Regular Banach space net and abstract-valued Orlicz space of range-varying type

Qinghua Zhang,Yueping Zhu

doi:10.1515/math-2019-0130

Abstract

Abstract This paper investigates the abstract-valued Orlicz space of range-varying type. We firstly give the notions and examples of partially continuous modular net and regular Banach space net of type (II), then deal with the definitions, constructions, and geometrical properties of the range-varying Orlicz spaces, including representation of the dual $\begin{array}{} L_{+}^{\varphi} \end{array}$(I, Xθ(⋅))*, and reflexivity of Lφ(I, Xθ(⋅)), under some reasonable conditions. As an application, we finally make another approach to the real interpolation spaces constructed by a generalized Φ-function.

Highlights

Introduction and preliminariesThis paper is devoted to studying the abstract-valued Orlicz space of range-varying type
This paper investigates the abstract-valued Orlicz space of range-varying type
Orlicz space is a special type of semimodular space, where the semimodular ρφ is commonly constructed by a generalized Φ−function φ, namely ρφ(f ) = φ(t, |f (t)|)dμ, f ∈ L (I, μ), I

Summary

Introduction and preliminaries

This paper is devoted to studying the abstract-valued Orlicz space of range-varying type. This work is licensed under the Creative net {Xα : α ∈ A}, and order-continuous map θ : I → A Based on these notions, they introduced a suitably measurable Xθ(·)−valued function space L (I, Xθ(·)) on an interval I. Continuity and successive assumption are revised slightly After these modi cations, except for (1.1), a lot of space families including the complex interpolation series {[X , X ]s : s ∈ [ , ]} and the real interpolation series {(X , X )p,s : < a ≤ s ≤ b < } become regular Banach space nets. Equivalence (1.2) is a natural but not trivial extension of the corresponding result from the scalar case to the vector-valued case Based on this extension, representation of the dual space of the range-varying Orlicz space constructed by the regular Banach space net (II) is derived, that is. Taking into account that φ is only a locally integral generalized Φ−function, and the extra assumption that X*α is norm-attainable is dropped here, (1.2) can be viewed as an improvement of that in [11]

Qinghua Zhang and Yueping Zhu

Application in real interpolation spaces

If p φ

Banach spaces

Denote by