Normal structure of Lorentz-Orlicz spaces

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

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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 ${\Lambda }_{\varphi ,w}\left(0,\infty \right)$ (respectively, ${\Lambda }_{\varphi ,w}\left(0,1\right)$) is an order continuous Lorentz-Orlicz space. (1) ${\Lambda }_{\varphi ,w}$ has normal structure if and only if u₀ = 0 (respectively, ${\int }^{v₀}\varphi \left(u₀\right)·w<2andu₀<\infty \right).$(2) ${\Lambda }_{\varphi ,w}$ has weakly normal structure if and only if ${\int }_{0}^{v₀}\varphi \left(u₀\right)·w<2$.

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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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
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