Some New Boundedness Results for Variable Marcinkiewicz Fractional Integral Operator on Herz-Morrey-Hardy Spaces

Abstract

In this paper, we define the idea of Herz-Morrey-Hardy spaces by using variable Herz-Morrey spaces and Hardy spaces. Then we give the atomic characterization of these spaces by using the grand maximal function.    Then our main objective is to prove   the boundedness of higher order commuatators of variable Marcinkiewicz fractional integral operator on Herz-Morrey-Hardy spaces where the exponents defining these spaces are variable. These results also holds for variable  Herz-Hardy spaces. The higher order commutators of variable Marcinkiewicz fractional integral operator is the generalization of Marcinkiewicz integral operators,  variable Marcinkiewicz fractional integral operator and commutators on Marcinkiewicz fractional integral operators, so these proofs generalize some previous results.