Advertisement

Analysis Mathematica

, Volume 44, Issue 2, pp 273–283 | Cite as

On Weighted Iterated Hardy-Type Operators

  • V. D. Stepanov
  • G. E. Shambilova
Article
  • 121 Downloads

Abstract

The weighted L v p L ρ r boundedness of a weighted iterated Hardy-type operator
$$Tf(x): = {(\int_x^\infty {u(t){{(\int_0^t f )}^q}dt} )^{1/q}}$$
for p = 1 is established.

Key words and phrases

iterated Hardy-type operator weighted Lebesgue space cone of monotone function 

Mathematics Subject Classification

26D10 46E20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Gogatishvili, R. Mustafayev and L.-E. Persson, Some new iterated Hardy-type inequalities, J. Funct. Spaces Appl. (2013), Art. ID 734194, 30 pp.Google Scholar
  2. [2]
    A. Gogatishvili, R. Mustafayev and L.-E. Persson, Some new iterated Hardy-type inequalities: the case θ = 1, J. Inequal. Appl. (2013), 2013:515.Google Scholar
  3. [3]
    A. Gogatishvili and R. Ch. Mustafayev, Weighted iterated Hardy-type inequalities, Math. Inequal. Appl., 20 (2017), 683–728.MathSciNetzbMATHGoogle Scholar
  4. [4]
    A. Gogatishvili and V. D. Stepanov, Reduction theorems for weighted integral inequalities on the cone of monotone functions, Russian Math. Surveys, 68 (2013), 597–664.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    A. Gogatishvili and L. Pick, Discretization and antidiscretization of rearrangementinvariant norms, Publ. Mat., 47 (2003), 311–358.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. L. Goldman, H. P. Heinig and V. D. Stepanov, On the principle of duality in Lorentz spaces, Canad. J. Math., 48 (1996), 959–979.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Johansson, V. D. Stepanov and E. P. Ushakova, Hardy inequality with three measures on monotone functions, Math. Inequal. Appl., 11 (2008), 393–413.MathSciNetzbMATHGoogle Scholar
  8. [8]
    R. Mustafayev, On weighted iterated Hardy-type inequalities, Positivity, 22 (2018), 275–299.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    R. Oinarov, Two-sided estimates of the norm of some classes of integral operators, Proc. Steklov Inst. Math., 204 (1994), 205–214.MathSciNetzbMATHGoogle Scholar
  10. [10]
    D. V. Prokhorov and V. D. Stepanov, On weighted Hardy inequalities in mixed norms, Proc. Steklov Inst. Math., 283 (2013), 149–164.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    D. V. Prokhorov and V. D. Stepanov, Weighted inequalities for quasilinear integral operators on the semi-axis and applications to Lorentz spaces, Mat. Sb., 207 (2016), 135–162 (in Russian); translated in Sb. Math., 207 (2016), 1159–1186.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D. V. Prokhorov, On a class of weighted inequalities involving quasilinear operators, Proc. Steklov Inst. Math., 293 (2016), 272–287.CrossRefzbMATHGoogle Scholar
  13. [13]
    H. L. Royden, Real Analysis, third edition, Macmillan Publishing Company (New York, 1988).Google Scholar
  14. [14]
    E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math., 96 (1990), 145–158.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    G. Sinnamon and V. D. Stepanov, The weighted Hardy inequality: new proofs and the case p = 1, J. London Math. Soc., 54 (1996), 89–101.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    G. Sinnamon, Transferring monotonicity in weighted norm inequalities, Collect. Math., 54 (2003), 181–216.MathSciNetzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of RASMoscowRussia
  2. 2.Peoples’ Friendship University of RussiaMoscowRussia

Personalised recommendations