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Inequalities for Hardy-type operators on the cone of decreasing functions in a weighted Orlicz space

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Abstract

Modular inequalities and inequalities for the norms of Hardy-type operators on the cone Ω of positive functions and on the cone of positive decreasing functions with common weight and common Young function in a weighted Orlicz space are considered. A reduction theorem for the norm of the Hardy operator on the cone Ω is obtained. It is shown that this norm is equivalent to the norm of a modified operator on the cone of all positive functions in the space under consideration. It is proved that the modified operator is a generalized Hardy-type operator. The equivalence of modular inequalities on the cone Ω and modified modular inequalities on the cone of all positive functions in the Orlicz space is shown. A criterion for the validity of such inequalities in general Orlicz spaces is obtained and refined for weighted Lebesgue spaces.

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References

  1. M. L. Goldman, Dokl. Math. 95 (3), 627–631 (2017).

    Article  MathSciNet  Google Scholar 

  2. C. Bennett and R. Sharpley, Interpolation of Operators (Academic, Boston, 1988).

    MATH  Google Scholar 

  3. S. Bloom and R. Kerman, Studia Math. 110 (1), 35–52 (1994).

    Article  MathSciNet  Google Scholar 

  4. E. Sawyer, Studia Math. 96, 145–158 (1990).

    Article  MathSciNet  Google Scholar 

  5. H. P. Heinig and A. Kufner, J. London Math. Soc. 53 (2), 256–270 (1996).

    Article  MathSciNet  Google Scholar 

  6. P. Drabek, H. P. Heinig, and A. Kufner, J. Inequal. Appl. 1, 183–197 (1997).

    MathSciNet  Google Scholar 

  7. M. L. Gol’dman and R. A. Kerman, in Function Spaces. Differential Operators. Problems of Mathematical International Conference Dedicated to 75th Birthday of L.D. Kudryavtsev (Moscow, 1998), Vol. 1, 179–183.

    Google Scholar 

  8. M. L. Goldman, Math. Notes 100 (1), 24–37 (2016).

    Article  MathSciNet  Google Scholar 

  9. M. L. Goldman, Dokl. Math. 94 (3), 627–631 (2016).

    Article  MathSciNet  Google Scholar 

  10. Jim Quile Sun, Hardy Type Inequalities on Weighted Orlicz Spaces, PhD Thesis (Univ. Western Ontario, London, Canada, 1995).

    Google Scholar 

  11. Jim Quile Sun, Proc. Am. Math. Soc. 125 (8), 2293–2305 (1997).

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. RUDN University, Moscow, 117198, Russia

    E. G. Bakhtigareeva & M. L. Gol’dman

  2. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, 119991, Russia

    M. L. Gol’dman

Authors
  1. E. G. Bakhtigareeva
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  2. M. L. Gol’dman
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Correspondence to E. G. Bakhtigareeva.

Additional information

Original Russian Text © E.G. Bakhtigareeva, M.L. Gol’dman, 2017, published in Doklady Akademii Nauk, 2017, Vol. 477, No. 2, pp. 133–137.

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Bakhtigareeva, E.G., Gol’dman, M.L. Inequalities for Hardy-type operators on the cone of decreasing functions in a weighted Orlicz space. Dokl. Math. 96, 553–557 (2017). https://doi.org/10.1134/S1064562417060059

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  • DOI: https://doi.org/10.1134/S1064562417060059

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