Improved A1 − A∞ and related estimates for commutators of rough singular integrals

Abstract

An $A_1-A_\infty$ estimate improving a previous result in [22] for $[b, T_\Omega]$ with $\Omega\in L^\infty(S^{n-1})$ and $b\in BMO$ is obtained. Also a new result in terms of the $A_\infty$ constant and the one supremum $A-A_\infty^{exp}$ constant is proved, providing a counterpart for commutators of the result obained in [19]. Both of the preceding results rely upon a sparse domination result in terms of bilinear forms which is established using techniques from [13].