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 $$[w{]}_{A_{\mathrm{\infty }}}^{\mathrm{\prime }}:=\underset{Q}{\sup }\frac{1}{w(Q)}{\int }_{Q}M(w{\chi }_Q)\mathrm{.}$$ In particular, we show the following improvement of the first author’s A_2 theorem for Calder'on-Zygmund operators T: $$\mathrm{\parallel }T{\mathrm{\parallel }}_{B(L^2(w))}\le c_T[w{]}_{A_2}^{1\mathrm{/}2}([w{]}_{A_{\mathrm{\infty }}}^{\mathrm{\prime }}+[w^{-1}{]}_{A_{\mathrm{\infty }}}^{\mathrm{\prime }}{)}^{1\mathrm{/}2}\mathrm{.}$$ 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: $$\mathrm{\parallel }T{\mathrm{\parallel }}_{B(L^p(w))}\le cp{p}^{\mathrm{\prime }}[w{]}_{A_1}^{1\mathrm{/}p}([w{]}_{A_{\mathrm{\infty }}}^{\mathrm{\prime }}{)}^{1\mathrm{/}{p}^{\mathrm{\prime }}},1<p<\mathrm{\infty },$$ $$\mathrm{\parallel }Tf{\mathrm{\parallel }}_{L^{1,\mathrm{\infty }}(w)}\le c[w{]}_{A_1}\log (e+[w{]}_{A_{\mathrm{\infty }}}^{\mathrm{\prime }})\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1(w)}\mathrm{.}$$ An estimate dual to the last one is also found, as well as new bounds for commutators of singular integrals.