Mathematische Annalen

, Volume 374, Issue 1–2, pp 907–929 | Cite as

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 


  1. 1.
    Andersen, K., Muckenhoupt, B.: Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions. Stud. Math. 72(1), 9–26 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Buckley, S.M.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Am. Math. Soc. 340, 253–272 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Coifman, R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Stud. Math. 51, 241–250 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103, 611–635 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cruz-Uribe, D., Pérez, C.: On the two-weight problem for singular integral operators. Ann. della Scuola Normale-Classe di Scienze I(5), 821–849 (2002)Google Scholar
  6. 6.
    Cruz-Uribe, D., Martell, J.M., Pérez, C.: Extrapolation results for \(A_\infty \) weights and applications. J. Funct. Anal. 213, 412–439 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cruz-Uribe, D., Martell, J., Pérez, C.: Weighted weak type inequalities and a conjecture of Sawyer. Int. Math. Res. Not. 30, 1849–1871 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cruz-Uribe, D., Martell, José, M., Pérez, C.: Weights, extrapolation and the theory of Rubio de Francia. Operator Theory: Advances and Applications, vol. 215. Birkhäuer/Springer Basel AG, Basel (2011)Google Scholar
  9. 9.
    Curbera, G.P., García-Cuerva, J., Martell, J.M., Pérez, C.: Extrapolation with weights, rearrangement invariant function spaces, modular inequalities and applications to singular integrals. Adv. Math. 203, 256–318 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duoandikoetxea, J.: Fourier Analysis, Grad. Stud. Math. vol. 29. American Math. Soc., Providence, RI (2000)Google Scholar
  11. 11.
    Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93, 107–115 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fefferman, C., Stein, E.M.: \(H^{p}\) spaces of several variables. Acta Math. 129, 137–193 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hagelstein, P.A., Parissis, I.: Weighted Solyanik estimates for the Hardy-littlewood maximal operator and embedding of \(A_\infty \) into \(A_p\). J. Geom. Anal. 26(2), 924–946 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hytönen, T.P., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE 6, 777–818 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hytönen, T.P., Pérez, C.: The \(L(\log L)^\varepsilon \) endpoint estimate for maximal singular integral operators. J. Math. Anal. Appl. 428(1), 605–626 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jones, P.: Factorization of \(A_p\) weights. Ann. Math. 111(3), 511–530 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory. Adv. Math. 220, 1222–1264 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lerner, AK., Ombrosi, S., Pérez, C.: Sharp \(A_1\) bounds for Calderón–Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden. Int. Math. Res. Not. (6), 11 (2008) (Art. ID rnm161)Google Scholar
  19. 19.
    Lerner, A.K., Ombrosi, S., Pérez, C.: Weak type estimates for singular integrals related to a dual problem of Muckenhoupt–Wheeden. J. Fourier Anal. Appl. 15, 394–403 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lerner, A.K., Ombrosi, S., Pérez, C.: \(A_{1}\) bounds for Calderón–Zygmund operators related to a problem of Muckenhoupt and Wheeden. Math. Res. Lett 16, 149–156 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Li, K., Ombrosi, S., Picardi, M.B.: Mixed weighted weak-type inequalities for multilinear operators. Stud. Math. 244, 203–215 (2019)CrossRefzbMATHGoogle Scholar
  22. 22.
    Li, K., Pérez, C., Rivera-Ríos, I.P., Roncal, L.: Weighted norm inequalities for rough singular integral operators. J. Geom. Anal (2018).
  23. 23.
    Martín-Reyes, F.J., Salvador, P.O., Gavilán, M., Sarrión, D.: Boundedness of operators of Hardy type in \(\Lambda ^{p,q}\) spaces and weighted mixed inequalities for singular integral operators. Proc. R. Soc. Edinb. Sect. A 127(1), 157–170 (1997)Google Scholar
  24. 24.
    Martín-Reyes, F.J.: Mixed weak type inequalities for one-sided operators and ergodic theorems. Rev. de la Unión Mat. Argent. 50(2), 51–61 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Muckenhoupt, B., Wheeden, R.: Some weighted weak-type inequalities for the Hardy-Littlewood maximal function and the Hilbert transform. Indiana Math. J. 26, 801–816 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ombrosi, S., Pérez, C.: Mixed weak type estimates: examples and counterexamples related to a problem of E. Sawyer. Colloq. Math. 145, 259–272 (2016)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ombrosi, S., Pérez, C., Recchi, J.: Quantitative weighted mixed weak-type inequalities for classical operators. Indiana Univ. Math. J. 65, 615–640 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pérez, C.: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128, 163–185 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Pérez, C., Rivera-Ríos, I.P.: Three Observations on commutators of singular integral operators with BMO functions. In: Pereyra, M., Marcantognini, S., Stokolos, A., Urbina, W. (eds.) Harmonic analysis, partial differential equations, banach spaces, and operator theory. Association for Women in Mathematics Series, vol 5. Springer, Cham (2017)Google Scholar
  31. 31.
    Sawyer, E.T.: A weighted weak type inequality for the maximal function. Proc. Am. Math. Soc. 93, 610–614 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar

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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