Variable exponents and grand Lebesgue spaces: Some optimal results

2015 ◽  
Vol 17 (06) ◽  
pp. 1550023 ◽  
Author(s):  
Alberto Fiorenza ◽  
Jean Michel Rakotoson ◽  
Carlo Sbordone
Keyword(s):  
Variable Exponent ◽  
Bounded Function ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Lebesgue Spaces ◽  
Bounded Set ◽  
Decreasing Rearrangement ◽  
Grand Lebesgue Spaces ◽  
Measurable Bounded Function ◽  
Grand Lebesgue Space

Consider p : Ω → [1, +∞[, a measurable bounded function on a bounded set Ø with decreasing rearrangement p* : [0, |Ω|] → [1, +∞[. We construct a rearrangement invariant space with variable exponent p* denoted by [Formula: see text]. According to the growth of p*, we compare this space to the Lebesgue spaces or grand Lebesgue spaces. In particular, if p*(⋅) satisfies the log-Hölder continuity at zero, then it is contained in the grand Lebesgue space Lp*(0))(Ω). This inclusion fails to be true if we impose a slower growth as [Formula: see text] at zero. Some other results are discussed.

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 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.

A Note on the Boundedness of the Hilbert Transform in Weighted Grand Lebesgue Spaces

2009 ◽  
Vol 16 (3) ◽  
pp. 547-551 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi
Keyword(s):  
Hilbert Transform ◽  
Lebesgue Space ◽  
Finite Interval ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces ◽  
Muckenhoupt Condition ◽  
The Hilbert Transform ◽  
Grand Lebesgue Space

Abstract It is proved that the Hilbert transform defined on a finite interval is bounded in the grand Lebesgue space if and only if 𝑤 satisfies the Muckenhoupt condition 𝐴𝑝.

Trace inequalities for fractional integrals in mixed norm grand lebesgue spaces

2020 ◽  
Vol 23 (5) ◽  
pp. 1452-1471
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi
Keyword(s):  
Riesz Potential ◽  
Fractional Integrals ◽  
Mixed Norm ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Grand Lebesgue Spaces ◽  
Measure Spaces ◽  
Necessary And Sufficient ◽  
Bounded Sets ◽  
Trace Inequalities

Abstract D. Adams type trace inequalities for multiple fractional integral operators in grand Lebesgue spaces with mixed norms are established. Operators under consideration contain multiple fractional integrals defined on the product of quasi-metric measure spaces, and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously necessary and sufficient for appropriate trace inequalities. The derived results are new even for multiple Riesz potential operators defined on the product of Euclidean spaces.

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