Paper detail

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&#39;s A_\infty constant \[ [w]_{A_\infty}&#39;:=\sup_Q\frac{1}{w(Q)}\int_Q M(wχ_Q). \] In particular, we show the following improvement of the first author&#39;s A_2 theorem for Calderón-Zygmund operators T: \[\|T\|_{B(L^2(w))}\leq c_T [w]_{A_2}^{1/2}([w]_{A_\infty}&#39;+[w^{-1}]_{A_\infty}&#39;)^{1/2}. \] 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&#39;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: \[\|T\|_{B(L^p(w))} \leq c pp&#39;[w]_{A_1}^{1/p}([w]_{A_{\infty}}&#39;)^{1/p&#39;}, 1<p<\infty, \] \[\|Tf\|_{L^{1,\infty}(w)} \leq c[w]_{A_1} \log(e+[w]&#39;_{A_{\infty}}) \|f\|_{L^1(w)}. \] An estimate dual to the last one is also found, as well as new bounds for commutators of singular integrals.