Notes on the Herz-type Hardy spaces of variable smoothness and integrability

The aim of this paper is twofold.First we give a new norm equivalents of the variable Herz spaces KSecondly we use these results to prove the atomic decomposition for Herz-type Hardy spaces of variable smoothness and integrability.Also, we prove the boundedness of a wide class of sublinear operators on these spaces, which includes maximal, potential and Calderón-Zygmund operators.