Weighted Gehring and Muckenhoupt classes and some inclusion properties with norm inequalities

Volume 24, Issue 3, pp 201--215 http://dx.doi.org/10.22436/jmcs.024.03.02

Publication Date: February 23, 2021 Submission Date: December 13, 2020 Revision Date: January 01, 2021 Accteptance Date: January 06, 2021

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Authors

S. H. Saker - Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. M. H. Hassan - Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.

Abstract

In this paper, we will prove some fundamental properties of the power mean operator \(\mathcal{H}_{\lambda }w^{p}\) of order \(p\), which is defined by% \[ \mathcal{H}_{\lambda }w^{p}\left( x\right) =\frac{1}{\Lambda \left( x\right) }\int_{0}^{x}\lambda \left( s\right) w^{p}\left( s\right) ds,\text{ for }% p\in \mathbb{R}^{+}\text{,} \] where \(\lambda \) and \(w\) are nonnegative functions and \(\Lambda \left( x\right) =\int_{0}^{x}\lambda \left( s\right) ds\). Then by using these properties we will establish some norm inequalities of the generalized Muckenhoupt and Gehring weights and prove some fundamental relations between them.

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Keywords

• Hardy type inequality
• generalized Muckenhoupt class
• generalized Gehring class
• inclusion properties

MSC

•  26D10
•  26D15

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