scholarly journals One-Sided Integral Operators with Homogeneous Kernels in Grand Lebesgue Spaces

2017 ◽  
Author(s):  
С.М. Умархаджиев
Keyword(s):  
Integral Operators ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces

Получены достаточные и необходимые условия на ядро и грандизатор для ограниченности односторонних интегральных операторов с однородными ядрами в гранд-пространствах Лебега на~$\mathbb{R}_+$ и $\mathbb{R}^n$, а также получены двусторонние оценки гранд-норм таких операторов. Кроме того, в~случае радиального ядра получены двусторонние оценки для норм многомерных операторов в~терминах сферических средних и показано, что этот результат сильнее, чем неравенства для норм операторов с нерадиальным ядром.

On integral operators in weighted grand Lebesgue spaces of Banach-valued functions

2020 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi
Keyword(s):  
Integral Operators ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces

Boundedness of Weighted Singular Integral Operators on a Carleson Curve in Grand Lebesgue Spaces

2010 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Stefan Samko ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces ◽  
Carleson Curve

scholarly journals Integral operators on fully measurable weighted grand Lebesgue spaces

2017 ◽  
Vol 28 (2) ◽  
pp. 516-526 ◽  
Author(s):  
Pankaj Jain ◽  
Monika Singh ◽  
Arun Pal Singh
Keyword(s):  
Integral Operators ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces

scholarly journals On integral operators in weighted grand Lebesgue spaces of Banach-valued functions

Authorea ◽  
2020 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi
Keyword(s):  
Integral Operators ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces

scholarly journals Гранд-пространства типа Морри

2020 ◽  
pp. 104-118
Author(s):  
S.G. Samko ◽  
S.M. Umarkhadzhiev
Keyword(s):  
Integral Operators ◽  
Finite Measure ◽  
Morrey Spaces ◽  
Function Spaces ◽  
Lebesgue Spaces ◽  
Hardy Type ◽  
Grand Lebesgue Spaces ◽  
Type Spaces

The so called grand spaces nowadays are one of the main objects in the theory of function spaces. Grand Lebesgue spaces were introduced by T. Iwaniec and C. Sbordone in the case of sets $\Omega$ with finite measure $|\Omega|<\infty$, and by the authors in the case $|\Omega|=\infty$. The latter is based on introduction of the notion of grandizer. The idea of "grandization" was also applied in the context of Morrey spaces. In this paper we develop the idea of grandization to more general Morrey spaces $L^{p,q,w}(\mathbb{R}^n)$, known as Morrey type spaces. We introduce grand Morrey type spaces, which include mixed and partial grand versions of such spaces. The mixed grand space is defined by the norm $$ \sup_{\varepsilon,\delta} \varphi(\varepsilon,\delta)\sup_{x\in E} \left(\int\limits_{0}^{\infty}{w(r)^{q-\delta}}b(r)^{\frac{\delta}{q}} \left(\,\int\limits_{|x-y|<r}\big|f(y)\big|^{p-\varepsilon} a(y)^{\frac{\varepsilon}{p}}\,dy\right)^{\frac{q-\delta}{p-\varepsilon}} \frac{dr}{r}\right)^{\frac{1}{q-\varepsilon}} $$ with the use of two grandizers $a$ and $b$. In the case of grand spaces, partial with respect to the exponent $q$, we study the boundedness of some integral operators. The class of these operators contains, in particular, multidimensional versions of Hardy type and Hilbert operators.

scholarly journals Weighted grand Lebesgue spaces with mixed-norms and integral operators

2019 ◽  
Vol 489 (4) ◽  
pp. 344-346
Author(s):  
V. M. Kokilashvili
Keyword(s):  
Integral Operators ◽  
Maximal Functions ◽  
Riesz Transforms ◽  
Lebesgue Spaces ◽  
Mixed Norms ◽  
Grand Lebesgue Spaces

In this paper, the weighted grand Lebesgue spaces with mixed-norms are introduced and boundedness criteria in these spaces of strong maximal functions and Riesz transforms are presented.

Integral operators with homogeneous kernels in grand Lebesgue spaces

2017 ◽  
Vol 102 (5-6) ◽  
pp. 710-721 ◽  
Author(s):  
S. M. Umarkhadzhiev
Keyword(s):  
Integral Operators ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces

Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications

2018 ◽  
Vol 9 (3) ◽  
pp. 413-425 ◽  
Author(s):  
Alberto Fiorenza ◽  
Vakhtang Kokilashvili
Keyword(s):  
Harmonic Analysis ◽  
Integral Operators ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces

Multilinear integral operators in weighted grand Lebesgue spaces

2016 ◽  
Vol 19 (3) ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Mieczysław Mastyło ◽  
Alexander Meskhi
Keyword(s):  
Singular Integrals ◽  
Type Theorem ◽  
Type Inequality ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Sobolev Type ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Grand Lebesgue Spaces ◽  
Measure Spaces

AbstractThe boundedness of multi(sub)linear Hardy&ndash;Littlewood maximal, Calder&oacute;n&ndash;Zygmund and fractional integral operators defined on metric measure spaces is established in weighted grand Lebesgue spaces. In particular, we derive the one-weight inequality for maximal and singular integrals under the Muckenhoupt type conditions, weighted Sobolev type theorem and trace type inequality for fractional integrals.

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