Skip to main content
SpringerLink
Log in

On weighted inequalities for a class of quasilinear integral operators

  • Original Paper
  • Published:
Banach Journal of Mathematical Analysis Aims and scope Submit manuscript

Abstract

In this paper, we establish necessary and sufficient conditions under which certain weighted inequalities hold for a class of quasilinear integral operators with kernels.

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

Availability of data and materials

Not applicable.

References

  1. Bernardis, A.L., Ortega Salvador, P.: Some new iterated Hardy-type inequalities and applications. J. Math. Inequal. 11(2), 577–594 (2017)

    Article  MATH  Google Scholar 

  2. Bloom, S., Kerman, R.: Weighted norm inequalities for operators of Hardy type. Proc. Am. Math. Soc. 1(113), 135–141 (1991)

    Article  MATH  Google Scholar 

  3. Burenkov, V.I., Oinarov, R.: Necessary and sufficient conditions for boundedness of the Hardy-type operator from a weighted Lebesgue space to a Morrey-type space. Math. Inequal. Appl. 16(1), 1–19 (2013)

    MATH  Google Scholar 

  4. Gogatishvili, A., Mustafayev, R., Persson, L.-E.: Some new iterated Hardy-type inequalities. J. Funct. Spaces Appl. (2012). https://doi.org/10.1155/2012/734194

    Article  MATH  Google Scholar 

  5. Gogatishvili, A., Mustafayev, R., Persson, L.-E.: Some new iterated Hardy-type inequalities: the case \(\theta =1\). J. Inequal. Appl. (2013). https://doi.org/10.1186/1029-242X-2013-515

    Article  MATH  Google Scholar 

  6. Gogatishvili, A., Mustafayev, R.: Weighted iterated Hardy-type inequalities. Math. Inequal. Appl. 203(3), 683–728 (2017)

    MATH  Google Scholar 

  7. Gogatishvili, A., Mihula, Z., Pick, L., Turčinová, H., Ünver, T.: Weighted inequalities for a superposition of the Copson operator and the Hardy operator. J. Fourier Anal. Appl. 28, 24 (2022). https://doi.org/10.1007/s00041-022-09918-6

    Article  MATH  Google Scholar 

  8. Kalybay, A.: Weighted estimates for a class of quasilinear integral operators. Sib. Math. J. 60(2), 291–303 (2019)

    Article  MATH  Google Scholar 

  9. Kufner, A., Persson, L.-E.: Weighted Inequalities of Hardy Type. World Scientific, London (2003)

    Book  MATH  Google Scholar 

  10. Lai, Q.: Weighted modular inequalities for Hardy type operators. Proc. Lond. Math. Soc. 79(3), 649–672 (1999)

    Article  MATH  Google Scholar 

  11. Maz’ya, V.: Sobolev Spaces. [in Russian]. LGU, Leningrad (1984)

  12. Maz’ya, V.: Basic properties of Sobolev spaces. In: Sobolev Spaces. Grundlehren der Mathematischen Wissenschaften, vol. 342. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-15564-2_1

  13. Muckenhoupt, B.: Hardy’s inequalities with weights. Stud. Math. 34(1), 31–38 (1972)

    Article  MATH  Google Scholar 

  14. Oinarov, R.: Weighted inequalities for one class of integral operators. Dokl. Akad. Nauk SSSR 5(319), 1076–1078 (1991)

    Google Scholar 

  15. Prokhorov, D.V.: Hardy’s inequality with three measures. Proc. Steklov Inst. Math. 255, 221–233 (2006)

    Article  MATH  Google Scholar 

  16. Prokhorov, D.V.: On the boundedness of a class of sublinear operators. Dokl. Math. 92(2), 602–605 (2015)

    Article  MATH  Google Scholar 

  17. Prokhorov, D.V.: On a class of weighted inequalities containing quasilinear operators. Proc. Steklov Inst. Math. 293, 272–287 (2016)

    Article  MATH  Google Scholar 

  18. Prokhorov, D.V., Stepanov, V.D.: Weighted estimates for a class of sublinear operators. Dokl. Math. 88(3), 721–723 (2013)

    Article  MATH  Google Scholar 

  19. Prokhorov, D.V., Stepanov, V.D.: Weighted inequalities for quasilinear integral operators on the semi-axis and applications to Lorentz spaces. Sb. Math. 207(8), 1159–1186 (2016)

    Article  MATH  Google Scholar 

  20. Prokhorov, D.V., Stepanov, V.D., Ushakova, E.P.: Hardy–Steklov integral operators: part I. Proc. Steklov Inst. Math. 300, 1–112 (2018). https://doi.org/10.1134/S008154381803001X

    Article  MATH  Google Scholar 

  21. Royden, H.L.: Real Analysis, 3rd edn. Macmillan Publishing Company, New York (1988)

    MATH  Google Scholar 

  22. Stepanov, V.D., Shambilova, G.E.: On weighted iterated Hardy-type operators. Anal. Math. 44(2), 273–283 (2018)

    Article  MATH  Google Scholar 

  23. Stepanov, V.D., Shambilova, G.E.: On iterated and bilinear integral Hardy-type operators. Math. Inequal. Appl. 22(4), 1505–1533 (2019)

    MATH  Google Scholar 

  24. Stepanov, V.D., Ushakova, E.P.: Hardy operator with variable limits on monotone functions. J. Funct. Spaces Appl. 1(1), 1–15 (2003)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the unknown referees for their generous suggestions and remarks, which have improved this paper. This work was completed with the support of the Ministry of Education and Science of the Republic of Kazakhstan, Grant no. AP09259084.

Author information

Authors and Affiliations

  1. KIMEP University, Abay Ave. 4, 050010, Almaty, Kazakhstan

    Aigerim Kalybay

  2. L. N. Gumilyov Eurasian National University, Satpaev Str. 2, 101008, Astana, Kazakhstan

    Ryskul Oinarov

Authors
  1. Aigerim Kalybay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ryskul Oinarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors contributed equally to this work.

Corresponding author

Correspondence to Aigerim Kalybay.

Ethics declarations

Conflict of interest

The authors have no competing interests to declare that are relevant to the content of this paper.

Additional information

Communicated by Sergey Astashkin.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kalybay, A., Oinarov, R. On weighted inequalities for a class of quasilinear integral operators. Banach J. Math. Anal. 17, 3 (2023). https://doi.org/10.1007/s43037-022-00226-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s43037-022-00226-1

Keywords

Mathematics Subject Classification

Search

Navigation