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Logarithmic bump conditions and the two weight boundedness of Calder'on-Zygmund operators | | David Cruz-Uribe
; Alexander Reznikov
; Alexander Volberg
; | | Date: |
3 Dec 2011 | | Abstract: | We prove that if a pair of weights $(u,v)$ satisfies a sharp $A_p$-bump
condition in the scale of log 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 Calder’on-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. | | Source: | arXiv, 1112.0676 | | Services: | Forum | Review | PDF | Favorites | |
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