Boundedness of Marcinkiewicz integral operator of variable order in grand Herz-Morrey spaces

Mehvish Sultan; Babar Sultan; Aziz Khan; Thabet Abdeljawad; Mehvish Sultan; Babar Sultan; Aziz Khan; Thabet Abdeljawad

doi:10.3934/math.20231139

AIMS Mathematics

2023, Volume 8, Issue 9: 22338-22353. doi: 10.3934/math.20231139

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Research article

Boundedness of Marcinkiewicz integral operator of variable order in grand Herz-Morrey spaces

1.
Department of Mathematics, Capital University Of Science and Technology, Islamabad, Pakistan
2.
Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan
3.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics, Kyung Hee University 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea

Received: 21 May 2023 Revised: 03 July 2023 Accepted: 04 July 2023 Published: 13 July 2023
MSC : 46E30, 47B38

Let $ \mathbb{S}^{n-1} $ denotes the unit sphere in $ \mathbb{R}^n $ equipped with the normalized Lebesgue measure. Let $ \Phi \in L^r(\mathbb{S}^{n-1}) $ be a homogeneous function of degree zero. The variable Marcinkiewicz fractional integral operator is defined as

$ \mu _{\Phi} (f)(z_1) = \left( \int \limits _0 ^ \infty \left|\int \limits _{|z_1-z_2| \leq s} \frac{\Phi(z_1-z_2)}{|z_1-z_2|^{n-1-\zeta(z_1)}}f(z_2)dz_2\right|^2 \frac{ds}{s^3}\right)^{\frac{1}{2}}. $

The Marcinkiewicz fractional operator of variable order $ \zeta(z_1) $ is shown to be bounded from the grand Herz-Morrey spaces $ {M\dot{K} ^{\alpha(\cdot), u), \theta}_{\beta, p(\cdot)}(\mathbb{R}^n)} $ with variable exponent into the weighted space $ {M\dot{K} ^{\alpha(\cdot), u), \theta}_{\beta, \rho, q(\cdot)}(\mathbb{R}^n)} $ where

$ \rho = (1+|z_1|)^{-\lambda} $

and

$ {1 \over q(z_1)} = {1 \over p(z_1)}-{\zeta(z_1) \over n} $

when $ p(z_1) $ is not necessarily constant at infinity.
- Lebesgue spaces,
- weighted estimates,
- Marcinkiewicz fractional,
- grand Herz-Morrey spaces
Citation: Mehvish Sultan, Babar Sultan, Aziz Khan, Thabet Abdeljawad. Boundedness of Marcinkiewicz integral operator of variable order in grand Herz-Morrey spaces[J]. AIMS Mathematics, 2023, 8(9): 22338-22353. doi: 10.3934/math.20231139

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Abstract

Let $ \mathbb{S}^{n-1} $ denotes the unit sphere in $ \mathbb{R}^n $ equipped with the normalized Lebesgue measure. Let $ \Phi \in L^r(\mathbb{S}^{n-1}) $ be a homogeneous function of degree zero. The variable Marcinkiewicz fractional integral operator is defined as

$ \mu _{\Phi} (f)(z_1) = \left( \int \limits _0 ^ \infty \left|\int \limits _{|z_1-z_2| \leq s} \frac{\Phi(z_1-z_2)}{|z_1-z_2|^{n-1-\zeta(z_1)}}f(z_2)dz_2\right|^2 \frac{ds}{s^3}\right)^{\frac{1}{2}}. $

The Marcinkiewicz fractional operator of variable order $ \zeta(z_1) $ is shown to be bounded from the grand Herz-Morrey spaces $ {M\dot{K} ^{\alpha(\cdot), u), \theta}_{\beta, p(\cdot)}(\mathbb{R}^n)} $ with variable exponent into the weighted space $ {M\dot{K} ^{\alpha(\cdot), u), \theta}_{\beta, \rho, q(\cdot)}(\mathbb{R}^n)} $ where

$ \rho = (1+|z_1|)^{-\lambda} $

and

$ {1 \over q(z_1)} = {1 \over p(z_1)}-{\zeta(z_1) \over n} $

when $ p(z_1) $ is not necessarily constant at infinity.

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