Title: Improved $A_1-A_\infty$ and related estimates for commutators of rough singular integrals,
Abstract: An $A_1-A_\infty$ estimate improving a previous result in arXiv:1607.06432 is
obtained. Also new a result in terms of the ${A_\infty}$ constant and the one
supremum $A_q-A_\infty^{\exp}$ 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 $[b,T_\Omega]$ with
$\Omega\in L^\infty(\mathbb{S}^{n-1})$ and $b\in BMO$ which is established
relying upon techniques from arXiv:1705.07397.