Best Constants for the Inequalities between Equivalent Norms in Orlicz Spaces (original) (raw)

Analytic norms in Orlicz spaces

Proceedings of the American Mathematical Society, 2001

It is shown that an Orlicz sequence space h M admits an equivalent analytic renorming if and only if it is either isomorphic to l 2n or isomorphically polyhedral. As a consequence, we show that there exists a separable Banach space admitting an equivalent C ∞-Fréchet norm, but no equivalent analytic norm.

Some Properties of Orlicz-Lorentz Spaces

2011

In this paper we study some fundamental properties of OrliczLorentz spaces defined on R such as finding their dual spaces, giving best constants for the inequalities between the Orlicz norm and the Luxemburg norm on Orlicz-Lorentz spaces and establishing the Kolmogorov inequality in these spaces. 1. Orlicz-Lorentz spaces Orlicz-Lorentz spaces as a generalization of Orlicz spaces Lφ and Lorentz spaces Λω have been studied by many authors (we refer to [9, 10, 11, 12, 14, 18, 19] for basic properties of OrliczLorentz spaces as well to the references therein). In this paper we study some fundamental properties of Orlicz-Lorentz spaces defined on the real line ΛRφ,ω. We first find the dual spaces of Λ R φ,ω. Note that the dual spaces of Orlicz-Lorentz spaces defined on (0,+∞) or (0, 1) were studied in [11]. Next we introduce the Orlicz norm on ΛRφ,ω which defined by using the M R φ∗,ω space and then we give a simple formula to calculate the Orlicz norm directly by φ,ω. On Orlicz spaces, ...

Local geometric properties in quasi-normed Orlicz spaces

Banach Center Publications, 2019

Several local geometric properties of Orlicz space L φ are presented for an increasing Orlicz function φ which is not necessarily convex, and thus L φ does not need to be a Banach space. In addition to monotonicity of φ it is supposed that φ(u 1/p) is convex for some p > 0 which is equivalent to that its lower Matuszewska-Orlicz index α φ > 0. Such spaces are locally bounded and are equipped with natural quasi-norms. Therefore many local geometric properties typical for Banach spaces can also be studied in those spaces. The techniques however have to be different, since duality theory cannot be applied in this case. In this article we present complete criteria, in terms of growth conditions of φ, for L φ to have type 0 < p ≤ 2, cotype q ≥ 2, to be (order) p-convex or q-concave, to have an upper p-estimate or a lower q-estimate, for 0 < p, q < ∞. We provide detailed proofs of most results, avoiding appealing to general not necessary theorems.

Sharp Conditions for Korn Inequalities in Orlicz Spaces

Journal of Mathematical Fluid Mechanics, 2012

We show that Korn's inequality in Orlicz spaces holds if and only if the Orlicz function satisfies the ∆ 2 -and the ∇ 2 -condition. This result applies to several types of Korn's inequality. In particular we show that Korn's inequality is false in L 1 , in L log L, in Exp and in L ∞ .

Uniformly convex and strictly convex Orlicz spaces

2016

In this paper we define the new norm ofOrlicz spaces on through a multiplication operator on an old Orlicz spaces. We obtain some necessary and sufficient conditions that the new norm to be a uniformly convex and strictly convex spaces.

On the WM property of Orlicz sequence spaces endowed with the Orlicz norm

Commentationes Mathematicae Universitatis Carolinae

We obtain the criterion of the WM property for Orlicz sequence spaces en- dowed with the Orlicz norm. Classification: 46B30, 46E30 B.B. Panda and O.P. Kapoor (1) introduced the concept of the WM property in 1975. The WM property is an important property in geometry of Banach spaces. Some criteria of WM properties for Orlicz function spaces endowed with the Lux- emburg norm and Orlicz norm have been discussed in (2) and (4), respectively. Moreover, the criterion of the WM property in Orlicz sequence spaces endowed with the Luxemburg norm has also been discussed in (3). A remained problem is the WM property of Orlicz sequence spaces endowed with the Orlicz norm. In this paper, we shall give the criterion of the WM property in Orlicz sequence spaces equipped with the Orlicz norm. Let X be a Banach space, and let B(X) and S(X) denote the unit ball and unit sphere of X, respectively. X is said to have the W M property if for any x ∈ S(X), xn ∈ B(X) (n ∈ N), k xn + xk → 2 implies that the...

On uniform convexity of Orlicz spaces

Indagationes Mathematicae (Proceedings), 1982

In the present paper we present criteria for uniform convexity of Orlicz spaces in the case of a nonatomic measure as well as in the case of a purely atomic measure.

Complete characterization of Kadec-Klee properties in Orlicz spaces

2003

We study for Orlicz function spaces, equipped with either the Luxemburg norm or the Orlicz norm, the connection between the Kadec-Klee property for local convergence in measure, the Kadec-Klee property for global convergence in measure, and some properties of the Orlicz function which defines the space.

On 2-Normed Space Valued Orlicz Space

2014

The aim of this paper is to introduce and study a new class  ((S, ||. , .|| ), , w – ) of 2normed space valued sequences using Orlicz function as a generalization of the basic space  of bounded complex sequences studied in Functional Analysis. Besides the investigation of conditions pertaining to the containment relation of the class  ((S, ||.,.|| ), , w – ) in terms of different w– , our primary interest is to explore the linear space structures of the class  ((S, ||. , .|| ), , w – ) with some topological properties.