^{1}

^{*}

^{2}

^{1}

The aim of this paper is to study the boundedness of Calderón-Zygmund operator and their commutator on Herz Spaces with two variable exponents
*p*(.),
*q*(.). By applying the properties of the Lebesgue spaces with variable exponent, the boundedness of the Calderón-Zygmund operator and the commutator generated by BMO function and Calderón-Zygmund operator is obtained on Herz space.

Definition 1.1. Let

1)

2) There exists a function

where

3)

A standard operator

if

The bounded mean oscillation BMO space and BMO norm are defined, respectively, by

The commutator of the Calderón-Zygmund operator is defined by

In 1983, J.-L. Jouné proved

Kovácik and Rákosník introduced Lebesgue spaces and Sobolev spaces with variable exponents (see [

Herz spaces play an important role in harmonic analysis. After they were introduced in [

In this paper, we will discuss the boundedness of the Calderón-Zygmund operator

In this section we recall some definitions. Let

Definition 2.1. [

For all compact

The Lebesgue spaces

We denote

Definition 2.2. [

Let

Let

Definition 2.3. [

Equipped the norm

Remark 2.1. [

1)

2) If

where

This implies that

Remark 2.2. Let

where

In this section, we recall some properties and some lemmas of variable exponent belonging to the class

Proposition 3.1. [

Hence we have

Lemma 3.1. [

where

Lemma 3.2. [

where

Proposition 3.2. [

Lemma 3.3. [

Lemma 3.4. [

1.

2.

Lemma 3.5. [

Lemma 3.6. [

Lemma 3.7. [

Theorem 4.1. Suppose that

Proof Let

By Definition 2.3, we have

Since

where

and

Thus,

We easily see that

This implies that we only need to prove

First, we consider

where,

In the above, we use the Proposition 3.2 and Remark 2.2. Since

Here

Let us now turn to estimate

By Lemmas 3.5-3.7 and the fact that

where

Therefore, if

where

If

where

Hence, we see that

Finally, we estimate

By Lemma 3.7 and

where

Then we have

Theorem 4.2. Let

Proof Let

By virtue of the definition of

Since

Let

and

Therefore, we can obtain

Thus it follows that,

Hence

So we can get that

Next we estimate

Thus, from Lemmas 3.4-3.7, We obtain that

Therefore, we get

where

This, for

where

If

where

This implies that

Finally we estimate

and

where

Hence, we arrive at that

This completes the proof of Theorem 4.2.

This paper is supported by National Natural Foundation of China (Grant No. 11561062).

Abdalrhman, O., Abdalmonem, A. and Tao, S.P. (2017) Boundedness of Calderón-Zygmund Operator and Their Commutator on Herz Spaces with Variable Exponent. Applied Mathematics, 8, 428-443. https://doi.org/10.4236/am.2017.84035