Skip to main content
Log in

Embeddings for Morrey–Lorentz Spaces

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, new classes of functions are defined. These spaces generalize Lorentz spaces and give a refinement of Lebesgue spaces, weak-Lebesgue spaces, and Morrey spaces. Some embeddings between these new classes are also proved.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Lorentz, G.G.: Some new functional spaces. Ann. Math. 51, 37–55 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  2. Lorentz, G.G.: On the theory of spaces Λ. Pac. J. Math. 1, 411–429 (1951)

    MathSciNet  MATH  Google Scholar 

  3. Zygmund, A.: Trigonometric Series, 2nd edn. Cambridge University Press, New York (1958)

    Google Scholar 

  4. Hunt, R.A., Weiss, G.: The Marcinkiewicz interpolation theorem. Proc. Am. Math. Soc. 15, 996–998 (1964)

    MathSciNet  MATH  Google Scholar 

  5. Miao, C.X., Yuang, B.Q.: Weak Morrey spaces and strong solutions to the Navier–Stokes equations. Sci. China Ser. A 50(10), 1401–1417 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ragusa, M.A.: Regularity for weak solutions to the Dirichlet problem in Morrey space. Riv. Mat. Univ. Parma 5(3), 355–369 (1994)

    MathSciNet  Google Scholar 

  7. Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Studia Math. 24, 113–190 (1964)

    MathSciNet  MATH  Google Scholar 

  8. Peetre, J.: Nouvelles proprietes d’espaces d’interpolation. C. R. Acad. Sci. 256, 1424–1426 (1963)

    MathSciNet  MATH  Google Scholar 

  9. Lions, J.-L., Peetre, J.: Sur une classe d’espaces d’interpolation. Inst. Hautes Études Sci. Publ. Math. 19, 5–68 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  10. Riviére, N.M., Sagher, Y.: Interpolation operator between L and H 1, the real method. J. Funct. Anal. 14, 401–409 (1973)

    Article  MATH  Google Scholar 

  11. Krein, S.G., Petunin, J.J., Yu, I., Semenov, E.M.: Interpolation of linear operators. Translation of Math. Monographs, vol. 54. Am. Math. Soc., Providence (1982)

    Google Scholar 

  12. Krein, S.G., Petunin, J.J.: Scales of Banach spaces. Usp. Math. Nauk 21(2), 89–168 (1966)

    MathSciNet  MATH  Google Scholar 

  13. Magenes, E.: Spazi d’interpolazione ed equazioni a derivate parziali. In: Atti Cong. Un. Mat. Ital., Genova, pp. 194–197 (1963)

    Google Scholar 

  14. Hardy, H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1934)

    Google Scholar 

  15. Hardy, H., Littlewood, J.E.: A maximal theorem with function-theoretic applications. Acta Math. 54, 81–116 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grothendieck, A.: Réarrengments de fonctions et ińegalités de convexité dans les algèbres de von Neumann munies d’une trace. In: Séminaire Boubaki, vol. 113 (1955)

    Google Scholar 

  17. Luxemburg, W.A.J.: Rearrangement-invariant Banach function spaces. In: Proc. Sympos. in Analysis, Queen’s Papers in Pure and Appl. Math., vol. 10, pp. 83–144 (1967)

    Google Scholar 

  18. Ryff, J.V.: Orbits of L 1-functions under doubly stochastic transformations. Trans. Am. Math. Soc. 117, 92–100 (1965)

    MathSciNet  MATH  Google Scholar 

  19. Ryff, J.V.: Measure preserving transformations and rearrangements. J. Math. Anal. Appl. 31, 449–458 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  20. Day, P.W.: Rearrangements of measurable functions. Thesis, California Institute of Technology (1970)

  21. Day, P.W.: Decreasing rearrangements and doubly stochastic operators. Trans. Am. Math. Soc. 178, 383–392 (1973)

    Article  MATH  Google Scholar 

  22. Chong, K.M., Rice, N.M.: Equimeasurable Rearrangements of Functions. Queen’s Papers in Pure and Appl. Math., vol. 28. Queen’s University, Kingston (1971)

    MATH  Google Scholar 

  23. Kokilashvili, V., Krbec, M.: Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific, Singapore (1991)

    Book  MATH  Google Scholar 

  24. Genebashvili, I., Gogatishvili, A., Kokilashvili, V., Krbec, M.: Weight theory for integral transforms on spaces of homogeneous type. In: Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 92. Longman, Harlow (1998)

    Google Scholar 

  25. Krbec, M., Lang, J.: On embeddings between weighted Orlicz–Lorentz spaces. Georgian Math. J. 4, 117–128 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Agarval, R., O’Regan, D., Shakhmurov, V.B.: Separable anisotropic differential operators in weighted abstract spaces and applications. J. Math. Anal. Appl. 338, 970–983 (2008)

    Article  MathSciNet  Google Scholar 

  27. Shakhmurov, V.B.: Embeddings and separable differential operators in spaces of Sobolev–Lions type. Math. Notes 84(6), 842–858 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shakhmurov, V.B., Shahmurova, A.: Nonlinear abstract boundary value problems modelling atmospheric dispersion of pollutants. Nonlinear Anal., Real World Appl. 11, 932–951 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Torchinsky, A.: Real Variable Methods in Harmonic Analysis. Academic Press, San Diego (1986)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Ragusa, M.A.: C (0,α)-regularity of the solution of the Dirichlet problem for elliptic equations in divergence form. Int. J. Differ. Equ. Appl. 1, 113–125 (2000)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is greatly indebted to the anonymous referee for the helpful ideas and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Maria Alessandra Ragusa.

Additional information

Communicated by Michel Théra.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ragusa, M.A. Embeddings for Morrey–Lorentz Spaces. J Optim Theory Appl 154, 491–499 (2012). https://doi.org/10.1007/s10957-012-0012-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0012-y

Keywords

Navigation