Acta Mathematica Hungarica

, Volume 104, Issue 3, pp 203–224 | Cite as

Multidimensional rearrangement and Lorentz spaces

  • Sorina Barza
  • Lars-Erik Persson
  • Javier Soria
Article

Abstract

We define a multidimensional rearrangement, which is related to classical inequalities for functions that are monotone in each variable. We prove the main measure theoretical results of the new theory and characterize the functional properties of the associated weighted Lorentz spaces.

rearrangement function spaces Lorentz spaces monotone functions weighted inequalities 

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References

  1. [1]
    M. A. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc., 320 (1990), 727–735.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    S. Barza, Weighted multidimensional integral inequalities and applications, Ph.D. Thesis, Luleâ University (1999).Google Scholar
  3. [3]
    S. Barza, L. E. Persson and J. Soria, Sharp weighted multidimensional integral inequalities for monotone functions, Math. Nachr., 210 (2000), 43–58.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    S. Barza, L. E. Persson and V. Stepanov, On weighted multidimensional embeddings for monotone functions, Math. Scand., 88 (2001), 303–319.MATHMathSciNetGoogle Scholar
  5. [5]
    C. Bennet and R. Sharpley, Interpolation of Operators, Academic Press (1988).Google Scholar
  6. [6]
    A. P. Blozinski, Multivariate rearrangements and Banach function spaces with mixed norms, Trans. Amer. Math. Soc., 263 (1981), 149–167.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    M. J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal., 112 (1993), 480–494.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14}. American Mathematical Society (Providence, RI, 200Google Scholar
  9. [9]
    G. G. Lorentz, On the theory of spaces Λ, Pacific J. Math., 1 (1951), 411–429.MATHMathSciNetGoogle Scholar
  10. [10]
    E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math., 96 (1990), 145–158.MATHMathSciNetGoogle Scholar
  11. [11]
    V. Stepanov, The weighted Hardy's inequality for nonincreasing functions, Trans. Amer. Math. Soc., 338 (1993), 173–186.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publisher/Akadémiai Kiadó 2004

Authors and Affiliations

  • Sorina Barza
    • 1
  • Lars-Erik Persson
    • 2
  • Javier Soria
    • 3
  1. 1.Department of MathematicsKarlstad UniversityKarlstadSweden
  2. 2.Department of MathematicsLuleå UniversityLuleåSweden
  3. 3.Department of Applied Mathematics and AnalysisUniversity of BarcelonaBarcelonaSpain

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