Skip to main content
SpringerLink
Log in

Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

Let denote a weight in which belongs to the Muckenhoupt class and let denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure . The sharp Tauberian constant of with respect to , denoted by , is defined by

In this paper, we show that the Solyanik estimate

$$\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}$$

holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator and a weight :

We show that we have if and only if . As a corollary of our methods we obtain a quantitative embedding of into .

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. Beznosova, O.V., Hagelstein, P.A.: Continuity of halo functions associated to homothecy invariant density bases. Colloquium Mathematicum 134(2), 235–243 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beznosova, O., Reznikov, A.: Sharp estimates involving \(A_\infty \) and \(L\log L\) constants, and their applications to PDE. Algebra i Anal. 26(1), 40–67 (2014). MR3234812

    MathSciNet  MATH  Google Scholar 

  3. Cabrelli, C., Lacey, M.T., Molter, U., Pipher, J.C.: Variations on the theme of Journé’s lemma. Houston J. Math. 32(3), 833–861 (2006). MR2247912 (2007e:42011)

    MathSciNet  MATH  Google Scholar 

  4. Córdoba, A., Fefferman, R.: A geometric proof of the strong maximal theorem. Ann. Math. 102(1), 95–100 (1975). MR0379785 (52 #690)

    Article  MathSciNet  MATH  Google Scholar 

  5. Córdoba, A., Fefferman, R.: On the equivalence between the boundedness of maximal and multiplier operators in Fourier analysis. Proc. Natl. Acad. Sci. USA 74(2), 423–425 (1977). MR0433117

    Article  MathSciNet  MATH  Google Scholar 

  6. Dindoš, M., Wall, T.: The sharp \(A_p\) constant for weights in a reverse-Hölder class. Rev. Mat. Iberoam. 25(2), 559–594 (2009). MR2569547 (2011b:42041)

    Article  MathSciNet  MATH  Google Scholar 

  7. Duoandikoetxea, J., Martín-Reyes, F.J., Ombrosi, S.: Calderón weights as Muckenhoupt weights. Indiana Univ. Math. J. 62(3), 891–910 (2013). MR3164849

    Article  MathSciNet  MATH  Google Scholar 

  8. Fujii, N.: Weighted bounded mean oscillation and singular integrals. Math. Japon. 22(5), 529–534 (1977/78). MR0481968 (58 #2058)

  9. Füredi, Z., Loeb, P.A.: On the best constant for the Besicovitch covering theorem. Proc. Am. Math. Soc. 121(4), 1063–1073 (1994). MR1249875 (95b:28003)

    Article  MathSciNet  MATH  Google Scholar 

  10. García-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. North-Holland Mathematics Studiesm, vol. 116. North-Holland Publishing Co., Amsterdam (1985). Notas de Matemática [Mathematical Notes], vol. 104 (1985) MR807149 (87d:42023)

  11. Garnett, J.B.: Bounded analytic functions. Graduate Texts in Mathematics, 1st edn. Springer, New York (2007). MR2261424 (2007e:30049)

    Google Scholar 

  12. de Guzmán, M.: Differentiation of integrals in \({\bf R}^{n}\). Measure Theory (Proc. Conf., Oberwolfach, 1975), Lecture Notes in Math., vol. 541, pp. 181–185. Springer, Berlin (1976). MR0476978 (57 #16523)

  13. Hagelstein, P. A., Luque, T., Parissis, I.: Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases. Trans. Am. Math. Soc. (To appear). arXiv:1304.1015

  14. Hagelstein, P.A., Parissis, I.: Solyanik estimates in harmonic analysis. In: Special Functions, Partial Differential Equations, and Harmonic Analysis. Springer Proceedings in Mathematics & Statistics, vol. 108, pp. 87–103. Springer, Heidelberg (2014)

  15. Hagelstein, P.A., Stokolos, A.: Tauberian conditions for geometric maximal operators. Trans. Am. Math. Soc. 361(6), 3031–3040 (2009). MR2485416 (2010b:42023)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hruščev, S.V.: A description of weights satisfying the \(A_{\infty }\) condition of Muckenhoupt. Proc. Am. Math. Soc. 90(2), 253–257 (1984). MR727244 (85k:42049)

    Google Scholar 

  17. Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE 6(4), 777–818 (2013). MR3092729

    Article  MathSciNet  MATH  Google Scholar 

  18. 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). MR2990061

    Article  MathSciNet  MATH  Google Scholar 

  19. Korey, M.B.: Ideal weights: asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation. J. Fourier Anal. Appl. 4(4–5), 491–519 (1998). MR1658636 (99m:42032)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lerner, A.K., Moen, K.: Mixed \(A_p\)-\(A_\infty \) estimates with one supremum. Studia Math. 219(3), 247–267 (2013). MR3145553

    Article  MathSciNet  MATH  Google Scholar 

  21. Mitsis, T.: Embedding \(B_\infty \) into Muckenhoupt classes. Proc. Am. Math. Soc. 133(4), 1057–1061 (2005). (electronic) MR2117206 (2005i:42031)

    Article  MathSciNet  MATH  Google Scholar 

  22. Politis, A.: Sharp results on the relation between weight spaces and BMO. ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.). The University of Chicago 1995. MR2716561

  23. Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973). MR0365062

    MATH  Google Scholar 

  24. Solyanik, A.A.: On halo functions for differentiation bases. Mat. Zametki, 54(6), 82–89, 160 (1993). (Russian, with Russian summary): English transl., Math. Notes, 54 (1993) (5–6), 1241–1245 (1994) MR1268374 (95g:42033)

  25. Wik, I.: On Muckenhoupt’s classes of weight functions. Studia Math. 94(3), 245–255 (1989). MR1019792 (90j:42029)

    MathSciNet  MATH  Google Scholar 

  26. Wilson, J.: Michael weighted inequalities for the dyadic square function without dyadic \(A_\infty \). Duke Math. J. 55(1), 19–50 (1987). MR883661 (88d:42034)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wilson, M.: Weighted Littlewood–Paley Theory and Exponential-Square Integrability. Lecture Notes in Mathematics, vol. 1924. Springer, Berlin (2008). MR2359017 (2008m:42034)

    MATH  Google Scholar 

Download references

Acknowledgments

Part of this work was carried out while the authors were visiting the Department of Mathematical Analysis at the University of Seville. We are indebted to Teresa Luque and Carlos Pérez for their warm hospitality. We wish to thank Teresa Luque and Alex Stokolos for pointing out some important references related to the subject of this paper. Finally, we would like to thank the referee for an expert reading that resulted in many improvements throughout the paper. P. H. is partially supported by a Grant from the Simons Foundation (#208831 to Paul Hagelstein). I. P. is supported by the Academy of Finland, Grant 138738.

Author information

Authors and Affiliations

  1. Department of Mathematics, Baylor University, Waco, TX, 76798, USA

    Paul Hagelstein

  2. Department of Mathematics, Aalto University, P. O. Box 11100, 00076, Espoo, Finland

    Ioannis Parissis

Authors
  1. Paul Hagelstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ioannis Parissis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul Hagelstein.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hagelstein, P., Parissis, I. Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into . J Geom Anal 26, 924–946 (2016). https://doi.org/10.1007/s12220-015-9578-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-015-9578-6

Keywords

Mathematics Subject Classification

Search

Navigation