Skip to main content
SpringerLink
Log in

On the \(A_{\infty }\) conditions for general bases

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We discuss several characterizations of the \(A_\infty \) class of weights in the setting of general bases. Although they are equivalent for the usual Muckenhoupt weights, we show that they can give rise to different classes of weights for other bases. We also obtain new characterizations for the usual \(A_\infty \) weights.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bateman, M.: Kakeya sets and directional maximal operators in the plane. Duke Math. J. 147(1), 55–77 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bekollé, D.: Inégalité à poids pour le projecteur de Bergman dans la boule unité de \({\bf C}^{n}\). Studia Math. 71(3), 305–323 (1981/82)

  3. Bekollé, D., Bonami, A.: Inégalités à poids pour le noyau de Bergman. C. R. Acad. Sci. Paris Sér. A-B 286(18), A775–A778 (1978)

    MATH  Google Scholar 

  4. Beznosova, O., Reznikov, A.: Sharp estimates involving \(A_\infty \) constants, and their applications to PDE. Algebra i Analiz, 26(1), 40–67 (2014). English transl., St. Petersburg Math. J. 26(1), 27–47 (2015)

  5. Beznosova, O., Reznikov, A.: Equivalent definitions of dyadic Muckenhoupt and reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadic \(L\log L\) and \(A_\infty \) constants. Rev. Mat. Iberoam. 30(4), 1191–1236 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coifman, R.R.: Distribution function inequalities for singular integrals. Proc. Nat. Acad. Sci. U.S.A. 69, 2838–2839 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  7. Coifman, R.R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51, 241–250 (1974)

    MathSciNet  MATH  Google Scholar 

  8. Cruz-Uribe, D.V., Martell, J.M., Pérez, C.: Weights, Extrapolation and the Theory of Rubio de Francia. Operator Theory: Advances and Applications, vol. 215. Birkhäuser/Springer Basel AG, Basel (2011)

    Book  MATH  Google Scholar 

  9. Duoandikoetxea, J.: Fourier Analysis. Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  10. Duoandikoetxea, J., Martín-Reyes, F.J., Ombrosi, S.: Calderón weights as Muckenhoupt weights. Indiana Univ. Math. J. 62(3), 891–910 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dyn’kin, E.M., Osilenker, B.P.: Weighted estimates for singular integrals and their applications. In: Mathematical analysis, vol. 21, Itogi Nauki i Tekhniki, pp. 42–129. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1983) (Russian)

  12. Fujii, N.: Weighted bounded mean oscillation and singular integrals. Math. Jpn. 22(5), 529–534 (1977/78)

  13. García-Cuerva, J., Rubio de Francia, J.: Weighted norm inequalities and related topics. North-Holland Mathematics Studies, vol. 116. North-Holland Publishing Co., Amsterdam (1985)

    Book  MATH  Google Scholar 

  14. Garnett, J.B.: Bounded Analytic Functions. Pure and Applied Mathematics, vol. 96. Academic Press Inc., New York (1981)

    MATH  Google Scholar 

  15. Gehring, F.W.: The \(L^{p}\)-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gogatishvili, A., Pick, L.: Weak and extra-weak type inequalities for the maximal operator and the Hilbert transform. Czechoslov. Math. J. 43(3), 547–566 (1993)

    MathSciNet  MATH  Google Scholar 

  17. Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics, vol. 250, 2nd edn. Springer, New York (2009)

    Book  MATH  Google Scholar 

  18. Gundy, R.F., Wheeden, R.L.: Weighted integral inequalities for the nontangential maximal function, Lusin area integral, and Walsh-Paley series. Studia Math. 49, 107–124 (1973/74)

  19. Hagelstein, P., Luque, T., Parissis, I.: Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases. Trans. Am. Math. Soc. 367(11), 7999–8032 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hruščev, S.V.: A description of weights satisfying the \(A_{\infty }\) condition of Muckenhoupt. Proc. Am. Math. Soc. 90(2), 253–257 (1984)

    Google Scholar 

  21. Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE 6(4), 777–818 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hytönen, T., Pérez, C., Rela, E.: Sharp Reverse Hölder property for \(A_\infty \) weights on spaces of homogeneous type. J. Funct. Anal. 263(12), 3883–3899 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kerman, R.A., Torchinsky, A.: Integral inequalities with weights for the Hardy maximal function. Studia Math. 71(3), 277–284 (1981/82)

  24. Lanzani, L., Stein, E.M.: Szegö and Bergman projections on non-smooth planar domains. J. Geom. Anal. 14(1), 63–86 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Muckenhoupt, B.: The equivalence of two conditions for weight functions. Studia Math. 49 101–106 (1973/74)

  26. Orobitg, J., Pérez, C.: \(A_p\) weights for nondoubling measures in \({{\bf R}}^n\) and applications. Trans. Am. Math. Soc. 354(5), 2013–2033 (2002). (electronic)

    Article  MATH  Google Scholar 

  27. Pérez, C.: Some topics from Calderón-Zygmund theory related to Poincaré-Sobolev inequalities, fractional integrals and singular integral operators. In: Function spaces, Nonlinear Analysis and Applications (Spring Lectures in Analysis, Paseky nad Jizerou, 1999), pp. 31–94. Charles University and Academy of Sciences, Prague (1999)

  28. Recchi, J.: Mixed \(A_1-A_\infty \) bounds for fractional integrals. J. Math. Anal. Appl. 403(1), 283–296 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Stein, E.M.: Note on the class \(L\) log \(L\). Studia Math. 32, 305–310 (1969)

    MathSciNet  MATH  Google Scholar 

  30. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  31. Strömberg, J.-O., Torchinsky, A.: Weighted Hardy Spaces. Lecture Notes in Mathematics, vol. 1381. Springer, Berlin (1989)

    MATH  Google Scholar 

  32. Strömberg, J.-O., Wheeden, R.L.: Fractional integrals on weighted \(H^p\) and \(L^p\) spaces. Trans. Am. Math. Soc. 287(1), 293–321 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  33. Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Pure and Applied Mathematics, vol. 123. Academic Press Inc., Orlando (1986)

    MATH  Google Scholar 

  34. Wilson, J.M.: Weighted inequalities for the dyadic square function without dyadic \(A_\infty \). Duke Math. J. 55(1), 19–50 (1987). doi:10.1215/S0012-7094-87-05502-5

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We would like to thank David Cruz-Uribe for calling to our attention reference [31] and the characterizations with the medians appearing in it. We thank also Amiran Gogatishvili for pointing out his paper [16] and Kabe Moen for indicating the recent characterization of [19].

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), 48080, Bilbao, Spain

    Javier Duoandikoetxea

  2. Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071, Málaga, Spain

    Francisco J. Martín-Reyes

  3. Departamento de Matemática, Universidad Nacional del Sur, 8000, Bahía Blanca, Argentina

    Sheldy Ombrosi

Authors
  1. Javier Duoandikoetxea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francisco J. Martín-Reyes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sheldy Ombrosi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Javier Duoandikoetxea.

Additional information

The first author is supported by Grant MTM2011-24054 of the Ministerio de Economía y Competitividad (Spain) and Grant IT-641-13 of the Basque Gouvernment. The second author is supported by Grant MTM2011-28149-C02-02 of the Ministerio de Economía y Competitividad (Spain) and Grants FQM-354 and FQM-01509 of the Junta de Andalucía. The third author is supported by Grant PICT-2008-356 (ANPCyT, Argentina).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duoandikoetxea, J., Martín-Reyes, F.J. & Ombrosi, S. On the \(A_{\infty }\) conditions for general bases. Math. Z. 282, 955–972 (2016). https://doi.org/10.1007/s00209-015-1572-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-015-1572-y

Keywords

Mathematics Subject Classification

Search

Navigation