Commutators of the Bilinear Hardy Operator on Herz Type Spaces with Variable Exponents

Shengrong Wang,Jingshi Xu

doi:10.1155/2019/7607893

Abstract

In this paper, we obtain the boundedness of bilinear commutators generated by the bilinear Hardy operator and BMO functions on products of Herz spaces and Herz-Morrey spaces with variable exponents.

Highlights

Denote by L1loc(Rn) integrable functions the set on Rn.of all complex-valued locally The Hardy operator was first considered in [1] as follows: Hf (x) fl 1 x x ∫ f (t) dt, x ≠ 0, (1)for f ∈ L1loc(R)
The boundeness of commutators of the Hardy operator was obtained in λcentral BMO spaces [31], Herz spaces Kpα(,q⋅)(Rn) with variable exponent p(⋅) [32]
In [34], Wu considered the boundedness for fractional Hardy type operator on HerzMorrey spaces MKpα,qλ(⋅)(Rn) with variable exponent q(⋅) but fixed α ∈ R and p ∈ (0, ∞)

Summary

Introduction

Denote by L1loc(Rn) integrable functions the set on Rn.of all complex-valued locally The Hardy operator was first considered in [1] as follows: Hf (x) fl 1 x x ∫ f (t) dt, x ≠ 0, (1)for f ∈ L1loc(R). We obtain the boundedness of bilinear commutators generated by the bilinear Hardy operator and BMO functions on products of Herz spaces and Herz-Morrey spaces with variable exponents. Denote by L1loc(Rn) integrable functions the set on Rn. of all complex-valued locally The Hardy operator was first considered in [1] as follows: Hf (x) The boundedness of the Hardy operator was considered in many function spaces, such as in variable Lebesgue spaces

Results

Conclusion