Commutators And Operators On The Herz-Hardy Space

In this paper,we first introduce the definition and simple properties of the variable Lebesgue space,the variable index Herz space,the variable Herz-type Hardy space.Using the atomic disappearance and the commonly used inequality,we obtain the boundedness of fractional integral commutator from the variable Herz-type Hardy space HKq1(·)?,p1(Rn)to the variable Herz space Kq1(·)?,p1(Rn).Based on the boundedness of fractional commu-tators,it is proved that Marcinkiewicz integral commutator from HKq1(·)?,p1(Rn)space to Kq1(·)?,p1(Rn)space is bounded.Finally,we study the boundedness of multi-linear fractional integral from HKq1(·)?,p1(Rn)space to Kq1(·)?,p1(Rn)space.