Hardy type inequalities in generalized grand Lebesgue spaces

AbstractWe study unboundedness properties of functions belonging to Lebesgue and Lorentz spaces with variable and mixed norms using growth envelopes. Our results extend the ones for the corresponding classical spaces in a natural way. In the case of spaces with mixed norms, it turns out that the unboundedness in the worst direction, i.e., in the direction where $$p_{i}$$ p i is the smallest, is crucial. More precisely, the growth envelope is given by $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p}}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},\min \{p_{1}, \ldots , p_{d} \})$$ E G ( L p → ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , min { p 1 , … , p d } ) for mixed Lebesgue and $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p},q}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},q)$$ E G ( L p → , q ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , q ) for mixed Lorentz spaces, respectively. For the variable Lebesgue spaces, we obtain $${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot )}(\varOmega )) = (t^{-1/p_{-}},p_{-})$$ E G ( L p ( · ) ( Ω ) ) = ( t - 1 / p - , p - ) , where $$p_{-}$$ p - is the essential infimum of $$p(\cdot )$$ p ( · ) , subject to some further assumptions. Similarly, for the variable Lorentz space, it holds$${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot ),q}(\varOmega )) = (t^{-1/p_{-}},q)$$ E G ( L p ( · ) , q ( Ω ) ) = ( t - 1 / p - , q ) . The growth envelope is used for Hardy-type inequalities and limiting embeddings. In particular, as a by-product, we determine the smallest classical Lebesgue (Lorentz) space which contains a fixed mixed or variable Lebesgue (Lorentz) space, respectively.

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Гранд-пространства типа Морри

Владикавказский математический журнал ◽

10.46698/c3825-5071-7579-i ◽

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.

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Some new Hardy-type inequalities on time scales

Advances in Difference Equations ◽

10.1186/s13662-020-02883-8 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Ahmed A. El-Deeb ◽

Hamza A. Elsennary ◽

Dumitru Baleanu

Keyword(s):

Time Scales ◽

Hardy Type Inequalities ◽

Hardy Type

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Trace inequalities for fractional integrals in mixed norm grand lebesgue spaces

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0072 ◽

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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