Skip to main content
SpringerLink
Log in

Hardy-type inequalities on the cones of monotone functions

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We establish necessary and sufficient conditions for various Hardy-type inequalities on the cones of monotone functions.

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. Arino M. and Muckenhoupt B., “Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for non-increasing functions,” Trans. Amer. Math. Soc., 320, 727–735 (1990).

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Stepanov V. D., “The weighted Hardy’s inequality for nonincreasing functions,” Trans. Amer. Math. Soc., 338, 173–186 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  5. Stepanov V. D., “Integral operators on the cone of monotone functions,” J. London Math. Soc., 48, 465–487 (1993).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Heinig H. P. and Maligranda L., “Weighted inequalities for monotone and concave functions,” Studia Math., 116, 133–165 (1995).

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. Gol’dman M. L., Heinig H. P., and Stepanov V. D., “On the principle of duality in Lorentz spaces,” Canad. J. Math., 48, 959–979 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  11. Carro M. J., Pick L., Soria J., and Stepanov V. D., “On embeddings between classical Lorentz spaces,” Math. Inequal. Appl., 4, 397–428 (2001).

    MathSciNet  MATH  Google Scholar 

  12. Gol’dman M. L., “Sharp estimates for the norms of Hardy-type operators on the cones of quasimonotone functions,” Proc. Steklov Inst. Math., 232, 109–137 (2001).

    Google Scholar 

  13. Sinnamon G., “Transferring monotonicity in weighted norm inequalities,” Collect. Math., 54, 181–216 (2003).

    MathSciNet  MATH  Google Scholar 

  14. Sinnamon G., “Hardy’s inequality and monotonicity,” in: Function Spaces and Nonlinear Analysis, Math. Inst. Acad. Sci. Czech Rep., Prague, 2005, pp. 292–310.

  15. Persson L.-E., Stepanov V. D., and Ushakova E. P., “Equivalence of Hardy-type inequalities with general measures on the cones of non-negative respective non-increasing functions,” Proc. Amer. Math. Soc., 134, 2363–2372 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  16. Bennett G. and Grosse-Erdmann K.-G., “Weighted Hardy inequalities for decreasing sequences and functions,” Math. Ann., 334, No. 3, 489–531 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  17. Johansson M., Stepanov V. D., and Ushakova E. P., “Hardy inequality with three measures on monotone functions,” Math. Inequal. Appl., 11, No. 3, 393–413 (2008).

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  19. Bloom S. and Kerman R., “Weighted norm inequalities for operators of Hardy type,” Proc. Amer. Math. Soc., 113, No. 1, 135–141 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  20. Stepanov V. D., “Weighted norm inequalities of Hardy type for a class of integral operators,” J. London Math. Soc., 50, No. 2, 105–120 (1994).

    MathSciNet  MATH  Google Scholar 

  21. Kufner A. and Perrson L.-E., Weighted Inequalities of Hardy Type, Word Sci. Publ. Co. Inc., New Jersey, London, Singapore, and Hong Kong (2003).

    MATH  Google Scholar 

  22. Kufner A., Maligranda L., and Persson L.-E., The Hardy Inequality. About Its History and Some Related Results, Vydavatelsky Servis, Pilsen (2007).

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Prokhorov D. V., “Inequalities of Hardy type for a class of integral operators with measures,” Anal. Math., 33, 199–225 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  25. Persson L.-E., Popova O. V., and Stepanov V. D., “Two-sided Hardy-type inequalities for monotone functions,” Complex Variables and Elliptic Equations, 55, No. 8–10, 973–989 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  26. Prokhorov D. V., “Weighted Hardy inequalities for negative indices,” Publ. Mat., 48, 423–443 (2004).

    MathSciNet  MATH  Google Scholar 

  27. Stepanov V. D., Persson L.-E., and Popova O. V., “Two-sided Hardy-type inequalities for monotone functions,” Dokl. Math., 80, No. 3, 814–817 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  28. Goldman M. L., “On equivalent criteria for the boundedness of Hardy type operators on the cone of decreasing functions,” Anal. Math., 37, 83–102 (2011).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Peoples’ Friendship University of Russia, Moscow, Russia

    O. V. Popova

Authors
  1. O. V. Popova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. V. Popova.

Additional information

Original Russian Text Copyright © 2012 Popova O. V.

The author was supported by the Russian Foundation for Basic Research (Grant 09-01-00093).

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 1, pp. 187–204, January–February, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Popova, O.V. Hardy-type inequalities on the cones of monotone functions. Sib Math J 53, 152–167 (2012). https://doi.org/10.1134/S0037446612010132

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446612010132

Keywords

Search

Navigation