Czechoslovak Mathematical Journal

, Volume 64, Issue 3, pp 581–597 | Cite as

Normability of Lorentz spaces—an alternative approach



We study normability properties of classical Lorentz spaces. Given a certain general lattice-like structure, we first prove a general sufficient condition for its associate space to be a Banach function space. We use this result to develop an alternative approach to Sawyer’s characterization of normability of a classical Lorentz space of type Λ. Furthermore, we also use this method in the weak case and characterize normability of Λ v . Finally, we characterize the linearity of the space Λ v by a simple condition on the weight v.


weighted Lorentz space weighted inequality non-increasing rearrangement Banach function space associate space 

MSC 2010



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  1. [1]
    C. Bennett, R. Sharpley: Interpolation of Operators. Pure and Applied Mathematics 129, Academic Press, Boston, 1988.MATHGoogle Scholar
  2. [2]
    M. Carro, L. Pick, J. Soria, V. D. Stepanov: On embeddings between classical Lorentz spaces. Math. Inequal. Appl. 4 (2001), 397–428.MATHMathSciNetGoogle Scholar
  3. [3]
    M. Cwikel, A. Kamińska, L. Maligranda, L. Pick: Are generalized Lorentz “spaces” really spaces? Proc. Am. Math. Soc. 132 (2004), 3615–3625.CrossRefMATHGoogle Scholar
  4. [4]
    A. Gogatishvili, L. Pick: Embeddings and duality theorem for weak classical Lorentz spaces. Can. Math. Bull. 49 (2006), 82–95.CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    A. Gogatishvili, L. Pick: Discretization and anti-discretization of rearrangement-invariant norms. Publ. Mat., Barc. 47 (2003), 311–358.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    G. G. Lorentz: On the theory of spaces Λ. Pac. J. Math. 1 (1951), 411–429.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    E. Sawyer: Boundedness of classical operators on classical Lorentz spaces. Stud. Math. 96 (1990), 145–158.MATHMathSciNetGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2014

Authors and Affiliations

  1. 1.Institute of MathematicsAcademy of Sciences of the Czech RepublicPraha 1Czech Republic
  2. 2.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles University in PraguePraha 8Czech Republic

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