Commutator of Marcinkiewicz Integral Operators on Herz-Morrey-Hardy Spaces with Variable Exponents

In this paper, our aim is to prove the boundedness of commutators generated by the Marcinkiewicz integrals operator [ ] , b µ Ω and obtain the result with Lipschitz function and BMO function f on the Herz-Morrey-Hardy spaces with variable exponents


Introduction
Firstly in 1938, Marcinkiewicz [1] introduced the Marcinkiewicz integral. Next, the Marcinkiewicz integral operator has been studied extensively by many mathematicians in various fields. For example, Stain in [2] introduced the Marcinkiewicz integral operator related to the littlewood-Paley g function on n  and proved that µ Ω is of type ( ) , p p for 1 2 p < ≤ and of week type ( ) 1,1 .
In [3], Ding, Fan and Pan improved the above result and obtained the ( ) Motivated by [6] and [7], the aim of this paper is to study the boundedness for .
for continuous function f and g on [ ] for continuous function f and g on [ ] , a b . for more general functions can be obtained naturally. A further generalization is:

Preliminaries
In this section, we give some preliminaries which we used to prove theorems.
, then there exist positive constants 0 C > , such that for all balls n B ⊂  and all measurable subset R B ⊂ , δ δ are constants with , then there exists a constant 0 C > such that for any balls B in n  , where infimum is taken over all above decomposition of f.

Lipschitz Boundedness for the Commutator of Marcikiewicz Integrals Operator
In this section, we prove the boundedness of the commutator of Marcikiewicz In beginning, we examine a function which we will use in proving : .
x j j n x y t j n x x y t  x y Noting that x y x −  . By the Minkowski's inequality, the generalized Hölder's inequality and the vanishing moments of j b we have x y Now, by using the generalized Hölder's inequality we get: Now by (3.3), (3.4), and lemmas 2.5 and 2.6, we have Firstly we estimate I. We need to show that there exists a positive constant C, [15]), we have the following. Therefore, when 0 Secondly we estimate II . We need to show that there exists a positive constant C, such that II C ≤ ∆ , we consider : .
When 0 q < < ∞ , let Thirdly, we estimate III , we need to show that there exists a positive constant C, such that III C ≤ ∆ .

BMO Boundedness for the Commutator of Marcikiewicz Integrals Operator
In this section, we prove the boundedness of the commutator of Marcikiewicz  : max , .
By lemma (2.12), when 2 , Similarly by lemma 2.4 we have By the boundedness of µ Ω in Firstly we estimate H. We need to show that there exists a positive constant C, such that H C ≤ ∆ Consider . 20]), when 0 1 q < ≤ we have When 1 q < < ∞ and 1 1 1 q q′ + = , and let Now we estimate Secondly, we estimate HH . We need to show that there exists a positive constant C, such that HH C ≤ ∆ Consider Now when 1 q < < ∞ , let 1 Thirdly, we estimate HHH , we need to show that there exists a positive constant C, such that HHH C : .
Now when 0 q    O. Khalil