$A_1$ theory of weights for rough homogeneous singular integrals and commutators

Abstract: Quantitative $A_1-A_\infty$ estimates for rough homogeneous singular integrals $T_{\Omega}$ and commutators of $BMO$ symbols and $T_{\Omega}$ are obtained. In particular the following estimates are proved: % [ |T_\Omega |{L^p(w)}\le c{n,p}|\Omega|{L^\infty} [w]{A_1}^{\frac{1}{p}}\,[w]{A{\infty}}^{1+\frac{1}{p'}}|f|_{L^p(w)} ] % and % [ | [b,T_{\Omega}]f|{L^{p}(w)}\leq c{n,p}|b|{BMO}|\Omega|{L^{\infty}} [w]{A_1}^{\frac{1}{p}}[w]{A_{\infty}}^{2+\frac{1}{p'}}|f|_{L^{p}\left(w\right)}, ] % for $1<p<\infty$ and $1/p+1/p'=1$.