Positivity

, Volume 10, Issue 3, pp 539–554 | Cite as

Mixed Norm and Multidimensional Lorentz Spaces

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

Abstract

In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved ([16], [7]). However, the question for multidimensional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the two-dimensional decreasing rearrangement, to be normable. To this end, it is also useful to study the mixed norm Lorentz spaces. Finally, we prove embeddings between all the classical, multidimensional, and mixed norm Lorentz spaces.

Mathematics Subject Classification (2000)

46E30 46B25 

Keywords

Function spaces Lorentz spaces mixed norm spaces rearrangement weighted inequalities 

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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  • Sorina Barza
    • 1
  • Anna Kamińska
    • 2
  • Lars-Erik Persson
    • 3
  • Javier Soria
    • 4
  1. 1.Dept. of Mathematics, Physics and Eng. SciencesKarlstad UniversityKarlstadSweden
  2. 2.Dept. of Mathematical SciencesThe University of MemphisMemphisUSA
  3. 3.Dept. of MathematicsLuleå University of TechnologyLuleåSweden
  4. 4.Dept. Appl. Math. and AnalysisUniversity of BarcelonaBarcelonaSpain

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