Skip to main content

On an Equivalency of Rare Differentiation Bases of Rectangles

Abstract

The paper considers differentiation properties of density bases formed of bounded open sets.We prove that two quasi-equivalent subbases of some density basis differentiate the same class of non-negative functions. Applications for bases formed of rectangles are discussed.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    M. Guzman, Differentiation of integrals in Rn (Springer, Berlin, 1975).

    Book  Google Scholar 

  2. 2.

    B. Jessen, J. Marcinkiewicz, A. Zygmund, “Note of differentiability of multiple integrals”, Fund. Math., 25, 217–237, 1935.

    Article  MATH  Google Scholar 

  3. 3.

    G. A. Karagulyan, “On equivalency of martingales and related problems”, Journal of Contemporary Mathematical Analysis, 48 (2), 51–65, 2013.

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    G. A. Karagulyan, D. A. Karagulyan, M. H. Safaryan, “On an equivalence for differentiation bases of dyadic rectangles”, Colloq. Math., 3506, 295–307, 2017.

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    G. Oniani, T. Zerekidze, “On differential bases formed of intervals”, GeorgianMath. J., 4 (1), 81–100, 1997.

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    S. Saks, “Remark on the differentiability of the Lebesgue indefinite integral”, Fund. Math., 22, 257–261, 1934.

    Article  MATH  Google Scholar 

  7. 7.

    K. Hare, A. Stokolos, “On weak type inequalities for rare maximal functions”, Colloq. Math., 83 (2), 173–182, 2000.

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    A. Stokolos, “On weak type inequalities for rare maximal function in Rn”, Colloq. Math., 104 (2), 311–315, 2006.

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    P. A. Hagelstein, “A note on rare maximal functions”, Colloq. Math.. 95 (1), 49–51, 2003.

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    T. Sh. Zerekidze, “Convergence of multiple Fourier-Haar series and strong differentiability of integrals”, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 76, 80–99, 1985.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    T. Sh. Zerekidze, “On some subbases of a strong differential basis”, Semin. I. Vekua Inst. Appl. Math. Rep., 35, 31–33, 2009.

    MathSciNet  Google Scholar 

  12. 12.

    T. Sh. Zerekidze, “On the equivalence and nonequivalence of some differential bases”, Proc. A. Razmadze Math. Inst., 133, 166–169, 2003.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. H. Safaryan.

Additional information

Original Russian Text © M. H. Safaryan, 2018, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2018, No. 1, pp. 68-73.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Safaryan, M.H. On an Equivalency of Rare Differentiation Bases of Rectangles. J. Contemp. Mathemat. Anal. 53, 56–60 (2018). https://doi.org/10.3103/S1068362318010090

Download citation

Keywords

  • Dyadic rectangles
  • differentiation basis
  • rare basis