pp 1–25 | Cite as

On weighted iterated Hardy-type inequalities



In this paper the inequality
$$\begin{aligned} \bigg ( \int _0^{\infty } \bigg ( \int _x^{\infty } \bigg ( \int _t^{\infty } h \bigg )^q w(t)\,dt \bigg )^{r / q} u(x)\,{ ds} \bigg )^{1/r}\le C \,\int _0^{\infty } h v, \quad h \in {\mathfrak {M}}^+(0,\infty ) \end{aligned}$$
is characterized. Here \(0< q ,\, r < \infty \) and \(u,\,v,\,w\) are weight functions on \((0,\infty )\).


Quasilinear operators Iterated Hardy inequalities Weights 

Mathematics Subject Classification

26D10 26D15 


  1. 1.
    Burenkov, V.I., Gogatishvili, A., Guliyev, V.S., Mustafayev, RCh.: Boundedness of the fractional maximal operator in local Morrey-type spaces. Complex Var. Elliptic Equ. 55(8–10), 739–758 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Burenkov, V.I., Gogatishvili, A., Guliyev, V.S., Mustafayev, RCh.: Boundedness of the Riesz potential in local Morrey-type spaces. Potential Anal. 35(1), 67–87 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 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)MathSciNetMATHGoogle Scholar
  4. 4.
    Gogatishvili, A., Mustafayev, RCh.: Weighted iterated Hardy-type inequalities. Math. Inequal. Appl. 20(3), 683–728 (2017)MathSciNetGoogle Scholar
  5. 5.
    Gogatishvili, A., Mustafayev, R. Ch., Persson, L.-E.: Some new iterated Hardy-type inequalities. J. Funct. Spaces Appl. Art. ID 734194, 30 (2012)Google Scholar
  6. 6.
    Gogatishvili, A., Mustafayev, RCh., Persson, L.-E.: Some new iterated Hardy-type inequalities: the case \(\theta = 1\). J. Inequal. Appl. (2013). doi:10.1186/1029-242X-2013-515 MathSciNetMATHGoogle Scholar
  7. 7.
    Gogatishvili, A., Stepanov, V.D.: Reduction theorems for operators on the cones of monotone functions. J. Math. Anal. Appl. 405(1), 156–172 (2013). doi:10.1016/j.jmaa.2013.03.046 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gogatishvili, A., Stepanov, V. D.: Reduction theorems for weighted integral inequalities on the cone of monotone functions. Uspekhi Mat. Nauk 68(4(412)), 3–68 (Russian, with Russian summary); English transl., Russian Math. Surveys, 68(4), 597–664 (2013)Google Scholar
  9. 9.
    Gogatishvili, A., Pick, L.: Discretization and anti-discretization of rearrangement-invariant norms. Publ. Mat. 47(2), 311–358 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Křepela, M.: Boundedness of Hardy-type operators with a kernel integral weighted conditions for the case \(0 < q < 1 \le p < \infty \). Preprint (2016)Google Scholar
  11. 11.
    Lai, Q.: Weighted modular inequalities for Hardy type operators. Proc. Lond. Math. Soc. (3) 79(3), 649–672 (1999). doi:10.1112/S0024611599012010 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Leindler, L.: Inequalities of Hardy–Littlewood type. Anal. Math. 2(2), 117–123 (1976) (English, with Russian summary) Google Scholar
  13. 13.
    Leindler, L.: On the converses of inequalities of Hardy and Littlewood. Acta Sci. Math. (Szeged) 58(1–4), 191–196 (1993)MathSciNetMATHGoogle Scholar
  14. 14.
    Oĭnarov, R.: Two-sided estimates for the norm of some classes of integral operators. Trudy Mat. Inst. Steklov. 204 (1993), no. Issled. po Teor. Differ. Funktsii Mnogikh Peremen. i ee Prilozh. 16, 240–250 (Russian); English transl.: Proc. Steklov Inst. Math. 3(204), 205–214 (1994)Google Scholar
  15. 15.
    Prokhorov, D.V., Stepanov, V.D.: On weighted Hardy inequalities in mixed norms. Proc. Steklov Inst. Math. 283, 149–164 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Sinnamon, G.: Transferring monotonicity in weighted norm inequalities. Collect. Math. 54(2), 181–216 (2003)MathSciNetMATHGoogle Scholar
  17. 17.
    Sinnamon, G., Stepanov, V.D.: The weighted Hardy inequality: new proofs and the case \(p=1\). J. Lond. Math. Soc. 54(1), 89–101 (1996). doi:10.1112/jlms/54.1.89 MathSciNetCrossRefMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Mathematics and MechanicsAcademy of Sciences of AzerbaijanBakuAzerbaijan

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