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

2019 ◽  
Vol 17 (1) ◽  
pp. 1680-1702
Author(s):  
Qinghua Zhang ◽  
Yueping Zhu
Keyword(s):  
Banach Space ◽  
Orlicz Space ◽  
Orlicz Spaces ◽  
Real Interpolation ◽  
Interpolation Spaces ◽  
Type Ii ◽  
The Real ◽  
Geometrical Properties

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.

scholarly journals On continuity properties of semigroups in real interpolation spaces

2020 ◽  
Author(s):  
Peer Christian Kunstmann
Keyword(s):  
Banach Space ◽  
Schrodinger Equation ◽  
Continuous Semigroup ◽  
Stokes Equations ◽  
Nonlinear Schrodinger Equation ◽  
Real Interpolation ◽  
Navier Stokes ◽  
Interpolation Spaces ◽  
Navier Stokes Equations ◽  
Continuity Properties

AbstractStarting from a bi-continuous semigroup in a Banach space X (which might actually be strongly continuous), we investigate continuity properties of the semigroup that is induced in real interpolation spaces between X and the domain D(A) of the generator. Of particular interest is the case $$(X,D(A))_{\theta ,\infty }$$ ( X , D ( A ) ) θ , ∞ . We obtain topologies with respect to which the induced semigroup is bi-continuous, among them topologies induced by a variety of norms. We illustrate our results with applications to a nonlinear Schrödinger equation and to the Navier–Stokes equations on $$\mathbb {R}^d$$ R d .

scholarly journals Reiteration Formulae for the Real Interpolation Method Including L or R Limiting Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Leo R. Ya. Doktorski
Keyword(s):  
Interpolation Method ◽  
Lorentz Spaces ◽  
Real Interpolation ◽  
Interpolation Spaces ◽  
The Real ◽  
Real Interpolation Method

We consider a real interpolation method defined by means of slowly varying functions. We present some reiteration formulae including so-called L or R limiting interpolation spaces. These spaces arise naturally in reiteration formulae for the limiting cases θ = 0 or θ = 1 . Applications to grand and small Lorentz spaces are given.

scholarly journals Functional Calculus on Real Interpolation Spaces for the Series Generators of C_0-groups

2019 ◽  
Vol 11 (5) ◽  
pp. 52
Author(s):  
Simon Joseph ◽  
Manal Juma ◽  
Isra Mukhtar ◽  
Nagat Suoliman ◽  
Fatin Saeed
Keyword(s):  
Banach Space ◽  
Functional Calculus ◽  
Real Interpolation ◽  
Interpolation Spaces ◽  
Transference Principle

In this paper, discus functional calculus properties of C_0-groups on real interpolation spaces using transference principles. Obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference principle for unbounded groups. Then showed in (Markus, H., & Jan, R. 2016) that all group sequence of generators on a Banach space has a bounded H_0^∞-calculus on real interpolation spaces. Additional results are derived from this.

scholarly journals Abstract-valued Orlicz spaces of range-varying type

2018 ◽  
Vol 16 (1) ◽  
pp. 924-954 ◽  
Author(s):  
Qinghua Zhang
Keyword(s):  
Banach Space ◽  
Elliptic Equation ◽  
Differential Inclusion ◽  
Sobolev Spaces ◽  
Nonlinear Elliptic Equation ◽  
Orlicz Spaces ◽  
Convex Functional ◽  
Geometrical Properties ◽  
Nonstandard Growth ◽  
Nonlinear Elliptic

AbstractThis paper mainly deals with the abstract-valued Orlicz spaces of range-varying type. Using notions of Banach space net and continuous modular net etc., we give definitions of Lϱθ(⋅)(I, Xθ(⋅)) and $\begin{array}{} L_{+}^{\varrho_{\theta(\cdot)}} \end{array} $(I, Xθ(⋅)), and discuss their geometrical properties as well as the representation of $\begin{array}{} L_{+}^{\varrho_{\theta(\cdot)}} \end{array} $(I, Xθ(⋅))*. We also investigate some functionals and operators on Lϱθ(⋅)(I, Xθ(⋅)), giving expression for the subdifferential of the convex functional generated by another continuous modular net. After making some investigations on the Bochner-Sobolev spaces W1, ϱθ(⋅)(I, Xθ(⋅)) and $\begin{array}{} W_{\textrm{per}}^{1,\varrho_{\theta(\cdot)}} \end{array} $(I, Xθ(⋅)), and the intersection space $\begin{array}{} W_{\textrm{per}}^{1,\varrho_{\theta(\cdot)}} \end{array} $(I, Xθ(⋅)) ∩ Lφϑ(⋅)(I, Vϑ(⋅)), a second order differential inclusion together with an anisotropic nonlinear elliptic equation with nonstandard growth are also taken into account.

scholarly journals SYMMETRIC FINITE REPRESENTABILITY OF ℓp IN ORLICZ SPACES

2021 ◽  
Vol 26 (4) ◽  
pp. 15-24
Author(s):  
S. V. Astashkin
Keyword(s):  
Banach Space ◽  
Orlicz Space ◽  
Orlicz Spaces ◽  
Large Dimension ◽  
Complete Proof ◽  
Finite Dimensional ◽  
Finite Representability

It is well known that a Banach space need not contain any subspace isomorphic to a space ℓp (1 6 p ) or c0 (it was shown by Tsirelson in 1974). At the same time, by the famous Krivines theorem, every Banach space X always contains at least one of these spaces locally, i.e., there exist finite-dimensional subspaces of X of arbitrarily large dimension n which are isomorphic (uniformly) to ℓnp for some 1 6 p or cn0 . In thiscase one says that ℓp (resp. c0) is finitely representable in X. The main purpose of this paper is to give a characterization (with a complete proof) of the set of p such that ℓp is symmetrically finitely representable in a separable Orlicz space.

scholarly journals Best Simultaneous Approximation in Orlicz Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
M. Khandaqji ◽  
Sh. Al-Sharif
Keyword(s):  
Banach Space ◽  
Simultaneous Approximation ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Unit Interval ◽  
Luxemburg Norm ◽  
Integrable Functions ◽  
Best Simultaneous Approximation ◽  
Distance Formula

LetXbe a Banach space and letLΦ(I,X)denote the space of OrliczX-valued integrable functions on the unit intervalIequipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, whereGis a closed subspace ofX, andf1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation inLΦ(I,X)are presented.

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