A note on the boundedness of Hardy operators in grand Herz spaces with variable exponent

Samia Bashir; Babar Sultan; Amjad Hussain; Aziz Khan; Thabet Abdeljawad; Samia Bashir; Babar Sultan; Amjad Hussain; Aziz Khan; Thabet Abdeljawad

doi:10.3934/math.20231130

AIMS Mathematics

2023, Volume 8, Issue 9: 22178-22191. doi: 10.3934/math.20231130

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Research article

A note on the boundedness of Hardy operators in grand Herz spaces with variable exponent

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

Received: 13 May 2023 Revised: 01 July 2023 Accepted: 06 July 2023 Published: 12 July 2023
MSC : 46E30, 47B38

The fractional Hardy-type operators of variable order is shown to be bounded from the grand Herz spaces $ {\dot{K} ^{a(\cdot), u), \theta}_{ p(\cdot)}(\mathbb{R}^n)} $ with variable exponent into the weighted space $ {\dot{K} ^{a(\cdot), u), \theta}_{\rho, q(\cdot)}(\mathbb{R}^n)} $, where $ \rho = (1+|z_1|)^{-\lambda} $ and

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

when $ p(z) $ is not necessarily constant at infinity.
- Lebesgue spaces,
- weighted estimates,
- Hardy operators,
- grand Herz spaces
Citation: Samia Bashir, Babar Sultan, Amjad Hussain, Aziz Khan, Thabet Abdeljawad. A note on the boundedness of Hardy operators in grand Herz spaces with variable exponent[J]. AIMS Mathematics, 2023, 8(9): 22178-22191. doi: 10.3934/math.20231130

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Abstract

The fractional Hardy-type operators of variable order is shown to be bounded from the grand Herz spaces $ {\dot{K} ^{a(\cdot), u), \theta}_{ p(\cdot)}(\mathbb{R}^n)} $ with variable exponent into the weighted space $ {\dot{K} ^{a(\cdot), u), \theta}_{\rho, q(\cdot)}(\mathbb{R}^n)} $, where $ \rho = (1+|z_1|)^{-\lambda} $ and

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

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

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