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

  • Kangwei Li
  • Sheldy Ombrosi
  • Carlos PérezEmail author


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.

Mathematics Subject Classification

Primary 42B25 Secondary 42B20 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.BCAM, Basque Center for Applied MathematicsBilbaoSpain
  2. 2.Department of MathematicsUniversidad Nacional del SurBahía BlancaArgentina
  3. 3.Department of MathematicsUniversity of the Basque Country, Ikerbasque and BCAMBilbaoSpain

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