Sharp weighted bounds involving A_\infty

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.