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Proof of an extension of E. Sawyer’s conjecture about weighted mixed weak-type estimates

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Abstract

We show that if \(v\in A_\infty \) and \(u\in A_1\), then there is a constant c depending on the \(A_1\) constant of u and the \(A_{\infty }\) constant of v such that

$$\begin{aligned} \left\| \frac{ T(fv)}{v}\right\| _{L^{1,\infty }(uv)}\le c\, \Vert f\Vert _{L^1(uv)}, \end{aligned}$$

where T can be the Hardy–Littlewood maximal function or any Calderón–Zygmund operator. This result was conjectured in Cruz-Uribe et al. (Int Math Res Not 30:1849–1871, 2005) and constitutes the most singular case of some extensions of several problems proposed by Sawyer and Muckenhoupt and Wheeden. We also improve and extends several quantitative estimates.

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Authors and Affiliations

  1. BCAM, Basque Center for Applied Mathematics, Mazarredo, 14, 48009, Bilbao, Basque, Spain

    Kangwei Li

  2. Department of Mathematics, Universidad Nacional del Sur, Bahía Blanca, Argentina

    Sheldy Ombrosi

  3. Department of Mathematics, University of the Basque Country, Ikerbasque and BCAM, Bilbao, Spain

    Carlos Pérez

Authors
  1. Kangwei Li
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  2. Sheldy Ombrosi
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  3. Carlos Pérez
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Corresponding author

Correspondence to Carlos Pérez.

Additional information

Communicated by Loukas Grafakos.

K.L. and C.P. are supported by the Basque Government through the BERC 2014-2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323. , S.O. is supported by CONICET PIP 11220130100329CO, Argentina. C.P. is supported by Spanish Ministry of Economy and Competitiveness MINECO through the project MTM2014-53850-P.

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Li, K., Ombrosi, S. & Pérez, C. Proof of an extension of E. Sawyer’s conjecture about weighted mixed weak-type estimates. Math. Ann. 374, 907–929 (2019). https://doi.org/10.1007/s00208-018-1762-0

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