Sharp weighted bounds involving A_\infty

Abstract: We improve on several weighted inequalities of recent interest by replacing a part of the A_p bounds by weaker A_\infty estimates involving Wilson's A_\infty constant [ [w]{A\infty}':=\sup_Q\frac{1}{w(Q)}\int_Q M(w\chi_Q). ] In particular, we show the following improvement of the first author's A_2 theorem for Calder\'on-Zygmund operators T: [|T|{B(L^2(w))}\leq c_T [w]{A_2}^{1/2}([w]{A\infty}'+[w^{-1}]{A\infty}')^{1/2}. ] Corresponding A_p type results are obtained from a new extrapolation theorem with appropriate mixed A_p-A_\infty bounds. This uses new two-weight estimates for the maximal function, which improve on Buckley's classical bound. We also derive mixed A_1-A_\infty type results of Lerner, Ombrosi and the second author (Math. Res. Lett. 2009) of the form: [|T|{B(L^p(w))} \leq c pp'[w]{A_1}^{1/p}([w]{A{\infty}}')^{1/p'}, 1<p<\infty, ] [|Tf|{L^{1,\infty}(w)} \leq c[w]{A_1} \log(e+[w]'{A{\infty}}) |f|_{L^1(w)}. ] An estimate dual to the last one is also found, as well as new bounds for commutators of singular integrals.