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

We prove that if a pair of weights (u, v) satisfies a sharp A(p)-bump condition in the scale of all log bumps or certain loglog bumps, then Haar shifts map L-p(v) into L-p(u) with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calderon-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 A(2) 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. (C) 2014 Elsevier Inc. All rights reserved.