  We give a self-contained proof of the $A_2$ conjecture, which claims that the
norm 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 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 $A_2$ 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 $A_2$ conjecture, we also
provide a simple variant of the representation, already in the original proof,
of an arbitrary Calderon-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 nonhomogeneous Harmonic Analysis.

Classical Analysis and ODEs