Equivalence of norms of the generalized fractional integral operator and the generalized fractional maximal operator on the generalized weighted Morrey spaces

Küçükaslan, Abdulhamit

doi:10.1007/s43034-020-00066-w

Cited by 9 publications

(5 citation statements)

References 29 publications

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“…Definition 2.8. (Kucukaslan, 2020) For nonnegative function 𝜌 taking value on [0, ∞), the generalized fractional maximal operator 𝑀 𝜌 is defined as…”

Section: Definitions and Notationsmentioning

confidence: 99%

On The Boundedness Properties of the Generalized Fractional Integrals on the Generalized Weighted Morrey Spaces

Ramadana

2022

JMathCoS

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Morrey Spaces were first introduced by C.B. Morrey in 1938. Morrey space can be considered as a generalization of the Lebesgue spaces. Morrey spaces were then generalized become the generalized Morrey spaces, the weighted Morrey spaces, and the generalized weighted Morrey spaces. One of the studies on Morrey spaces is the boundedness of certain operators on the spaces. One of the operators is the fractional integral. The boundedness of fractional integrals on the classical Morrey spaces, the weighted Morrey spaces, the generalized Morrey spaces, and the generalized weighted Morrey spaces had been known. One of the extensions of fractional integrals is generalized fractional integral. The operator was bounded on the generalized Morrey spaces. The purpose of this study is to investigate the boundedness of generalized fractional integrals on the generalized weighted Morrey spaces. The weight used is Muckenhoupt class. The results obtained show that the generalized fractional integral is bounded from generalized weighted Morrey space to another generalized weighted Morrey space under some assumptions. The main result obtained then implies the boundedness of the generalized fractional maximal operator on generalized weighted Morrey spaces under the same assumptions.

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“…Definition 2.8. (Kucukaslan, 2020) For nonnegative function 𝜌 taking value on [0, ∞), the generalized fractional maximal operator 𝑀 𝜌 is defined as…”

Section: Definitions and Notationsmentioning

confidence: 99%

On The Boundedness Properties of the Generalized Fractional Integrals on the Generalized Weighted Morrey Spaces

Ramadana

2022

JMathCoS

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“…Note that various variants of the generalized fractional-maximal function were previously considered in [9][10][11][12][13][14][15]. For such variants of a generalized fractional-maximal function, the questions of boundedness in Lorentz spaces were considered in [8], [11].…”

Section: Introductionmentioning

confidence: 99%

On the Boundedness of a Generalized Fractional-Maximal Operator in Lorentz Spaces

Abek,

Turgumbayev,

Suleimenova

2023

JMMCS

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In this paper considers a generalized fractional-maximal operator, a special case of which is the classical fractional-maximal function. Conditions for the function Φ, which defines a generalized fractional-maximal function, and for the weight functions w and v, which determine the weighted Lorentz spaces Λ p (v) and Λ q (w) (1 < p ≤ q < ∞) under which the generalized maximal-fractional operator is bounded from one Lorentz space Λ p (v) to another Lorentz space Λ q (w) are obtained. For the classical fractional maximal operator and the classical maximal Hardy-Littlewood function such results were previously known. When proving the main result, we make essential use of an estimate for a nonincreasing rearrangement of a generalized fractional-maximal operator. In addition, we introduce a supremal operator for which conditions of boundedness in weighted Lebesgue spaces are obtained. This result is also essentially used in the proof of the main theorem.

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“…Nakai [24] introduced the the generalized Morrey spaces M p,φ and proved the boundedness of the generalized fractional integral operator I ρ in these spaces. Nowadays many authors have been culminating important observations about these twooperators M ρ and I ρ especially in connection with Morrey-type spaces (see [3,10,[14][15][16]).…”

Section: Introductionmentioning

confidence: 99%

An Extension of the Adams-type Theorem to the Vanishing Generalized Weighted Morrey Spaces

Küçükaslan

2022

Turkish Journal of Mathematics and Computer Science

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In this paper, we generalize Adams-type theorems given in [1,13] (which are the following Theorem A and Theorem B, respectively) to the vanishing generalized weighted Morrey spaces. We prove the Adams-type boundedness of the generalized fractional maximal operator $M_{\rho}$ from the vanishing generalized weighted Morrey spaces $\mathcal{\mathcal{VM}}_{p,\varphi^{\frac{1}{p}}}(\mathbb{R}^n, w)$ to another one $\mathcal{\mathcal{VM}}_{q,\varphi^{\frac{1}{q}}}(\mathbb{R}^n, w)$ with $w \in A_{p,q}$ for $1$

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Equivalence of norms of the generalized fractional integral operator and the generalized fractional maximal operator on the generalized weighted Morrey spaces

Cited by 9 publications

References 29 publications

On The Boundedness Properties of the Generalized Fractional Integrals on the Generalized Weighted Morrey Spaces

On The Boundedness Properties of the Generalized Fractional Integrals on the Generalized Weighted Morrey Spaces

On the Boundedness of a Generalized Fractional-Maximal Operator in Lorentz Spaces

An Extension of the Adams-type Theorem to the Vanishing Generalized Weighted Morrey Spaces

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