The Riemann and Dirichlet Problems with Data from the Grand Lebesgue Spaces

2013 ◽  
pp. 233-251 ◽  
Author(s):  
V. Kokilashvili ◽  
V. Paatashvili
Keyword(s):  
Dirichlet Problems ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces

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.

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.

Grand Lebesgue Spaces on Sets of Infinite Measure

2016 ◽  
pp. 851-870
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi ◽  
Humberto Rafeiro ◽  
Stefan Samko
Keyword(s):  
Lebesgue Spaces ◽  
Infinite Measure ◽  
Grand Lebesgue Spaces

Approximation in weighted generalized grand Lebesgue spaces

2015 ◽  
pp. 1-14
Author(s):  
Daniyal M. Israfilov ◽  
Ahmet Testici
Keyword(s):  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces

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.

Weighted fully measurable grand Lebesgue spaces and the maximal theorem

2016 ◽  
Vol 65 (1) ◽  
pp. 221-233 ◽  
Author(s):  
Giuseppina Anatriello ◽  
Maria Rosaria Formica
Keyword(s):  
Lebesgue Spaces ◽  
Maximal Theorem ◽  
Grand Lebesgue Spaces
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