, Volume 13, Issue 3, pp 497–518 | Cite as

Boundedness and compactness of positive integral operators on cones of homogeneous groups

  • Usman Ashraf
  • Muhammad Asif
  • Alexander Meskhi


Necessary and sufficient conditions on a weight function v guaranteeing the boundedness/compactness of integral operators with positive kernels defined on cones of homogeneous groups from L p to L v q are established, where \(1< p,q < \infty\) or \(0 < q \leq 1< p < \infty\). Behavior of singular numbers for these operators is also studied.


Operators with positive kernels potentials homogeneous groups trace inequality weights singular numbers of kernel operators 

Mathematics Subject Classification (2000)

Primary 26A33, 42B25 Secondary 43A15, 46B50, 47B10, 47B34 


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  1. 1.
    T. Ando, On the compactness of integral operators, Indag. Math., (N.S.) 24 (1962), 235–239.Google Scholar
  2. 2.
    J. Bergh, J. Löfström, Interpolation spaces: An introduction, Grundlehren Math.Wiss. vol. 223, Springer, Berlin (1976).Google Scholar
  3. 3.
    J.S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405–408.Google Scholar
  4. 4.
    M.Sh. Birman, M. Solomyak, Estimates for the singular numbers of integral operators, Uspekhi Mat. Nauk. 32(1) (1977), 17–82. English transl. Russian Math. Surveys 32 (1977) (Russian).Google Scholar
  5. 5.
    D.E. Edmunds, W.D. Evans, D.J. Harris, Two-sided estimates of the approximation numbers of certain Voltera integral operators, Stud. Math., 124(1) (1997), 59–80.Google Scholar
  6. 6.
    D.E. Edmunds, V. Kokilashvili, A. Meskhi, Bounded and compact integral operators, Kluwer, Dordrecht (2002).Google Scholar
  7. 7.
    D.E. Edmunds, D. Stepanov, The measure of non-compactness and approximation numbers of certain Voltera integral operators, Math. Ann. 298 (1994), 41–66.Google Scholar
  8. 8.
    G.B. Folland, E.M. Stein, Hardy spaces on homogeneous groups, Princeton Univ. Press, University of Tokyo Press, Princeton, New Jersey (1982).Google Scholar
  9. 9.
    I. Genebashvili, A. Gogatishvili, V. Kokilashvili, Solution of two-weight problems for integral transforms with positive kernels, Georgian Math., J. 3(1) (1996), 319–342.Google Scholar
  10. 10.
    I. Genebashvili, A. Gogatishvili, V. Kokilashvili, M. Krbec, Weight theory for integral transforms on spaces of homogeneous type, Pitman Monographs and Surveys in Pure and Applied Mathematics 92, Longman, Harlow (1998).Google Scholar
  11. 11.
    P. Jain, P.K. Jain, B. Gupta, Higher dimensional compactness of Hardy operators involving Oinarov-type kernels, Math. Ineq. Appl. 9(4) (2002), 739–748.Google Scholar
  12. 12.
    L.P. Kantorovich, G.P. Akilov, Functional analysis, Pergamon, Oxford (1982).Google Scholar
  13. 13.
    H. König, Eigenvalue distribution of compact operators, Birkäuser, Boston (1986).Google Scholar
  14. 14.
    V.M. Kokilashvili, On Hardy’s inequality in weighted spaces, Soobsch. Akad. Nauk. SSR, 96 (1979), 37–40 (Russian).Google Scholar
  15. 15.
    M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustilnik, P.E. Sobolevskii, Integral operators in spaces of summable functions, (Nauka, Moscow, 1966), Engl. Transl. Noordhoff International Publishing, Leiden (1976) (Russian).Google Scholar
  16. 16.
    A. Kufner, L.E. Persson, Integral inequalities with weights, Academy of Sciences of the Czech Republic (2000).Google Scholar
  17. 17.
    F.J. Martin-Reyes, E. Sawyer, Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater, Proc. Amer. Math. Soc. 106 (1989), 727–733.Google Scholar
  18. 18.
    V.G. Maz’ya , Sobolev spaces, Springer, Berlin (1985).Google Scholar
  19. 19.
    A. Meskhi, Solution of some weight problems for the Riemann-Liouville and Weyl operators, Georgian Math. J. 5(6), (1998), 565–574.Google Scholar
  20. 20.
    A. Meskhi, Criteria for the boundedness and compactness of integral transforms with positive kernels, Proc. Edinb. Math. Soc. 44(2) (2001), 267–284.Google Scholar
  21. 21.
    A. Meskhi, On singualar numbers for some integral operators, Revista Mat. Compl., 14 (2001), 379–393.Google Scholar
  22. 22.
    B. Muckenhoupt, Hardy’s inequality with weights, Stud. Math., 44 (1972), 31–38.Google Scholar
  23. 23.
    K. Nowak, Schatten ideal behavior of a generalized Hardy operator, Proc. Am. Math. Soc., 118 (1993), 479–483.Google Scholar
  24. 24.
    J. Newman, M. Solomyak, Two-sided estimates on singular values for a class of integral operators on the semi-axis. Int. Equ. Operat. Theory, 20 (1994), 335–349.Google Scholar
  25. 25.
    R. Oinarov, Two-sided estimate of certain classes of integral operators. Trans. Math. Inst. Steklova 204 (1993), 240–250 (Russian).Google Scholar
  26. 26.
    B. Opic, A. Kufner, Hardy-type inequalities, Pitman Research Notes in Math. Series, 219, Longman Sci. and Tech. Harlow (1990).Google Scholar
  27. 27.
    A. Pietsch, Eigenvalues and s-numbers, Cambridge University Press, Cambridge (1987).Google Scholar
  28. 28.
    D.V. Prokhorov, On the boundedness of a class of integral operators, J. Lond. Math. Soc., 61(2) (2000), 617–628.Google Scholar
  29. 29.
    G. Sinnamon, One-dimensional Hardy-inequalities in many dimensions, Proc. Royal Soc. Edinburgh Sect, A 128A (1998), 833–848.Google Scholar
  30. 30.
    G. Sinnamon, V. Stepanov, The weighted Hardy inequality: new proof and the case p = 1, J, Lond. Math. Soc. 54 (1996), 89–101.Google Scholar
  31. 31.
    V. Stepanov, Two-weight estimates for the Riemann–Livouville operators, Izv. Akad. Nauk SSSR. 54 (1990), 645–656 (Russian).Google Scholar
  32. 32.
    V. Stepanov, Weighted norm inequalities for integral operators and related topics, In: Nonlinear Analysis. Function Spaces and Applications, 5 Olympia Press, Prague (1994) 139–176.Google Scholar
  33. 33.
    H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, (1978), 2nd ed. Johann Ambrosius Barth, Heidelberg (1995).Google Scholar
  34. 34.
    A. Wedestig, Weighted inequalities of Hardy-type and their limiting inequalities, Doctoral Thesis, Lulea University of Technology, Department of Mathematics (2003).Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  1. 1.Abdus Salam School of Mathematical SciencesGC UniversityLahorePakistan
  2. 2.Georgian Academy of SciencesA. Razmadze Mathematical InstituteTbilisiGeorgia

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