previous result in arXiv:1607.06432 is obtained. Also new a result in terms of the $A_{\mathrm{\infty }}$ constant and the one supremum $A_q-A_{\mathrm{\infty }}^{\exp }$ constant, is proved, providing a counterpart for the result obained in arXiv:1705.08364. Both of the preceding results rely upon a sparse domination in terms of bilinear forms for $[b,T_{\mathrm{\Omega }}]$ with $\mathrm{\Omega }\in L^{\mathrm{\infty }}({\mathbb{S}}^{n-1})$ and $b\in BMO$ which is established relying upon techniques from arXiv:1705.07397.