Boundedness of Rough Singular Integral Operators on Homogeneous Herz Spaces with Variable Exponents

Jinling Cai,Baohua Dongand Canqin Tang,*

1College of Science,Dalian Maritime University,Dalian 116026,China;

2Department of Mathematics,Nanjing University of Information Science and Technology,Nanjing 210044,China.

Abstract.We establish the boundedness of rough singular integral operators on homogeneous Herz spaces with variable exponents.As an application,we obtain the boundedness of related commutators with BMO functions on homogeneous Herz spaces with variable exponents.

Key words:Rough singular integral operator,commutator,Herz spaces with variable exponents,BMO spaces.

1 Introduction

Let Sn-1be the unit sphere in Rn(n≥2)and Ω be a measurable function defined on Sn-1.The rough singular integral operatorTΩis defined by

In 1952,Calder′on and Zygmund first studied the operatorTΩin[3]and proved thatTΩis bounded inLp(Rn)forp∈(1,∞)if Ω∈C∞(Sn-1)is a homogeneous function of degree zero and

The condition on Ω can be weakened or revised;see[4,8,27].Later,the results were further extended to the weight Lebesgue spaces by Duoandikoetxea[14].Further development of the topic in other function spaces with constant exponents can be found in[5–7,15,16,23]and the references therein.

Variable exponent function spaces received considerable attentions in recent decades[37].They are important not only in theory as generalizations of classical function spaces,but also for their wide applications in the fields of fluid dynamics,elasticity dynamics,the differential equations with nonstandard growth.We refer to[1,2,12,28]for the details.The rich development can be found in many research works of the theory of variable exponent function spaces.For example,Lebesgue spaces with variable exponent were studied in[10,12,19],Herz spaces with variable exponent were studied in[17,26,29],Morrey spaces with variable exponent were studied in[20,32,34],and some other type of function spaces with variable exponent can be found in[12,13,21,25,35,36,38–41].

Along with the development of the theory of variable exponent function spaces,the theories of the rough singular integral operators and their commutators on these function spaces with variable exponents have attracted many researchers’attentions.In the variable exponent Lebesgue spaces,Cruz-Uribe et al obtained the boundedness of the rough singular integral operator[10].The related works were generalized to Herz spaces with variable exponents.For example,Wang proved the rough singular integral operatorTΩand its commutators are bounded from the variable exponent Herz spaceto the variable exponent Herz space[31].Besides it,Wang et al considered the parameterized Littlewood-Paley operators and their commutators on Herz spaces with variable exponents[33].

Motivated by the above works,in the paper,we devote to solve the boundedness of the rough singular integral operators and their commutators on the homogeneous Herz spaces with variable exponentsThe rest of this paper is arranged as follows.In Section 2 we recall the definition of the homogeneous Herz spaces with variable exponentsand state our main results.The proofs of the main theorems will be proved in Sections 3 and 4,respectively.

Finally,some conventions should be explained.Cis denoted by a positive constant whose value may be different from line to line.The symbolA?Bstands for the inequalityA≤CB.Other notations will be explained when we meet it.

2 Preliminary and main results

Letλ∈ (0,∞)andp(·):Rn→ [1,∞)be a measurable function.The Lebesgue space with variable exponentp(·)is defined by

3 Proof of Theorem 2.1

To give the proof of Theorem 2.1,we need the following lemmas.

4 Proof of Theorem 2.2

The following lemma will play an important role in our proof.

Lemma 4.1([17]).Let p(·)∈B(Rn),k be a positive integer and b∈BMO(Rn),then there is a positive C such that for all balls B inRnand all j,i∈Zwith j＞i,

Acknowledgments

The authors would like to thank the referee for careful reading and valuable suggestions.The work is supported by the National Natural Science Foundation of China(Grant Nos 11971402 and 11901303).