Amemiya Norm Equals Orlicz Norm in Musielak–Orlicz Spaces

2006 ◽  
Vol 23 (2) ◽  
pp. 281-288 ◽  
Author(s):  
Xian Ling Fan
Keyword(s):  
Orlicz Spaces ◽  
Orlicz Norm ◽  
Amemiya Norm

scholarly journals Smoothness of Orlicz function spaces equipped with the p-Amemiya norm

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Xiaoyan Li ◽  
Yunan Cui ◽  
Marek Wisla
Keyword(s):  
Explicit Form ◽  
Necessary Conditions ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Luxemburg Norm ◽  
Orlicz Norm ◽  
Dual Norm ◽  
Sufficient And Necessary Conditions ◽  
Supporting Functional ◽  
Amemiya Norm

AbstractIn this paper, we will use the convex modular $$\rho ^{*}(f)$$ ρ ∗ ( f ) to investigate $$\Vert f\Vert _{\Psi ,q}^{*}$$ ‖ f ‖ Ψ , q ∗ on $$(L_{\Phi })^{*}$$ ( L Φ ) ∗ defined by the formula $$\Vert f\Vert _{\Psi ,q}^{*}=\inf _{k>0}\frac{1}{k}s_{q}(\rho ^{*}(kf))$$ ‖ f ‖ Ψ , q ∗ = inf k > 0 1 k s q ( ρ ∗ ( k f ) ) , which is the norm formula in Orlicz dual spaces equipped with p-Amemiya norm. The attainable points of dual norm $$\Vert f\Vert _{\Psi ,q}^{*}$$ ‖ f ‖ Ψ , q ∗ are discussed, the interval for dual norm $$\Vert f\Vert _{\Psi ,q}^{*}$$ ‖ f ‖ Ψ , q ∗ attainability is described. By presenting the explicit form of supporting functional, we get sufficient and necessary conditions for smooth points. As a result, criteria for smoothness of $$L_{\Phi ,p}~(1\le p\le \infty )$$ L Φ , p ( 1 ≤ p ≤ ∞ ) is also obtained. The obtained results unify, complete and extended as well the results presented by a number of paper devoted to studying the smoothness of Orlicz spaces endowed with the Luxemburg norm and the Orlicz norm separately.

scholarly journals Extreme points of the Besicovitch-Orlicz space of almost periodic functions equipped with Orlicz norm

2021 ◽  
Vol 41 (5) ◽  
pp. 629-648
Author(s):  
Fatiha Boulahia ◽  
Slimane Hassaine
Keyword(s):  
Orlicz Space ◽  
Extreme Points ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Orlicz Norm ◽  
Amemiya Norm

In the present paper, we give criteria for the existence of extreme points of the Besicovitch-Orlicz space of almost periodic functions equipped with Orlicz norm. Some properties of the set of attainable points of the Amemiya norm in this space are also discussed.

Maximal inequalities of Kahane–Khintchine's type in Orlicz spaces

1994 ◽  
Vol 115 (1) ◽  
pp. 175-190 ◽  
Author(s):  
Goran Peškir
Keyword(s):  
Invariance Principle ◽  
Orlicz Spaces ◽  
Orlicz Norm ◽  
Numerical Constant ◽  
Maximal Inequalities ◽  
Sharp Estimates ◽  
Bernoulli Sequence ◽  
Image Position ◽  
Donsker’S Invariance Principle

AbstractSeveral maximal inequalities of Kahane–Khintchine's type in certain Orlicz spaces are proved. The method relies upon Lévy's inequality and the technique established in [14] which is obtained by Haagerup–Young–Stechkin's best possible constants in the classical Khintchine inequalities. Moreover by using Donsker's invariance principle it is shown that the numerical constant in the inequality deduced by the method presented is nearly optimal: If is a Bernoulli sequence, and ‖ · ‖ψ denotes the Orlicz norm induced by the function then the following inequality is satisfied:for all a1,…, an and all n ≥ 1, and the best possible numerical constant which can take the place of lies in the interval ]. Sharp estimates of this type are also deduced for some other maximal inequalities in Orlicz spaces discovered in this paper.

Non-squareness properties of Orlicz spaces equipped with the -Amemiya norm

2012 ◽  
Vol 75 (10) ◽  
pp. 3973-3993 ◽  
Author(s):  
Yunan Cui ◽  
Henryk Hudzik ◽  
Marek Wisła ◽  
Karol Wlaźlak
Keyword(s):  
Orlicz Spaces ◽  
Amemiya Norm

scholarly journals On the weakly strongly exposed property and some smoothness properties of Orlicz spaces

1996 ◽  
Vol 54 (3) ◽  
pp. 431-440
Author(s):  
Yunan Cui ◽  
Henry K. Hudzik ◽  
Hongwei Zhu
Keyword(s):  
Banach Space ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Function Spaces ◽  
Luxemburg Norm ◽  
Orlicz Norm ◽  
Dual Property ◽  
Order Continuous

The notion of a weakly strongly exposed Banach space is introduced and it is shown that this property is the dual property of very smoothness. Criteria for this property in Orlicz function spaces equipped with the Orlicz norm are presented. Criteria for strong smoothness and very smoothness of their subspaces of order continuous elements in the case of the Luxemburg norm are also given.

scholarly journals Amemiya norm equals Orlicz norm in general

2000 ◽  
Vol 11 (4) ◽  
pp. 573-585 ◽  
Author(s):  
Henryk Hudzik ◽  
Lech Maligranda
Keyword(s):  
Orlicz Norm ◽  
Amemiya Norm
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