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Geometric properties of Orlicz spaces equipped with \(p\)-Amemiya norms − results and open questions

100%

Wisła M.

Commentationes Mathematicae

2015

tom 55

nr 2

The classical Orlicz and Luxemburg norms generated by an Orlicz function \(\Phi\) can be defined with the use of the Amemiya formula [H. Hudzik and L. Maligranda, Amemiya norm equals Orlicz norm in general, Indag. Math. 11 (2000), no. 4, 573-585]. Moreover, in this article Hudzik and Maligranda suggested investigating a family of p-Amemiya norms defined by the formula \(\|x\|_{\Phi,p}=\inf_{k>0} \frac{1}{k} (1+I_\Phi^p(kx))^{1/p}\), where \(1\le p\le\infty\) (under the convention: \((1+u^\infty)^{1/\infty}=\lim_{p\to\infty}(1+u^p)^{1/p}=\max{1,u}\) for all \(u\ge 0\)). Based on this idea, a number of papers have been published in the past few years. In this paper, we present some major results concerning the geometric properties of Orlicz spaces equipped with p-Amemiya norms. In the last section, a more general case of Amemiya type norms is investigated. A few open questions concerning this theory will be stated as well.

Non-square points of Orlicz-Lorentz function spaces

86%

Kończak J.

Commentationes Mathematicae

2019

tom 59

nr 1-2

In this paper, criteria for non-square points in Orlicz−Lorentz function spaces \(\Lambda_{\varphi, \omega}\) endowed with the Luxemburg norm are given. The widest possible classes of convex Orlicz functions and weight functions are admitted. In consequence, criteria for non-square points in Orlicz spaces \(L^{\varphi}\), which generalize the already known results, are presented.

Ograniczanie wyników

2 Commentationes Mathematicae

1 Kończak J.

1 Wisła M.

1 2019

1 2015

Wyniki wyszukiwania

Geometric properties of Orlicz spaces equipped with \(p\)-Amemiya norms − results and open questions

Non-square points of Orlicz-Lorentz function spaces