Skip to main content
SpringerLink
Log in

Hardy Type Operators In Local Vanishing Morrey Spaces On Fractal Sets

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

We obtain two-weighted estimates for the Hardy type operators from local generalized Morrey spaces ℒp,φloc(X, w1) defined on an arbitrary underlying quasi-metric measure space (X, μ, ϱ) with the growth condition, to ℒq,Ψloc(X, w2)(X,w2), where w1 = w1[ϱ(x, x0)], x0], x) ∈ X is a weight of radial type, while w2 = w2(x) may be an arbitrary weight. The proof allows to simultaneously treat a similar boundedness V ℒp,φloc(X, w1) → V ℒq,φloc(X, w2) for vanishing Morrey spaces. We obtain sufficient conditions for such estimates in terms of some integral inequalities imposed on φ, Ψ and w1.w2. We also specially treat the one weight case where w2(x) is also of radial type. We do not impose doubling condition on the measure μ, but base our result on the growth condition.

The obtained results show the explicit dependence of the mapping properties of the Hardy type operators on the fractional dimension of the set (X, μ, ?). An application to spherical Hardy type operators is also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D.R. Adams, A note on Riesz potentials. Duke Math. J. 42, No 4 (1975), 765–778.

    Article  MathSciNet  Google Scholar 

  2. D.R. Zhou and J. Xiao, Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53, No 6 (2004), 1631–1666.

    MathSciNet  Google Scholar 

  3. J. Alvarez, The distribution function in the Morrey space. Proc. Amer. Math. Soc. 83 (1981), 693–699.

    MathSciNet  MATH  Google Scholar 

  4. H. Zhou and T. Mizuhara, Morrey spaces on spaces of homogeneous type and estimates for ⃞b and the Cauchy-Szego projection. Math. Nachr. 185, No 1 (1997), 5–20.

    MathSciNet  MATH  Google Scholar 

  5. N.K. Zhou and S.B. Stechkin, Best approximations and differential properties of two conjugate functions (in Russian). Proc. Moscow Math. Soc. 5 (1956), 483–522.

    Google Scholar 

  6. F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function. Rend. Math. 7 (1987), 273–279.

    MathSciNet  MATH  Google Scholar 

  7. D.E. Zhou, V. Zhou and A. Meskhi, Bounded and Compact Integral Operators. Mathematics and its Applications, Vol. 543, Kluwer Academic Publ., 2002.

    Google Scholar 

  8. M. Giaquinta, Multiple Integrals in the Calculus of Variations and Non-linear Elliptic Systems. Princeton Univ. Press, 1983.

    MATH  Google Scholar 

  9. N.K. Zhou and N. Samko, Weighted theorems on fractional integrals in the generalized Hölder spaces Hw 0 (ρ) via the indices mw and Mw. Fract. Calc. Appl. Anal. 7, No 4 (2004), 437–458.

    MathSciNet  Google Scholar 

  10. V. Zhou and A. Meskhi, Maximal and potential operators in variable Morrey spaces defined on nondoubling quasimetric measure spaces. Bull. Georgian Natl. Acad. Sci. (N.S.) 2, No 3 (2008), 18–21.

    MathSciNet  MATH  Google Scholar 

  11. V. Zhou and A. Meskhi, Boundedness of maximal and singular operators in Morrey spaces with variable exponent. Armenian J. Math. 1, No 1 (2008), 18–28.

    MathSciNet  MATH  Google Scholar 

  12. V. Zhou, A. Zhou and L.E. Persson, Weighted Norm Inequalities for Integral Transforms with Product Weights. Nova Scientific Publ. Inc., New York, 2010.

    Google Scholar 

  13. S.G. Krein, Yu.I. Zhou and E.M. Semenov, Interpolation of Linear Operators. Nauka, Moscow, 1978.

    Google Scholar 

  14. A. Zhou, O. John and S. Fuˇcik, Function Spaces. Noordhoff International Publ., 1977.

    Google Scholar 

  15. A. Zhou and L.E. Persson, Weighted Inequalities of Hardy Type. World Sci. Publ. Co. Inc., River Edge - NJ, 2003.

    Google Scholar 

  16. K. Zhou, S. Zhou and S. Sugano, Boundedness of integral operators on generalized Morrey spaces and its application to Schrödinger operators. Proc. Amer. Math. Soc. 128, No 4 (1999), 1125–1134.

    Google Scholar 

  17. D. Zhou, A. Zhou, L.E. Zhou and N. Samko, Hardy and singular operators in weighted generalized Morrey spaces with applications to singular integral equations. Math. Methods Appl. Sci. 35, No 11 (2012), 1300–1311.

    MathSciNet  Google Scholar 

  18. W. Zhou and W. Orlicz, On some classes of functions with regard to their orders of growth. Studia Math. 26 (1965), 11–24.

    MathSciNet  MATH  Google Scholar 

  19. C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43 (1938), 126–166.

    MathSciNet  MATH  Google Scholar 

  20. E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166 (1994), 95–103.

    MathSciNet  MATH  Google Scholar 

  21. J. Peetre, On convolution operators leaving Lp,λ spaces invariant. Annali di Mat. Pura ed Appl. 72, No 1 (1966), 295–304.

    MathSciNet  MATH  Google Scholar 

  22. J. Peetre, On the theory of Lp,λ spaces. J. Func. Anal. 4 (1969), 71–87.

    Google Scholar 

  23. L.E. Zhou and N. Samko, Weighted Hardy and potential operators in the generalized Morrey spaces. J. Math. Anal. Appl. 377 (2011), 792–806.

    MathSciNet  MATH  Google Scholar 

  24. L.E. Persson and N. Samko, ”What should have happened if Hardy had discovered this?” J. Inequal. Appl. 2012 (2012), # 29, 20 p.

    MathSciNet  MATH  Google Scholar 

  25. L.E. Zhou, N. Samko and P. Wall, Quasi-monotone weight functions and their characteristics and applications. Math. Inequal. Appl. 15, No 3 (2012), 685–705.

    MathSciNet  MATH  Google Scholar 

  26. H. Zhou and S. Samko, Variable exponent Campanato spaces. J. Math. Sci. 172, No 1 (2011), 143–164.

    MathSciNet  MATH  Google Scholar 

  27. H. Zhou, N. Samko and S. Samko, Morrey-Campanato spaces: An overview. Oper. Theory Adv. Appl. 228 (2013), 293–323.

    MathSciNet  MATH  Google Scholar 

  28. M. Zhou and H. Triebel, Morrey spaces, their duals and preduals. Revista Mat. Complutenze 28, No 1 (2015), 1–30; DOI: 10.1007/s13163-013-0145-z.

    MathSciNet  MATH  Google Scholar 

  29. N. Samko, Singular integral operators in weighted spaces with generalized Hölder condition. Proc. A. Razmadze Math. Inst. 120 (1999), 107–134.

    MathSciNet  MATH  Google Scholar 

  30. N. Samko, On non-equilibrated almost monotonic functions of the Zygmund-Bary-Stechkin class. Real Anal. Exch. 30, No 2 (2004), 727–745.

    MathSciNet  MATH  Google Scholar 

  31. N. Samko, Weighted Hardy and singular operators in Morrey spaces. J. Math. Anal. Appl. 350 (2009), 56–72.

    MathSciNet  MATH  Google Scholar 

  32. N. Samko, Weighted Hardy operators in the local generalized vanishing Morrey spaces. Positivity 17, No 3 (2013), 683–706.

    MathSciNet  MATH  Google Scholar 

  33. N. Samko, Maximal, potential and singular operators in vanishing generalized Morrey spaces. J. of Global Optimization 57, No 4 (2013), 1385–1399.

    MathSciNet  MATH  Google Scholar 

  34. N. Samko, On a Muckenhoupt-type condition for Morrey spaces. Mediterr. J. Math. 10, No 2 (2013), 941–951.

    MathSciNet  MATH  Google Scholar 

  35. N. Zhou, S. Zhou and B. Vakulov, Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces. J. Funct. Spaces Appl. 8, No 3 (2010), 215–244.

    MathSciNet  MATH  Google Scholar 

  36. N. Zhou, S. Zhou and B. Vakulov, Weighted Sobolev theorem in Lebesgue spaces with variable exponent. J. Math. Anal. Appl. 335 (2007), 560–583.

    MathSciNet  MATH  Google Scholar 

  37. S. Samko, Hypersingular Integrals and Their Applications. Ser. ”Analytical Methods and Special Functions”, Vol. 5, Taylor & Francis, London-New York, 2002.

    MATH  Google Scholar 

  38. S. Zhou, A. Zhou and O. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach Sci. Publ., London-New York, 1993 (Russian Ed.: Fractional Integrals and Derivatives and Some of Their Applications, Nauka i Tekhnika, Minsk, 1987).

    Google Scholar 

  39. Y. Zhou, S. Zhou and H. Tanaka, A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces. Boundary Value Problems 2009 (2009), # 835865; doi:10.1155/2009/835865.

    MathSciNet  MATH  Google Scholar 

  40. S. Shirai, Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on classical Morrey spaces. Hokkaido Math. J. 35, No 3 (2006), 683–696.

    MathSciNet  MATH  Google Scholar 

  41. S. Spanne, Some function spaces defined by using the mean oscillation over cubes. Ann. Scuola Norm. Sup. Pisa 19 (1965), 593–608.

    MathSciNet  MATH  Google Scholar 

  42. G. Stampacchia, The spaces Lp,λ, N(p,λ) and interpolation. Ann. Scuola Norm. Super. Pisa 3, No 19 (1965), 443–462.

    Google Scholar 

  43. M. E. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials. Vol. 81 of Math. Surveys and Monogr., AMS, Providence - R.I., 2000.

    MATH  Google Scholar 

  44. C. Vitanza, Functions with vanishing Morrey norm and elliptic partial differential equations. In: Proc. of Methods of Real Analysis and Partial Differential Equations, Capri, Springer, 1990, 147–150.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Narvik University College and Norut Narvik, P.O. Box 385, N-8505, Narvik, Norway

    Dag Lukkassen & Lars-Erik Persson

  2. Luleå University of Technology, SE 971 87, Luleå, Sweden

    Natasha Samko

Authors
  1. Dag Lukkassen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lars-Erik Persson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Natasha Samko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dag Lukkassen.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lukkassen, D., Persson, LE. & Samko, N. Hardy Type Operators In Local Vanishing Morrey Spaces On Fractal Sets. FCAA 18, 1252–1276 (2015). https://doi.org/10.1515/fca-2015-0072

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2015-0072

MSC 2010

Key Words and Phrases

Search

Navigation