Higher-Order Commutators of Parametric Marcinkiewicz Integrals on Herz Spaces with Variable Exponent

Hongbin Wang,Dunyan Yan

doi:10.1155/2018/7319093

Abstract

LetΩ∈Ls(Sn-1)fors⩾1be a homogeneous function of degree zero andbbe BMO functions. In this paper, we obtain some boundedness of the parametric Marcinkiewicz integral operatorμΩρand its higher-order commutator[bm,μΩρ]on Herz spaces with variable exponent.

Highlights

Function spaces with variable exponent are being concerned with strong interest in harmonic analysis and in applied mathematics
In the past 27 years, the theory of function spaces with variable exponent has made great progress since some elementary properties were given by Kovacik and Rakosnık [1] in 1991
In [2,3,4,5,6], the authors proved the boundedness of some integral operators on variable Lp spaces, respectively

Summary

Introduction

Function spaces with variable exponent are being concerned with strong interest in harmonic analysis and in applied mathematics. In [2,3,4,5,6], the authors proved the boundedness of some integral operators on variable Lp spaces, respectively. Lebesgue and Sobolev spaces with integrability exponent have been widely studied; see [3, 5] and the references therein. In 2011, Izuki [7] studied the Herz spaces with variable exponent and proved the boundedness of some sublinear operators on the spaces. Motivated by [10, 11], we will study the boundedness for its the parametric Marcinkiewicz integral operator commutator [bm, μΩρ ] on the Herz spaces with μΩρ and variable exponent, where Ω ∈ Ls(Sn−1) for s ⩾ 1.

Preliminaries

Boundedness of the Parametric Marcinkiewicz Integral Operator