Boundedness of Integral Operators in Generalized Weighted Grand Lebesgue Spaces with Non-doubling Measures

2021 ◽  
Vol 18 (2) ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi
Keyword(s):  
Integral Operators ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces ◽  
Doubling Measures

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

A note on fractional integral operators defined by weights and non-doubling measures

2010 ◽  
Vol 106 (2) ◽  
pp. 283 ◽  
Author(s):  
Oscar Blasco ◽  
Vicente Casanova ◽  
Joaquín Motos
Keyword(s):  
Measure Space ◽  
Fractional Integral ◽  
Integral Operators ◽  
Metric Measure Space ◽  
Integral Type ◽  
Lebesgue Spaces ◽  
Lipschitz Spaces ◽  
Fractional Integral Operators ◽  
Doubling Measures

Given a metric measure space $(X,d,\mu)$, a weight $w$ defined on $(0,\infty)$ and a kernel $k_w(x,y)$ satisfying the standard fractional integral type estimates, we study the boundedness of the operators $K_w f(x)=\int_X k_w(x,y)f(y)\,d\mu(y)$ and $\tilde K_w f(x)=\int_X (k_w(x,y)-k_w(x_0,y))f(y)\,d\mu(y)$ on Lebesgue spaces $L^p(\mu)$ and generalized Lipschitz spaces $\mathrm{Lip}_\phi$, respectively, for certain range of the parameters depending on the $n$-dimension of $\mu$ and some indices associated to the weight $w$.

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 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.

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.

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