Boundedness of fractional Marcinkiewicz integrals with variable kernel on Hardy type spaces

The boundedness of the fractional type Marcinkiewicz integrals μ Ω,α with variable kernels on Hardy spaces and Herz type Hardy spaces are established.Some properties of μ Ω,α which kernel satisfies certain condition are given by the atomic decomposition theorem.Also,the integral μ Ω,α is an operator of type H1,Ln/(n(α)) and of type(Hp,Lq).The boundedness from Herz type Hardy spaces to Herz spaces about the integral μ are presented.