^{1}

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In this article, we consider the boundedness of

A function defined on is said to belong to, if it satisfies the following three conditions:

1) for any and any;

2)

3), for any.

In [

where

When, we set, which is the parametric Marcinkiewicz integral with variable kernels considered in [

For the homogenous Lipschitz space is the space of function such that

where denotes -th difference operator (see [

In 2006, Lu and Xu studied the boundedness of the commutator of in [

Theorem A [

where

Let

In this article, we mainly consider the commutator defined by

where

Given any positive integer, for all, we denote by the family of all finite subsets of of different elements. For any, we associate the complementary sequence given by, (see [

For any, we will denote and the product When, we haveby definition, we have. Similarly, when, we have and. With this notation, if

we write

When, we write

Definition1.1.Let, be defined as above such that.

A function on is called a -atom if 1), for some and;

2)

3) for any

and

Definition 1.2. Let, we say that a distribution on belongs to if and only if

can be written as in the distributional sensewhere each is a -atom and

Moreover,

with the infimum taken over all the above decompositions of as above Definition 1.3. A function is said to satisfy the -Dini condition, if

where denotes the integral modulus of continuity of order of defined by

We will denote simply -Dini condition for - Dini condition when.

Now let us formulate our main results as follows.

Theorem 2.1. Suppose that is the commutator

(2), and let, then is bounded from into. That is,

Theorem 2.2. Suppose that is the commutator

(2), and let If

satisfies the following two conditions:

1) satisfies -Dini condition (3);

2) there exists

such that then is bounded from into. That is

Remark Obviously, is the commutator of the operator in [

In order to prove our Theorems, we need several preliminary lemmas.

Lemma 2.1. [

,

where the constant is independent of and.

lemma 2.2. [

, such that then

is bounded from into. That is

Applying the Minkowski’ inequality, we can get

By Lemma 2.2 , we have

This completes the proof of Theorem 2.1.

Noting that, we can choose such that. It is easy to see that. Next , we choose such that It follows from Theorem that is bounded from into. That is

By the atomic decomposition theory on Hardy type space, it suffices to prove that there is a constant such that for all -atom the following holds

Without loss of generality we may assume that

. We write We split

into two parts as follows:

We can easily see that. By (4) and the size condition of atom, we have

Next we estimate. Let us consider:

for

. By the mean value theorem, we have

Thus, by the Minkowski’s inequality for integrals,

Applying the Hölder inequality and the size condition of, we have

So we can get

Noting that we have

For, we write

So is dominated by

Now let us estimate. By the vanishing condition of, we have

where

Since we get from Hölder’s inequality and Lemma,

Now we estimate. Applying Minkowski's inequality, the size condition of, we obtain

So we have

Thus

So when, we have

Combining the estimates for and, we have

This completes the proof of Theorem 2.2.

The authors would like to thank anonymous reviewers for their comments and suggestions. The authors are partially supported by project 11226108, 11071065, 11171306 funded by NSF of China, project 20094306110004 funded by RFDP of high education of China.