Weighted and Unweighted Solyanik Estimates for the Multilinear Strong Maximal Function


Let \(\displaystyle \omega \) be a weight in \(A^*_\infty \) and let \(\displaystyle \mathcal {M}^m_n(\mathbf {f})\) be the multilinear strong maximal function of \(\displaystyle \mathbf {f}=\left( f_1,\ldots ,f_m\right) \), where \(f_1,\ldots ,f_m\) are functions on \(\mathbb R^n\). In this paper, we consider the asymptotic estimates for the distribution functions of \(\displaystyle \mathcal {M}^m_n\). We show that, for \(\displaystyle \lambda \in (0,1)\), if \(\displaystyle \lambda \rightarrow 1^-\), then the multilinear Tauberian constant \(\displaystyle \mathcal C^m_n\) and the weighted Tauberian constant \(\displaystyle \mathcal C^m_{n,\omega }\) associated with \(\displaystyle \mathcal M^m_n\) enjoy the properties that

$$\begin{aligned} \displaystyle \mathcal C^m_n(\lambda )-1\simeq m\left( 1-\lambda \right) ^{\frac{1}{n}}\quad \hbox {and }\quad \mathcal C^m_{n,\omega }(\lambda )-1\lesssim m(1-\lambda )^{\left( 4n[\omega ]_{A^*_\infty }\right) ^{-1}}. \end{aligned}$$

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

Fig. 1
Fig. 2


  1. 1.

    Bagby, R.J.: Maximal functions and rearrangements: some new proofs. Indiana Univ. Math. J. 32(6), 879–891 (1983)

    MathSciNet  Article  Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 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)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Cao, M., Xue, Q., Yabuta, K.: On multilinear fractional strong maximal operator associated with rectangles and multiple weights. Rev. Mat. Iberoam. 33(2), 555–572 (2017)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Córdoba, A., Fefferman, R.: A geometric proof of the strong maximal theorem. Ann. Math. (2) 102(1), 95–100 (1975)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Córdoba, A., Fefferman, R.: On differentiation of integrals. Proc. Nat. Acad. Sci. USA 74(6), 2211–2213 (1977)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Fava, N.A., Gatto, E.A., Gutiérrez, C.: On the strong maximal function and Zygmund’s class \(L(\log ^+L)\). Stud. Math. 69, 155–158 (1980)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Fefferman, R.: Strong differentiation with respect to measures. Am. J. Math. 103(1), 33–40 (1981)

    MathSciNet  Article  Google Scholar 

  9. 9.

    García-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. In: North-Holland Mathematics Studies 116, Notas de Matemática 104. North-Holland Publishing Co., Amsterdam (1985)

  10. 10.

    Grafakos, L., Liu, L., Pérez, C., Torres, R.H.: The multilinear strong maximal function. J. Geom. Anal. 21, 118–149 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Hagelstein, P., Parissis, I.: Solyanik Estimates in Harmonic Analysis, Springer Proc. Math. Stat., vol. 108, pp. 87–103. Springer, Cham (2014)

  12. 12.

    Hagelstein, P., Parissis, I.: Solyanik estimates and local Hölder continuity of halo functions of geometric maximal operators. Adv. Math. 285, 434–453 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Hagelstein, P., Parissis, I.: Solyanik estimates in ergodic theory. Colloq. Math. 145(2), 193–207 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Hagelstein, P., 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)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Hagelstein, P., Parissis, I.: Hölder continuity of Tauberian constants associated with discrete and ergodic strong maximal operators. N. Y. J. Math. 23, 1219–1236 (2017)

    MATH  Google Scholar 

  16. 16.

    Hagelstein, P., Parissis, I.: Weighted Solyanik estimates for the strong maximal function. Publ. Mat. 62, 133–159 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    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)

    MathSciNet  Article  Google Scholar 

  18. 18.

    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 

  19. 19.

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

    MathSciNet  Article  Google Scholar 

  20. 20.

    Jessen, B., Marcinkiewicz, J., Zygmund, A.: Note on the differentiability of multiple integrals. Fund. Math. 25, 217–234 (1935)

    Article  Google Scholar 

  21. 21.

    Liu, F., Torres, R.H., Xue, Q., Yabuta, K.: The multilinear strong maximal operators on mixed Lebesgue spaces. Preprint

  22. 22.

    Solyanik, A.A.: Bordering functions for differentiation bases, (Russian). Mat. Zametki 54(6) (1993), 82-89, 160. [translation in Math. Notes 54(5-6) (1993), 1241-1245 (1994)]

  23. 23.

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

    Article  Google Scholar 

  24. 24.

    Zhang, J., Xue, Q.: Multilinear strong maximal operators on mixed norm spaces. Publ. Math. Debrecen. 96(3–4), 347–361 (2020)

    MathSciNet  Article  Google Scholar 

Download references


The authors want to express their sincere thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable.

Author information



Corresponding author

Correspondence to Qingying Xue.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Q. Xue was supported partly by NSFC (Nos. 11671039, 11871101) and NSFC-DFG (No. 11761131002).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Qin, M., Xue, Q. Weighted and Unweighted Solyanik Estimates for the Multilinear Strong Maximal Function. Results Math 76, 6 (2021). https://doi.org/10.1007/s00025-020-01317-x

Download citation


  • Solyanik estimates
  • Tauberian constant
  • multilinear strong maximal function

Mathematics Subject Classification

  • Primary 42B20
  • Secondary 42B35