SHARP WEIGHTED BOUNDS INVOLVING A∞

We improve on several weighted inequalities of recent interest by replacing a part of the A(p) bounds by weaker A 1 estimates involving Wilson's A 1 constant [w](A infinity)' := sup(Q) 1/w(Q) integral(M)(Q) (w chi(Q)) In particular, we show the following improvement of the first author's A(2) theorem for Calderon-Zygmund ;operators T: parallel to T parallel to(B)(L-2(w)) <= c(T) [w](A2)(1/2) ([w](A infinity)' + [w(-1)](A infinity)')(1/2). Corresponding A(p) type results are obtained from a new extrapolation theorem with appropriate mixed A(p)-A 1 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(infinity) type results of Lerner, Ombrosi and Perez (2009) of the form parallel to T parallel to(B)(L-P(w)) <= cpp' [w](A1)(1/)p ([w](A infinity)')(1/p'), 1 < p < infinity, parallel to Tf parallel to(L1,infinity) <= c[w](A1) log (e + [w](A infinity)')parallel to f parallel to(L1)(w). An estimate dual to the last one is also found, as well as new bounds for commutators of singular integrals.