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.

Rearrangement Inequalities

1972 ◽  
Vol 24 (5) ◽  
pp. 930-943 ◽  
Author(s):  
Peter W. Day
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Function Spaces ◽  
Banach Function Spaces ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Rearrangement Inequalities

In recent years a number of inequalities have appeared which involve rearrangements of vectors in Rn and of measurable functions on a finite measure space. These inequalities are not only interesting in themselves, but also are important in investigations involving rearrangement invariant Banach function spaces and interpolation theorems for these spaces [2; 8; 9].

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.

Some Pointwise Convergence Results in Lp(μ), 1 < p < ∞

1977 ◽  
Vol 20 (3) ◽  
pp. 277-284 ◽  
Author(s):  
Richard Duncan
Keyword(s):  
Probability Theory ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Almost Everywhere Convergence ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Convergence Results ◽  
Convergence In Norm

The theory of almost everywhere convergence has its roots in the poineering work of A. Kolmogorov, and today it constitutes one of the most captivating and challenging chapters in modern probability theory and analysis. Whereas some modes of convergence for sequences of measurable functions, e.g. convergence in norm, can be readily obtained by an intelligent exploitation of the various properties of the function spaces involved, a.e. convergence invariably requires a rather high, and sometimes surprising, degree of mathematical virtuosity.

Weak-type inequalities and convergence of the ergodic averages in variable Lebesgue spaces with weights

2009 ◽  
Vol 139 (4) ◽  
pp. 673-683 ◽  
Author(s):  
M. Isabel Aguilar Cañestro ◽  
Pedro Ortega Salvador
Keyword(s):  
Lebesgue Space ◽  
Weak Type ◽  
Suitable Condition ◽  
Variable Exponents ◽  
Ergodic Averages ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Almost Everywhere ◽  
Variable Lebesgue Space ◽  
Measure Preserving Transformation

We characterize the weighted weak-type inequalities with variable exponents for the maximal operator associated with an ergodic, invertible, measure-preserving transformation and prove the almost everywhere convergence of the ergodic averages for all functions in a variable Lebesgue space with a weight verifying a suitable condition.

scholarly journals Littlewood-Paley spaces

2011 ◽  
Vol 108 (1) ◽  
pp. 77 ◽  
Author(s):  
Kwok-Pun Ho
Keyword(s):  
Hardy Spaces ◽  
Orlicz Spaces ◽  
Morrey Spaces ◽  
Function Spaces ◽  
Unified Framework ◽  
Lebesgue Spaces ◽  
Banach Function Spaces

We introduce the Littlewood-Paley spaces in which the Lebesgue spaces, the Hardy spaces, the Orlicz spaces, the Lorentz-Karamata spaces, the r.-i. quasi-Banach function spaces and the Morrey spaces reside. The Littlewood-Paley spaces provide a unified framework for the study of some important function spaces arising in analysis.

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 Compactness in quasi-Banach function spaces and applications to compact embeddings of Besov-type spaces

2016 ◽  
Vol 146 (5) ◽  
pp. 905-927 ◽  
Author(s):  
António Caetano ◽  
Amiran Gogatishvili ◽  
Bohumír Opic
Keyword(s):  
Besov Spaces ◽  
Function Spaces ◽  
Lebesgue Spaces ◽  
Banach Function Spaces ◽  
Compact Embeddings ◽  
Type Spaces ◽  
Besov Type Spaces

There are two main aims of the paper. The first is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second is to extend the criterion for the precompactness of sets in the Lebesgue spaces Lp(ℝn), 1 ⩽ p < ∞, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces , into Lorentz-type spaces.

scholarly journals Spaces of measurable functions

2013 ◽  
Vol 11 (7) ◽  
Author(s):  
Piotr Niemiec
Keyword(s):  
Hilbert Space ◽  
Measure Space ◽  
Finite Measure ◽  
Metrizable Space ◽  
Equivalence Classes ◽  
Convergence In Measure ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Completely Metrizable

AbstractFor a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

scholarly journals Relationship Between an Atomic Decomposition of Double and Unary Systems in Grand-Lebesgue Spaces

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Lebesgue Space ◽  
Atomic Decomposition ◽  
Shift Operator ◽  
Continuous Functions ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces ◽  
Separable Subspace ◽  
Grand Lebesgue Space

The grand-Lebesgue space is defined. Based on the shift operator, a separable subspace is determined in which continuous functions are dense. The concepts of frame and atomic decomposition are defined. An atomic decomposition of double and unary systems of functions in grand-Lebesgue spaces is considered. Relationship between atomic decomposition of these systems in grand-Lebesgue spaces is established

Boundedness of weighted singular integral operators in grand Lebesgue spaces

2011 ◽  
Vol 18 (2) ◽  
pp. 259-269
Author(s):  
Vakhtang Kokilashvili ◽  
Stefan Samko
Keyword(s):  
Singular Integral ◽  
Lebesgue Space ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces ◽  
Carleson Curve ◽  
Power Weights ◽  
Necessary And Sufficient ◽  
Grand Lebesgue Space

Abstract We obtain the necessary and sufficient conditions for the boundedness of the weighted singular integral operator with power weights in grand Lebesgue spaces. Because of applications to singular integral equations, the underlying set on which the functions are defined is a Carleson curve in the complex plane. Note that weighted boundedness of an operator in grand Lebesgue space is not the same as the boundedness in weighted grand Lebesgue space.

Sign in / Sign up

Export Citation Format

Share Document