Commutators of the Fractional Hardy Operator on Weighted Variable Herz-Morrey Spaces

Amjad Hussain,Fahd Jarad,Muhammad Asim,Muhammad Aslam,Sarfraz Nawaz Malik

doi:10.1155/2021/9705250

Abstract

In the present paper, our aim is to establish the boundedness of commutators of the fractional Hardy operator and its adjoint operator on weighted Herz-Morrey spaces with variable exponents M K ̇ p , q ⋅ α ⋅ , λ w .

Highlights

Hardy operators and related commutators play an indispensable role in the theory of partial differential equations [1, 2] and the characterization of function spaces [3,4,5]
By taking into account the generalization of function spaces with variable exponents and the same with weights, many results like duality, boundedness of sublinear operators, the wavelet characterization, and commutators of fractional and singular integrals have been studied [29,30,31,32,33,34,35,36,37,38]
Journal of Function Spaces the weighted versions of Herz-Morrey spaces were introduced recently in [43, 44]. In this piece of work, our main focus is on establishing the boundedness of commutators of fractional Hardy operators on a class of function spaces called the weighted HerzMorrey space with variable exponents

Summary

Introduction

Hardy operators and related commutators play an indispensable role in the theory of partial differential equations [1, 2] and the characterization of function spaces [3,4,5]. By taking into account the generalization of function spaces with variable exponents and the same with weights, many results like duality, boundedness of sublinear operators, the wavelet characterization, and commutators of fractional and singular integrals have been studied [29,30,31,32,33,34,35,36,37,38]. Journal of Function Spaces the weighted versions of Herz-Morrey spaces were introduced recently in [43, 44] In this piece of work, our main focus is on establishing the boundedness of commutators of fractional Hardy operators on a class of function spaces called the weighted HerzMorrey space with variable exponents. Before turning to our key results, let us first define the relevant variable exponent function spaces

Preliminaries

Some Useful Lemmas

Main Results and their Proofs