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We give a self-contained proof of the $A_2$ conjecture, which claims that thenorm of any Calderon-Zygmund operator is bounded by the first degree of the$A_2$ norm of the weight. The original proof of this result by the first authorrelied on a subtle and rather difficult reduction to a testing condition by thelast three authors. Here we replace this reduction by a new weighted norm boundfor dyadic shifts - linear in the $A_2$ norm of the weight and quadratic in thecomplexity of the shift -, which is based on a new quantitative two-weightinequality for the shifts. These sharp one- and two-weight bounds for dyadicshifts are the main new results of this paper. They are obtained by rethinkingthe corresponding previous results of Lacey-Petermichl-Reguera andNazarov-Treil-Volberg. To complete the proof of the $A_2$ conjecture, we alsoprovide a simple variant of the representation, already in the original proof,of an arbitrary Calderon-Zygmund operator as an average of random dyadic shiftsand random dyadic paraproducts. This method of the representation amounts tothe refinement of the techniques from nonhomogeneous Harmonic Analysis.

Sharp weighted estimates for dyadic shifts and the $A_2$ conjecture