Some fundamental geometric and topological properties of generalized Orlicz‐Lorentz function spaces

Abstract

Generalized Orlicz-Lorentz function spaces Λϕ generated by Musielak-Orlicz functions ϕ satisfying some growth and regularity conditions (cf. [34] and [38]) are investigated. A regularity condition Δ Λ 2 for ϕ is defined in such a way that it guarantees many positive topological and geometric properties of Λϕ . The problems of the Fatou property, order continuity (separability) and the Kadec-Klee property with respect to the local convergence in measure of Λϕ are considered. Moreover, some embeddings between Λϕ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of Λϕ are presented. This paper generalizes the results from [20]. Analogous results in the sequence case were presented in [10] and [11], but the techniques in the function case are different. Investigations of structural topological and geometric properties of the generalized Orlicz-Lorentz function spaces Λ ϕ generated by special Musielak-Orlicz functions, not necessarily being weighted Orlicz functions, are initiated in this paper. Examples 2.6-2.9 on pages 1012-1014 and Examples 6.2-6.3 on page 1019 show that the class of generalized Orlicz-Lorentz spaces is much more wide than the class of Orlicz-Lorentz spaces. One of the main problems in the investigations of generalized Orlicz-Lorentz spaces Λ ϕ is to find a regularity condition of "Δ 2 -type" for the generating Musielak-Orlicz function ϕ which guarantees "good properties" of Λ ϕ . In this paper we introduce the condition Δ Λ 2 , essentialy weaker than the condition Δ 2 used in the theory of MusielakOrlicz spaces (see [7]). Unfortunately, it is not judged if this condition is the simplest possible among these ones that can guarantee the desired "good properties". The essential difficulty here is the fact that the generalized Orlicz-Lorentz spaces, as opposed to the Orlicz-Lorentz spaces, are not Calderón-Lozanovskiǐ spaces (see [16] and [36]). In consequence, some new techniques were developed in this paper.Recall that investigations of generalized Orlicz-Lorentz sequence spaces were initiated in [10] and [11]. This paper is organized as follows. In Section 1 condition (L1), which is necessary and sufficient for convexity of the functional ϕ x = γ 0 ϕ(t, x * (t)) dt is defined. Thanks to this property the generalized Orlicz-Lorentz function space Λ ϕ is a symmetric Banach space. It is also shown that if a Musielak-Orlicz function ϕ does not satisfy condition (L2), then the space Λ ϕ contains an order linearly isometric copy of l

Non-squareness properties of Orlicz–Lorentz sequence spaces

“…These spaces were introduced by A. Kamińska (see [34][35][36]) at the beginning of 1990s. Her investigations gave an impulse to further investigations of the spaces, results of which have been published among others in the papers [27,38,28,7,30,39,44,45,16,17,40,20,14,15].…”

Kadec–Klee properties of some quasi-Banach function spaces

“…4 and 6 for definitions, the generalized Orlicz-Lorentz space which is a symmetrization of the Musielak-Orlicz space-see [12,13]) the respective space E satisfies trivially this condition.…”

“…The spaces E ( * ) have been studied among others in the papers [12,13,21,22,[24][25][26]. Kamińska and Raynaud studied the relationships between the structure of E ( * ) and the structure of E (see [22]).…”

“…The geometric structure of a separated point of a Banach space (Banach lattice) has been intensively investigated recently (see [7,8,14,15,19,24,25,27,28]). On the other hand, a symmetrization E ( * ) of a quasi-Banach function space E is an important construction covering several classical classes of Banach function spaces (see [12,13,21,22,[24][25][26] We study Kadec-Klee properties with respect to global (local) convergence in measure. We concentrate on the structure of the separated point.…”

On some geometric properties of generalized Orlicz–Lorentz function spaces

“… is a symmetric Banach space (see [29]). As usual, by (Λ ϕ ) + , B(Λ ϕ ) and S(Λ ϕ ) we will denote the positive cone of Λ ϕ , the unit ball of Λ ϕ and the unit sphere of Λ ϕ , respectively.…”