Comparison of Orlicz-Lorentz spaces

S. Montgomery-Smith

Studia Mathematica (1992)

  • Volume: 103, Issue: 2, page 161-189
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Montgomery-Smith, S.. "Comparison of Orlicz-Lorentz spaces." Studia Mathematica 103.2 (1992): 161-189. <http://eudml.org/doc/215943>.

@article{Montgomery1992,
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.},
author = {Montgomery-Smith, S.},
journal = {Studia Mathematica},
keywords = {Orlicz-Lorentz spaces; comparing the Orlicz-Lorentz norms; Lorentz- Sharpley space; rearrangement invariant space},
language = {eng},
number = {2},
pages = {161-189},
title = {Comparison of Orlicz-Lorentz spaces},
url = {http://eudml.org/doc/215943},
volume = {103},
year = {1992},
}

TY - JOUR
AU - Montgomery-Smith, S.
TI - Comparison of Orlicz-Lorentz spaces
JO - Studia Mathematica
PY - 1992
VL - 103
IS - 2
SP - 161
EP - 189
AB - 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.
LA - eng
KW - Orlicz-Lorentz spaces; comparing the Orlicz-Lorentz norms; Lorentz- Sharpley space; rearrangement invariant space
UR - http://eudml.org/doc/215943
ER -

References

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  1. [B-R] C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. 175 (1980). Zbl0456.46028
  2. [B-S] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988. Zbl0647.46057
  3. [H] R. A. Hunt, On L(p,q) spaces, Enseign. Math. (2) 12 (1966), 249-275. Zbl0181.40301
  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. Zbl0636.46064
  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. Zbl0838.46024
  16. [O] W. Orlicz, Über eine gewisse Klasse von Räumen vom Typus B, Bull. Intern. Acad. Pol. 8 (1932), 207-220. Zbl0006.31503
  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 Λ α ( X ) and interpolation, J. Funct. Anal. 11 (1972), 479-513. Zbl0245.46043
  19. [T] A. Torchinsky, Interpolation of operations and Orlicz classes, Studia Math. 59 (1976), 177-207. Zbl0348.46027

Citations in EuDML Documents

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  1. Miroslav Krbec, Thomas Schott, Superposition of imbeddings and Fefferman's inequality
  2. Beata Randrianantoanina, Isometric classification of norms in rearrangement-invariant function spaces
  3. P. Hitczenko, S. Montgomery-Smith, K. Oleszkiewicz, Moment inequalities for sums of certain independent symmetric random variables
  4. N. Kalton, Calderón couples of rearrangement invariant spaces
  5. Jesús Bastero, Francisco Ruiz, Interpolation of operators when the extreme spaces are L ∞
  6. Jeff Hogan, Chun Li, Alan McIntosh, Kewei Zhang, Global higher integrability of jacobians on bounded domains
  7. Andrea Cianchi, Optimal integrability of the Jacobian of orientation preserving maps

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