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A Fractional Muckenhoupt–Wheeden Theorem and its Consequences

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Abstract

In the 1970s Muckenhoupt and Wheeden made several conjectures relating two weight norm inequalities for the Hardy-Littlewood maximal operator to such inequalities for singular integrals. Using techniques developed for the recent proof of the A 2 conjecture we prove a related pair of conjectures linking the Riesz potential and the fractional maximal operator. As a consequence we are able to prove a number of sharp one and two weight norm inequalities for the Riesz potential.

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Correspondence to Kabe Moen.

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D. Cruz-Uribe was supported by the Stewart-Dorwart faculty development fund at Trinity College and by grant MTM2009-08934 from the Spanish Ministry of Science and Innovation. K. Moen was supported by NSF Grant 1201504.

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Cruz-Uribe, D., SFO. & Moen, K. A Fractional Muckenhoupt–Wheeden Theorem and its Consequences. Integr. Equ. Oper. Theory 76, 421–446 (2013). https://doi.org/10.1007/s00020-013-2059-z

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