, Volume 17, Issue 3, pp 683–706 | Cite as

Weighted Hardy operators in the local generalized vanishing Morrey spaces



In this paper we study \(p\rightarrow q\)-boundedness of the multi-dimensional Hardy type operators in the vanishing local generalized Morrey spaces \(V\mathcal L ^{p,\varphi }_\mathrm{{loc}}(\mathbb R ^n,w)\) defined by an almost increasing function \(\varphi (r)\) and radial type weight \(w(|x|)\). We obtain sufficient conditions, in terms of some integral inequalities imposed on \(\varphi \) and \(w\), for such a boundedness. In the case where the function \(\varphi (r)\) and the weight are power functions, these conditions are also necessary.


Generalized weighted Morrey space Local Morrey spaces  Weighted Hardy inequalities Weighted Hardy operators Bary-Stechkin classes Matuszewska-Orlicz indices 

Mathematics Subject Classification (2000)

46E30 42B35 42B25 47B38 


  1. 1.
    Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42(4), 765–778 (1975)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Adams, D.R., Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53(6), 1631–1666 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alvarez, J.: The distribution function in the Morrey space. Proc. Amer. Math. Soc. 83, 693–699 (1981)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Arai, H., Mizuhara, T.: Morrey spaces on spaces of homogeneous type and estimates for \(\square _b\) and the Cauchy-Szego projection. Math. Nachr. 185(1), 5–20 (1997)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bari, N.K., Stechkin, S.B.: Best approximations and differential properties of two conjugate functions (in Russian). Proc. of Moscow Math. Soc. 5, 483–522 (1956)MATHGoogle Scholar
  6. 6.
    Chiarenza, F., Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function. Rend. Math. 7, 273–279 (1987)MathSciNetMATHGoogle Scholar
  7. 7.
    Giaquinta, M.: Multiple integrals in the calculus of variations and non-linear elliptic systems. Princeton University Press, Princeton (1983)Google Scholar
  8. 8.
    Karapetiants, N.K., Samko, N.G.: Weighted theorems on fractional integrals in the generalized Hölder spaces \({H}_0^w (\rho )\) via the indices \(m_w \) and \({M}_w \). Fract. Calc. Appl. Anal. 7(4), 437–458 (2004)MathSciNetGoogle Scholar
  9. 9.
    Kokilashvili, V., Meshki, A., Persson, L.-E.: Weighted Norm Inequalities for Integral Transforms with Product Weights. Nova Scientific Publishers Inc., New York (2010)Google Scholar
  10. 10.
    Krein, S.G., Petunin, Yu.I., Semenov, E.M.: Interpolation of linear operators. Nauka, Moscow (1978)Google Scholar
  11. 11.
    Kufner, A., John, O., Fucik, S.: Function Spaces. Noordhoff International Publishing, Leyden (1977)MATHGoogle Scholar
  12. 12.
    Kufner, A., Maligranda, L., Persson, L.E.: The Hardy Inequality. About Its History and Some Related Results. Vydavatelsky Servis Publishing House, Pilsen (2007)MATHGoogle Scholar
  13. 13.
    Kufner, A., Persson, L.-E.: Weighted inequalities of Hardy type. World Scientific Publishing Co. Inc., River Edge (2003)MATHCrossRefGoogle Scholar
  14. 14.
    Lukkassen, D., Medell, A., Persson, L.-E., Samko, N.: Hardy and singular operators in weighted generalized Morrey spaces with applications to singular integral equations. Math. Methods Appl. Sci. 35(11), 1300–1311 (2012)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Maligranda, L.: Indices and interpolation. Diss. Math. (Rozprawy Mat.) 234, 49 (1985)MathSciNetGoogle Scholar
  16. 16.
    Maligranda. L.: Orlicz spaces and interpolation. Departamento de Matemática, Universidade Estadual de Campinas, Campinas SP Brazil (1989)Google Scholar
  17. 17.
    Matuszewska, W., Orlicz, W.: On some classes of functions with regard to their orders of growth. Studia Math. 26, 11–24 (1965)MathSciNetMATHGoogle Scholar
  18. 18.
    Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43, 126–166 (1938)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Opic, B., Kufner, A.: Hardy-type Inequalities. Pitman Research Notes in Mathematics, 219. Longman Scientific & Technical, Harlow (1990)Google Scholar
  21. 21.
    Peetre, J.: On convolution operators leaving \({\cal {L}}^{p,\lambda }\) spaces invariant. Annali di Mat. Pura ed Appl. 72(1), 295–304 (1966)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Peetre, J., Peetre, J.: On the theory of \({\cal {L}}_{p,\lambda }\) spaces. J. Func. Anal. 4, 71–87 (1969)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Persson, L.E., Samko, N.: Weighted Hardy and potential operators in the generalized Morrey spaces. J. Math. Anal. Appl. 377, 792–806 (2011)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Persson, L.-E., Samko, N., Wall, P.: Quasi-monotone weight functions and their characteristics and applications. Math. Inequal. Appl. (MIA), 12(3), 685–705 (2012)Google Scholar
  25. 25.
    Samko, N., Samko, S., Vakulov, B.: Weighted Sobolev theorem in Lebesgue spaces with variable exponent. J. Math. Anal. Appl. 335, 560–583 (2007)Google Scholar
  26. 26.
    Samko, N.: Singular integral operators in weighted spaces with generalized Hölder condition. Proc. A. Razmadze Math. Inst. 120, 107–134 (1999)MathSciNetMATHGoogle Scholar
  27. 27.
    Samko, N.: On non-equilibrated almost monotonic functions of the Zygmund–Bary–Stechkin class. Real Anal. Exch. 30(2), 727–745 (2004)MathSciNetGoogle Scholar
  28. 28.
    Samko, N.: Weighted Hardy and singular operators in Morrey spaces. J. Math. Anal. Appl. 350, 56–72 (2009)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Samko, N.G.: Weighted Hardy and potential operators in Morrey spaces. J. Funct. Spaces Appl. 2012 (2012). doi:10.1155/2012/678171
  30. 30.
    Shirai, S.: Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on classical Morrey spaces. Hokkaido Math. J. 35(3), 683–696 (2006)MathSciNetMATHGoogle Scholar
  31. 31.
    Spanne, S.: Some function spaces defined by using the mean oscillation over cubes. Ann. Scuola Norm. Sup. Pisa 19, 593–608 (1965)MathSciNetMATHGoogle Scholar
  32. 32.
    Stampacchia, G.: The spaces \({L}^{p,\lambda }, {N}^{(p, \lambda )}\) and interpolation. Ann. Scuola Norm. Super. Pisa 3(19), 443–462 (1965)MathSciNetGoogle Scholar
  33. 33.
    Taylor, M.E.: Tools for P D E: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Math. Surveys and Monogr., vol. 81, AMS, Providence, R.I. (2000)Google Scholar
  34. 34.
    Vitanza, C.: Functions with vanishing Morrey norm and elliptic partial differential equations. In: Proceedings of Methods of Real Analysis and Partial Differential Equations, Capri, pp. 147–150. Springer, Berlin (1990)Google Scholar

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© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Departamento de Matematica, Centro CEAFInstituto Superior TécnicoLisboaPortugal
  2. 2.Luleå University of TechnologyLuleåSweden

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