Variable Exponent Herz Type Hardy Spaces and Their Applications

Variable Exponent Herz Type Hardy Spacesand Their

Applications

Baohua Dong1and Jingshi Xu2，

【期刊名称】分析、理论与应用（英文版）

【年(卷),期】2015(031)004

【总页数】33

【关键词】Key Words：Hardy space，Herz space，variable exponent，maximal

function，atom，molecule. AMS Subject Classifications:42B35，42B25，42B20

Abstract.In this paper，the authors introduce certain Herz type Hardy spaces with variableexponents andestablish the characterizationsof these spacesin terms ofatomic and molecular http://doc.xuehai.neting these decompositions，the authors obtain the boundedness of some singular integral operators on the Herz type Hardy spaces with variable exponents.

1　Introduction

In 1991，Kov´aˇcik and R´akosn´ık introduced basic properties of variable Lebesgue and Sobolev spaces in［23］.After that many spaces with variable exponents appeared，for example:Bessel potential spaces with variable exponent，Besov and Triebel-Lizorkin spaces with variable exponents，Morrey spaces with variable exponents，Hardy spaces with variable exponent and so on，see［2，3，9，11，15，17，21，22，31，41］and reference therein. Indeed，the atomic，molecular and wavelet decompositions of variable exponent Besov and Triebel-Lizorkin spaces were given in［3，9，21，22，42］.The duality and reflexivity of spaces Bsp（·），qand Fsp（·），qwere discussed in［33］.The atomic and molecular decompositions of Hardy spaces with variable exponent and their applications for the boundedness of singular integral operators were obtained

in［31，34］.Variable exponent spaces have many applications，for instance in differential equations，see the article［16］.

In parallel to the above，there are other type function spaces，Herz type spaces，have attractedmany authors'interestsforlast threedecades.Actually，many propertiesofclassic Lebesgue spaces have been generalized to Herz type spaces.We outline some recent results we have concerned.In 2010，Izuki proved the boundednessof sublinear

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