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Boundedness of Calderón-Zygmund Operator and Their Commutator on Herz Spaces with Variable Exponent

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ABSTRACT

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.

1. Introduction

Definition 1.1. Let be a bounded linear operator from to (see [1] , [2] ). is called a standard operator if satisfies the following conditions:

1) extends to a bounded linear operator on.

2) There exists a function defined by satisfies

(1.1)

where.

3) for with

A standard operator is called a -CalderónZygmund operator if is a standard kernel satisfies:

(1.2)

(1.3)

if for some.

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

(1.4)

(1.5)

The commutator of the Calderón-Zygmund operator is defined by

(1.6)

In 1983, J.-L. Jouné proved -CalderónZygmund operator is bounded on in [3] . Coifman, Rochberg and Weiss proved that commutator [b,T] is bounded on (see [4] ).

Kovácik and Rákosník introduced Lebesgue spaces and Sobolev spaces with variable exponents (see [5] ). The function spaces with variable exponent has been recently obtained an increasing interest by a number of authors since many applications are found in many different fields, for example, in fluid dynamics (see [6] ), image restoration (see [7] [8] [9] ) and differential equations.

Herz spaces play an important role in harmonic analysis. After they were introduced in [10] , the boundedness of some operators and some characteriza- tions of Herz spaces with variable exponents were studied extensively (see [11] - [16] ). In 2015, Wang and Tao introduced the Herz spaces with two variable exponents, and studied the parameterized Littlewood-Paley operators and their commutators on Herz spaces with variable exponents in [17] .

In this paper, we will discuss the boundedness of the Calderón-Zygmund operator and their commutator are bounded on Herz spaces with two variable exponents.

2. Definitions of Function Spaces with Variable Exponent

In this section we recall some definitions. Let be a measurable set in with. We firstly recall the definition of the Lebesgue spaces with variable exponent.

Definition 2.1. [5] Let be a measurable function. The Lebesgue space with variable exponent is defined by

(2.1)

For all compact, the space is defined by

(2.2)

The Lebesgue spaces is a Banach spaces with the norm defined by

(2.3)

We denote. Then consists of all satisfying and. Let be the Hardy-Littlewood maximal operator. We denote to be the set of all function satisfying the is bounded on.

Definition 2.2. [18] Let. The mixed Lebesgue sequence space with variable exponent is the collection of all sequences of the measurable functions on such that

(2.4)

Let, , for, we have that

(2.5)

Let,

Definition 2.3. [17] Let. The homogeneous Herz space with variable exponent is defined by

Equipped the norm

Remark 2.1. [17] Let satisfying and satisfy the following results:

1)

2) If and. For any, by using Lemma 3.7 and Remark 2.2, we have

where

This implies that.

Remark 2.2. Let. Then we have

where

3. Properties and Lemmas of Variable Exponent

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

Proposition 3.1. [19] If satisfies

(3.1)

(3.2)

Hence we have.

Lemma 3.1. [5] Given have that for all functions and,

(3.3)

where.

Lemma 3.2. [5] Suppose that, for any , when, we get

(3.4)

where.

Proposition 3.2. [20] Let and be a CalderónZygmund operator. Then we have

(3.5)

Lemma 3.3. [20] Let function and be a CalderónZygmund operator.Then

(3.6)

Lemma 3.4. [11] Let. If with, then we have

1.

2.

Lemma 3.5. [21] Let, then there exist constants, and such that for all balls and all measurable subset,

(3.7)

Lemma 3.6. [11] If, there exist a constant such that for any balls B in, we have

(3.8)

Lemma 3.7. [17] Suppose that. If, then

(3.9)

4. The Main Theorems and Their Proofs

Theorem 4.1. Suppose that with . If with as defined in Lemma 3.5, then the operator is bounded from to.

Proof Let. We write

By Definition 2.3, we have

(4.1)

Since

(4.2)

where

(4.3)

(4.4)

and

Thus,

We easily see that

(4.6)

This implies that we only need to prove. Denote

First, we consider. By virtue of Lemma 3.7, we get

(4.7)

where,

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

Here and. That is

(4.8)

Let us now turn to estimate. Noting that and, by the generalized Hölder's inequality and the Minkowski’s inequality, we get

(4.9)

By Lemmas 3.5-3.7 and the fact that, we easily see that

(4.10)

where

Therefore, if and, we can get

where.

If and. By Remark 2.2 and applying the generalized Hölder’s inequality, we obtain

where.

Hence, we see that

(4.11)

Finally, we estimate. Noting that for each and, we have

(4.12)

By Lemma 3.7 and, we get

(4.13)

where

Then we have, by using the same argument in. Thus, we prove Theorem 4.1.

Theorem 4.2. Let. Suppose that with. If with as defined in lemma 3.5, then the commutator is bounded from to.

Proof Let.We write

By virtue of the definition of, we have

(4.14)

Since

(4.15)

Let

(4.16)

(4.17)

(4.18)

and

Therefore, we can obtain

Thus it follows that,

(4.20)

Hence. Denoting, firstly we estimate as in Theorem 4.1. Applying Lemma 3.3, we imme- diately arrive at

So we can get that

(4.21)

Next we estimate, Let.

(4.22)

Thus, from Lemmas 3.4-3.7, We obtain that

Therefore, we get

(4.23)

where

This, for, , along with Remark 2.2, tells us that

where

If, it is follows from Remark 2.2 and Hölder’s inequality that

where.

This implies that

(4.24)

Finally we estimate, for any, by the same way to argument in, we obtain that

(4.25)

and

(4.26)

where

Hence, we arrive at that by the similar argument in the proof Theorem 4.1.

This completes the proof of Theorem 4.2.

Acknowledgements

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

Cite this paper

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

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