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Description of interpolation spaces for local Morrey-type spaces

  • V. I. Burenkov
  • E. D. Nursultanov
Article

Abstract

We consider the real interpolation method and prove that for local Morrey spaces, in the case when they have the same integrability parameter, the interpolation spaces are again local Morrey-type spaces with appropriately chosen parameters. Thus, in contrast to the standard Morrey-type spaces, these local spaces form an interpolation scale. In particular, this is true in the so-called nondiagonal case.

Keywords

STEKLOV Institute Maximal Operator Singular Integral Operator Interpolation Space Morrey Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    D. R. Adams, “A Note on Riesz Potentials,” Duke Math. J. 42, 765–778 (1975).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    O. Blasco, A. Ruiz, and L. Vega, “Non Interpolation in Morrey-Campanato and Block Spaces,” Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. 4, 28(1), 31–40 (1999).zbMATHMathSciNetGoogle Scholar
  3. 3.
    V. I. Burenkov and H. V. Guliyev, “Necessary and Sufficient Conditions for Boundedness of the Maximal Operator in Local Morrey-Type Spaces,” Stud. Math. 163(2), 157–176 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    V. I. Burenkov, H. V. Guliyev, and V. S. Guliyev, “Necessary and Sufficient Conditions for the Boundedness of Fractional Maximal Operators in Local Morrey-Type Spaces,” J. Comput. Appl. Math. 208, 280–301 (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    V. I. Burenkov and V. S. Guliyev, “Necessary and Sufficient Conditions for the Boundedness of the Riesz Potential in Local Morrey-Type Spaces,” Potential Anal. 30(3), 211–249 (2009).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    V. I. Burenkov, V. S. Guliev, T. V. Tararykova, and A. Serbetci, “Necessary and Sufficient Conditions for the Boundedness of Genuine Singular Integral Operators in Local Morrey-Type Spaces,” Dokl. Akad. Nauk 422(1), 11–14 (2008) [Dokl. Math. 78 (2), 651–654 (2008)].MathSciNetGoogle Scholar
  7. 7.
    S. Campanato and M. K. V. Murthy, “Una generalizzazione del teorema di Riesz-Thorin,” Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., Ser. 3, 19, 87–100 (1965).zbMATHMathSciNetGoogle Scholar
  8. 8.
    F. Chiarenza and M. Frasca, “Morrey Spaces and Hardy-Littlewood Maximal Function,” Rend. Mat. Appl., Ser. 7, 7, 273–279 (1987).zbMATHMathSciNetGoogle Scholar
  9. 9.
    F. Chiarenza and A. Ruiz, “Uniform L 2-Weighted Sobolev Inequalities,” Proc. Am. Math. Soc. 112, 53–64 (1991).zbMATHMathSciNetGoogle Scholar
  10. 10.
    S. Chanillo and E. Sawyer, “Unique Continuation for Δ+υ and the C. Fefferman-Phong Class,” Trans. Am. Math. Soc. 318, 275–300 (1990).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. E. Gilbert, “Interpolation between Weighted L p-Spaces,” Ark. Mat. 10(2), 235–249 (1972).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    T. Mizuhara, “Boundedness of Some Classical Operators on Generalized Morrey Spaces,” in Harmonic Analysis: ICM 90 Satell. Conf. Proc., Ed. by S. Igari (Springer, Tokyo, 1991), pp. 183–189.Google Scholar
  13. 13.
    C. B. Morrey, Jr., “On the Solution of Quasi-linear Elliptic Partial Differential Equations,” Trans. Am. Math. Soc. 43, 126–166 (1938).zbMATHMathSciNetGoogle Scholar
  14. 14.
    E. Nakai, “Hardy-Littlewood Maximal Operator, Singular Integral Operators and Riez Potentials on Generalized Morrey Spaces,” Math. Nachr. 166, 95–103 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    J. Peetre, “On an Interpolation Theorem of Foiaş and Lions,” Acta Sci. Math. 25(3–4), 255–261 (1964).zbMATHMathSciNetGoogle Scholar
  16. 16.
    J. Peetre, “On the Theory of L p Spaces,” J. Funct. Anal. 4, 71–87 (1969).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    A. Ruiz and L. Vega, “Corrigenda to ‘Unique Continuation for Schrödinger Operators’ and a Remark on Interpolation on Morrey Spaces,” Publ. Mat., Barc. 39, 405–411 (1995).zbMATHMathSciNetGoogle Scholar
  18. 18.
    G. Stampacchia, “L p,λ-Spaces and Interpolation,” Commun. Pure Appl. Math. 17, 293–306 (1964).zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    E. M. Stein and G. Weiss, “Interpolation of Operators with Change of Measures,” Trans. Am. Math. Soc. 87(1), 159–172 (1958).zbMATHMathSciNetGoogle Scholar
  20. 20.
    M. E. Taylor, “Analysis on Morrey Spaces and Applications to Navier-Stokes and Other Evolution Equations,” Commun. Part. Diff. Eqns. 17, 1407–1456 (1992).zbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.L.N. Gumilyov Eurasian National UniversityAstanaKazakhstan

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