scholarly journals A Decomposition of the Dual Space of Some Banach Function Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Claudia Capone ◽  
Maria Rosaria Formica
Keyword(s):  
Lebesgue Space ◽  
Dual Space ◽  
Function Spaces ◽  
Zygmund Space ◽  
Marcinkiewicz Space ◽  
Banach Function Spaces ◽  
Integrable Functions ◽  
Grand Lebesgue Space

We give a decomposition for the dual space of some Banach Function Spaces as the Zygmund space of the exponential integrable functions, the Marcinkiewicz space , and the Grand Lebesgue Space .

scholarly journals Topological Dual Systems for Spaces of Vector Measurep-Integrable Functions

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
P. Rueda ◽  
E. A. Sánchez Pérez
Keyword(s):  
Type Theorem ◽  
Classical Case ◽  
Function Spaces ◽  
Dual Systems ◽  
Banach Function Spaces ◽  
Infinite Dimensional ◽  
Local Compactness ◽  
Finite Dimensional ◽  
Integrable Functions ◽  
Vector Valued

We show a Dvoretzky-Rogers type theorem for the adapted version of theq-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given.

scholarly journals BMO Functions Generated by A X ℝ n Weights on Ball Banach Function Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ruimin Wu ◽  
Songbai Wang
Keyword(s):  
Linear Operator ◽  
Function Space ◽  
Maximal Function ◽  
Lebesgue Space ◽  
Variable Exponent ◽  
Banach Function Space ◽  
Function Spaces ◽  
Banach Function Spaces ◽  
Bmo Functions ◽  
Littlewood Maximal Function

Let X be a ball Banach function space on ℝ n . We introduce the class of weights A X ℝ n . Assuming that the Hardy-Littlewood maximal function M is bounded on X and X ′ , we obtain that BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A X ℝ n . As a consequence, we have BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A L p · ℝ n ℝ n , where L p · ℝ n is the variable exponent Lebesgue space. As an application, if a linear operator T is bounded on the weighted ball Banach function space X ω for any ω ∈ A X ℝ n , then the commutator b , T is bounded on X with b ∈ BMO ℝ n .

scholarly journals On Generalized Morse-Transue Function Spaces

1959 ◽  
Vol 11 ◽  
pp. 416-426
Author(s):  
H. W. Ellis
Keyword(s):  
Continuous Functions ◽  
Function Spaces ◽  
Length Function ◽  
Integral Type ◽  
Banach Function Spaces ◽  
Integrable Functions ◽  
Bounded Functions

Marston Morse and William Transue (6, 8) have introduced and studied function spaces, called MT-spaces, for which the elements of the topological dual are of integral type. Their theory does not admit certain classical Banach function spaces including spaces of bounded functions and spaces. The theory of function spaces determined by a length function (λ-spaces) (4, 5), which depends on a fixed measure, admits many of the maximal MT-spaces, the spaces and spaces of locally integrable functions but does not admit certain maximal MT-spaces including the space of complex continuous functions with compact supports.

scholarly journals Identification of Fully Measurable Grand Lebesgue Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-3 ◽  
Author(s):  
Giuseppina Anatriello ◽  
Ralph Chill ◽  
Alberto Fiorenza
Keyword(s):  
Lebesgue Space ◽  
Measure Space ◽  
Function Spaces ◽  
Lebesgue Spaces ◽  
Banach Function Spaces ◽  
Measurable Functions ◽  
Grand Lebesgue Spaces ◽  
Almost Everywhere ◽  
Function Norm

We consider the Banach function spaces, called fully measurable grand Lebesgue spaces, associated with the function norm ρ(f)=ess supx∈X⁡δ(x)ρp(x)(f), where ρp(x) denotes the norm of the Lebesgue space of exponent p(x), and p(·) and δ(·) are measurable functions over a measure space (X,ν), p(x)∈[1,∞], and δ(x)∈(0,1] almost everywhere. We prove that every such space can be expressed equivalently replacing p(·) and δ(·) with functions defined everywhere on the interval (0,1), decreasing and increasing, respectively (hence the full measurability assumption in the definition does not give an effective generalization with respect to the pointwise monotone assumption and the essential supremum can be replaced with the simple supremum). In particular, we show that, in the case of bounded p(·), the class of fully measurable Lebesgue spaces coincides with the class of generalized grand Lebesgue spaces introduced by Capone, Formica, and Giova.

scholarly journals Non-commutative Banach function spaces

1989 ◽  
Vol 201 (4) ◽  
pp. 583-597 ◽  
Author(s):  
Peter G. Dodds ◽  
Theresa K. -Y. Dodds ◽  
Ben de Pagter
Keyword(s):  
Function Spaces ◽  
Banach Function Spaces

Approximation by trigonometric polynomials in the framework of variable exponent grand Lebesgue spaces

2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Nina Danelia ◽  
Vakhtang Kokilashvili
Keyword(s):  
Lebesgue Space ◽  
Variable Exponent ◽  
Trigonometric Polynomials ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Space ◽  
Grand Lebesgue Spaces ◽  
Direct And Inverse Theorems ◽  
Inverse Theorems ◽  
Grand Lebesgue Space

AbstractIn this paper we establish direct and inverse theorems on approximation by trigonometric polynomials for the functions of the closure of the variable exponent Lebesgue space in the variable exponent grand Lebesgue space.

scholarly journals The weak topology on q-convex Banach function spaces

2011 ◽  
Vol 285 (2-3) ◽  
pp. 136-149 ◽  
Author(s):  
L. Agud ◽  
J. M. Calabuig ◽  
E. A. Sánchez Pérez
Keyword(s):  
Weak Topology ◽  
Function Spaces ◽  
Banach Function Spaces
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