Normal structure of Lorentz-Orlicz spaces

Pei-Kee Lin; Huiying Sun

Annales Polonici Mathematici (1997)

  • Volume: 67, Issue: 2, page 147-168
  • ISSN: 0066-2216

Abstract

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Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that Λ ϕ , w ( 0 , ∞ ) (respectively, Λ ϕ , w ( 0 , 1 ) ) is an order continuous Lorentz-Orlicz space. (1) Λ ϕ , w has normal structure if and only if u₀ = 0 (respectively, ∫ v ₀ ϕ ( u ₀ ) · w < 2 a n d u ₀ < ∞ ) . (2) Λ ϕ , w has weakly normal structure if and only if ∫ 0 v ₀ ϕ ( u ₀ ) · w < 2 .

How to cite

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Pei-Kee Lin, and Huiying Sun. "Normal structure of Lorentz-Orlicz spaces." Annales Polonici Mathematici 67.2 (1997): 147-168. <http://eudml.org/doc/270441>.

@article{Pei1997,
abstract = {Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that $Λ_\{ϕ,w\}(0,∞)$ (respectively, $Λ_\{ϕ,w\}(0,1)$) is an order continuous Lorentz-Orlicz space. (1) $Λ_\{ϕ,w\}$ has normal structure if and only if u₀ = 0 (respectively, $∫_^\{v₀\} ϕ(u₀) · w < 2 and u₀ <∞). $(2) $Λ_\{ϕ,w\}$ has weakly normal structure if and only if $∫_0^\{v₀\} ϕ(u₀)· w < 2$.},
author = {Pei-Kee Lin, Huiying Sun},
journal = {Annales Polonici Mathematici},
keywords = {Lorentz-Orlicz space; normal sturcture; order continuous; Young function; normal structure; weight function; order continuous Lorentz-Orlicz space},
language = {eng},
number = {2},
pages = {147-168},
title = {Normal structure of Lorentz-Orlicz spaces},
url = {http://eudml.org/doc/270441},
volume = {67},
year = {1997},
}

TY - JOUR
AU - Pei-Kee Lin
AU - Huiying Sun
TI - Normal structure of Lorentz-Orlicz spaces
JO - Annales Polonici Mathematici
PY - 1997
VL - 67
IS - 2
SP - 147
EP - 168
AB - Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that $Λ_{ϕ,w}(0,∞)$ (respectively, $Λ_{ϕ,w}(0,1)$) is an order continuous Lorentz-Orlicz space. (1) $Λ_{ϕ,w}$ has normal structure if and only if u₀ = 0 (respectively, $∫_^{v₀} ϕ(u₀) · w < 2 and u₀ <∞). $(2) $Λ_{ϕ,w}$ has weakly normal structure if and only if $∫_0^{v₀} ϕ(u₀)· w < 2$.
LA - eng
KW - Lorentz-Orlicz space; normal sturcture; order continuous; Young function; normal structure; weight function; order continuous Lorentz-Orlicz space
UR - http://eudml.org/doc/270441
ER -

References

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  9. [9] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. Zbl0141.32402
  10. [10] T. Landes, Permanence properties of normal structure, Pacific J. Math. 110 (1984), 125-143. Zbl0534.46015
  11. [11] T. Landes, Normal structure and weakly normal structure of Orlicz sequence spaces, Trans. Amer. Math. Soc. 285 (1984), 523-534. Zbl0594.46010
  12. [12] P.-K. Lin and H. Y. Sun, Some geometric properties of Lorentz-Orlicz spaces, Arch. Math. (Basel) 64 (1995), 500-511. Zbl0823.46019
  13. [13] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, 1979. Zbl0403.46022
  14. [14] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, 1991. 

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