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arXiv · 1112.0676

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

David Cruz-Uribe·Alexander Reznikov·Alexander Volberg
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Abstract

We prove that if a pair of weights $(u,v)$ satisfies a sharp $A_p$-bump condition in the scale of log bumps and 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 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 $A_2$ 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.

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David Cruz-Uribe, Alexander Reznikov, Alexander Volberg. 2013-01-04. Logarithmic bump conditions and the two weight boundedness of Calderón-Zygmund operators. https://arxiv.org/abs/1112.0676

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