Logarithmic bump conditions and the two-weight boundedness of Calderón–Zygmund operators

We prove that if a pair of weights (u,v)(u,v) satisfies a sharp ApAp-bump condition in the scale of all log bumps or certain loglog bumps, then Haar shifts map Lp(v)Lp(v) into Lp(u)Lp(u) with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calderón–Zygmund operators. This gives a partial answer to a long-standing conjecture. We also give a partial result for a related conjecture for weak-type inequalities. To prove our main results we combine several different approaches to these problems; in particular we use many of the ideas developed to prove the A2A2 conjecture. As a byproduct of our work we also disprove a conjecture by Muckenhoupt and Wheeden on weak-type inequalities for the Hilbert transform. This is closely related to the recent counterexamples of Reguera, Scurry and Thiele.