Two operators related to Marcinkiewicz integrals

In this paper, the authors consider the boundedness of generalized higher commutator of Marcinkiewicz integral $ \mu _\Omega ^{\vec b} $ \mu _\Omega ^{\vec b} , multilinear Marcinkiewicz integral μ _Ω ^A and its variation $ \tilde \mu _\Omega ^A $ \tilde \mu _\Omega ^A on Herz-type Hardy spaces, here Ω is homogeneous of degree zero and satisfies a class of L ^s -Dini condition. And as a special case, they also get the boundedness of commutators of Marcinkiewicz integrals on Herz-type Hardy spaces.