Sharp weighted estimates for dyadic shifts and the A2 conjecture

Abstract

We give a self-contained proof of the A2 conjecture, which claims that the norm of any Calderón–Zygmund operator is bounded by the first degree of the A2 norm of the weight. The original proof of this result by the first author relied on a subtle and rather difficult reduction to a testing condition by the last three authors. Here we replace this reduction by a new weighted norm bound for dyadic shifts – linear in the A2 norm of the weight and quadratic in the complexity of the shift –, which is based on a new quantitative two-weight inequality for the shifts. These sharp one- and two-weight bounds for dyadic shifts are the main new results of this paper. They are obtained by rethinking the corresponding previous results of Lacey-Petermichl-Reguera and Nazarov-Treil-Volberg. To complete the proof of the A2 conjecture, we also provide a simple variant of the representation, already in the original proof, of an arbitrary Calderón-Zygmund operator as an average of random dyadic shifts and random dyadic paraproducts. This method of the representation amounts to the refinement of the techniques from non-homogeneous Harmonic Analysis.