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Geometric properties of Orlicz spaces equipped with p-Amemiya norms : results and open questions

Wisła M.

Commentationes Mathematicae

2015

Vol 55, No. 2

183--209

The classical Orlicz and Luxemburg norms generated by an Orlicz function Φ 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∥Φ,p=infk>011/k(1+IpΦ(kx))1/p, where 1≤p≤∞ (under the convention: (1+u∞)1/∞=limp→∞(1+up)1/p=max1,u for all u≥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.

On Property β of Rolewicz in Köthe–Bochner Function Spaces

Kolwicz P.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 1

75-85

It is proved that the Köthe–Bochner function space E(X) has property β if and only if X is uniformly convex and E has property β. In particular, property β does not lift from X to E(X) in contrast to the case of Köthe–Bochner sequence spaces.

Orthogonal uniform convexity in Köthe spaces and Orlicz spaces

Kolwicz P.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 4

395-411

We study a geometric property in Köthe spaces which is called orthogonal uniform convexity (UC┴). It was introduced in [19]. We prove that the class of Köthe spaces with property (UC┴) is a proper subset of the class of uniformly monotone and P-convex Köthe spaces. Next we consider connections between (UC┴) and property (β) of Rolewicz. We shown that the implication (UC┴) → (β) is true in any Köthe sequence space. Moreover, we find criteria for Orlicz function (sequence) spaces to be orthogonally uniformly convex. As a corollary we get that there holds (UC) → (UC┴) → (β) in any Köthe sequence space and the converse of any of these implications is not true. Furthermore, the implications (UC) → (β) → (UC┴) hold in any Köthe function space and the second one cannot be reversed.