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ArticleOriginal scientific text

Title

Comparison of Orlicz-Lorentz spaces

Authors 1

Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, Missouri 65211, U.S.A.

Abstract

Orlicz-Lorentz spaces provide a common generalization of Orlicz spaces and Lorentz spaces. They have been studied by many authors, including Mastyło, Maligranda, and Kamińska. In this paper, we consider the problem of comparing the Orlicz-Lorentz norms, and establish necessary and sufficient conditions for them to be equivalent. As a corollary, we give necessary and sufficient conditions for a Lorentz-Sharpley space to be equivalent to an Orlicz space, extending results of Lorentz and Raynaud. We also give an example of a rearrangement invariant space that is not an Orlicz-Lorentz space.

Bibliography

  1. [B-R] C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. 175 (1980).
  2. [B-S] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
  3. [H] R. A. Hunt, On L(p,q) spaces, Enseign. Math. (2) 12 (1966), 249-275.
  4. [Ka1] A. Kamińska, Some remarks on Orlicz-Lorentz spaces, Math. Nachr., to appear.
  5. [Ka2] A. Kamińska, Extreme points in Orlicz-Lorentz spaces, Arch. Math. (Basel), to appear.
  6. [Ka3] A. Kamińska, Uniform convexity of generalized Lorentz spaces, ibid., to appear.
  7. [K-R] M. A. Krasnosel'skiĭ and Ya. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff, 1961.
  8. [Lo1] G. G. Lorentz, Some new function spaces, Ann. of Math. 51 (1950), 37-55.
  9. [Lo2] G. G. Lorentz, On the theory of spaces Λ, Pacific J. Math. 1 (1951), 411-429.
  10. [Lo3] G. G. Lorentz, Relations between function spaces, Proc. Amer. Math. Soc. 12 (1961), 127-132.
  11. [Lu] W. A. J. Luxemburg, Banach Function Spaces, Thesis, Delft Technical Univ., 1955.
  12. [Ma] L. Maligranda, Indices and interpolation, Dissertationes Math. 234 (1984).
  13. [My] M. Mastyło, Interpolation of linear operators in Calderón-Lozanovskii spaces, Comment. Math. 26 (2) (1986), 247-256.
  14. [Mo1] S. J. Montgomery-Smith, The Cotype of Operators from C(K), Ph.D. thesis, Cambridge, 1988.
  15. [Mo2] S. J. Montgomery-Smith, Boyd indices of Orlicz-Lorentz spaces, in preparation.
  16. [O] W. Orlicz, Über eine gewisse Klasse von Räumen vom Typus B, Bull. Intern. Acad. Pol. 8 (1932), 207-220.
  17. [R] Y. Raynaud, On Lorentz-Sharpley spaces, in: Proc. of the Workshop "Interpolation Spaces and Related Topics", Haifa, June 1990, Amer. Math. Soc., to appear.
  18. [S] R. Sharpley, Spaces Λα(X) and interpolation, J. Funct. Anal. 11 (1972), 479-513.
  19. [T] A. Torchinsky, Interpolation of operations and Orlicz classes, Studia Math. 59 (1976), 177-207.
Studia Mathematica
1992/103/2
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Pages:
161-189
Main language of publication
English
Received
1991-03-04
Accepted
1991-08-17
Published
1992
Exact and natural sciences