Skip to main content
SpringerLink
Log in

Boundedness of quasilinear integral operators on the cone of monotone functions

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We study the problem of characterizing weighted inequalities on Lebesgue cones of monotone functions on the half-axis for one class of quasilinear integral operators.

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. Oinarov R., “Two-sided norm estimates for certain classes of integral operators,” Proc. Steklov Inst. Math., 204, 205–214 (1994).

    MathSciNet  MATH  Google Scholar 

  2. Sawyer E., “Boundedness of classical operators on classical Lorentz spaces,” Stud. Math., 96, 145–158 (1990).

    MathSciNet  MATH  Google Scholar 

  3. Stepanov V. D., “Boundedness of linear integral operators on a class of monotone functions,” Sib. Math. J., 32, No. 3, 540–542 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  4. Stepanov V. D., “Integral operators on the cone of monotone functions,” J. London Math. Soc., 48, No. 3, 465–487 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  5. Carro M. and Soria J., “Weighted Lorentz spaces and the Hardy operator,” J. Funct. Anal., 112, 480–494 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  6. Carro M. and Soria J., “Boundedness of some integral operators,” Canad. J. Math., 45, 1155–1166 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  7. Carro M. and Soria J., “The Hardy–Littlewood maximal function and weighted Lorentz spaces,” J. London Math. Soc., 55, 146–158 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  8. Gol’dman M. L., Heinig H. P., and Stepanov V. D., “On the principle of duality in Lorentz spaces,” Canad. J. Math., 48, No. 5, 959–979 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  9. Sinnamon G., “Embeddings of concave functions and duals of Lorentz spaces,” Publ. Mat., 46, 489–515 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  10. Goldman M. L. and Sorokina M. V., “Three-weighted Hardy-type inequalities on the cone of quasimonotone functions,” Dokl. Math., 71, No. 2, 209–213 (2005).

    MATH  Google Scholar 

  11. Popova O. V., “Hardy-type inequalities on the cones of monotone functions,” Sib. Math. J., 53, No. 1, 187152–167 (2012).

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Shambilova G. E., “The weighted inequalities for a certain class of quasilinear integral operators on the cone of monotone functions,” Sib. Math. J., 55, No. 4, 745–767 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  14. Gogatishvili A. and Stepanov V. D., “Operators on cones of monotone functions,” Dokl. Math., 86, No. 1, 562–565 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  15. Gogatishvili A. and Stepanov V. D., “Integral operators on cones of monotone functions,” Dokl. Math., 86, No. 2, 650–653 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  16. Gogatishvili A. and Stepanov V. D., “Reduction theorems for operators on the cones of monotone functions,” J. Math. Anal. Appl., 405, No. 1, 156–172 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  17. Gogatishvili A. and Stepanov V. D., “Reduction theorems for weighted integral inequalities on the cone of monotone functions,” Russian Math. Surveys, 68, No. 4, 597–664 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  18. Gogatishvili A., Mustafayev R., and Persson L.-E., “Some new iterated Hardy-type inequalities,” J. Funct. Spaces Appl. 2012. V. 2012. Art. ID 734194. 30 p.

    Google Scholar 

  19. Gogatishvili A., Mustafayev R., and Persson L.-E., “Some new iterated Hardy-type inequalities: the case 1,” J. Inequal. Appl. 2013. 2013:515. 29 p. arXiv:1302.3436[math.CA].

  20. Prokhorov D. V. and Stepanov V. D., “On weighted Hardy inequalities in mixed norms,” Proc. Steklov Inst. Math., 283, Issue 1, 149–164 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  21. Prokhorov D. V. and Stepanov V. D., “Weighted estimates for a class of sublinear operators,” Dokl. Math., 88, No. 3, 721–723 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  22. Prokhorov D. V. and Stepanov V. D., “Estimates for a class of sublinear integral operators,” Dokl. Math., 89, No. 3, 372–377 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  23. Prokhorov D. V., “On the boundedness of a class of sublinear integral operators,” Dokl. Math., 92, No. 2, 602–605 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  24. Prokhorov D. V., “On a class of weighted inequalities containing quasilinear operators,” Proc. Steklov Inst. Math., 293, Issue 1, 272–287 (2016).

    Article  Google Scholar 

  25. Burenkov V. I., Gogatishvili A., Guliev V. S., and Mustafaeyev R. Ch., “Boundedness of the fractional maximal operator in local Morrey-type spaces,” Complex Var. Elliptic Equ., 55, No. 8–10, 739–758 (2010).

    Article  MathSciNet  Google Scholar 

  26. Burenkov V. I., Gogatishvili A., Guliev V. S., and Mustafaeyev R. Ch., “Boundedness of the Riesz potential in local Morrey-type spaces,” Potential Anal., 35, No. 1, 67–87 (2011).

    Article  MathSciNet  Google Scholar 

  27. Persson L.-E., Shambilova G. E., and Stepanov V. D., “Hardy-type inequalities on the weighted cones of quasi-concave functions,” Banach J. Math. Anal., 9, No. 2, 21–34 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  28. Gogatishvili A., Opic B., and Pick L., “Weighted inequalities for Hardy-type operators involving suprema,” Collect. Math., 57, 227–255 (2006).

    MathSciNet  MATH  Google Scholar 

  29. Sinnamon G. and Stepanov V. D., “The weighted Hardy inequality: new proofs and the case p = 1,” J. London Math. Soc., 54, No. 2, 89–101 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  30. Stepanov V. D. and Ushakova E. P., “Kernel operators with variable intervals of integration in Lebesgue spaces and applications,” Math. Inequal. Appl., 13, No. 3, 449–510 (2010).

    MathSciNet  MATH  Google Scholar 

  31. Prokhorov D. V., “On a weighted inequality for a Hardy-type operator,” Proc. Steklov Inst. Math., 284, Issue 1, 208–215 (2014).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Peoples’ Friendship University of Russia Steklov Institute of Mathematics, Moscow, Russia

    V. D. Stepanov

  2. Financial University Under the Government of the Russian Federation, Moscow, Russia

    G. E. Shambilova

Authors
  1. V. D. Stepanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. E. Shambilova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. D. Stepanov.

Additional information

This research was carried out at the Peoples’ Friendship University of Russia and financially supported by the Russian Science Foundation (Grant 16–41–02004).

Moscow. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 5, pp. 1131–1155, September–October, 2016; DOI: 10.17377/smzh.2016.57.519.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stepanov, V.D., Shambilova, G.E. Boundedness of quasilinear integral operators on the cone of monotone functions. Sib Math J 57, 884–904 (2016). https://doi.org/10.1134/S0037446616050190

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446616050190

Keywords

Search

Navigation