In 2012, Hongbin Wang and Zongguang Liu [1] discussed boundedness Calderón-Zygmund operator on Herz- type Hardy space with variable exponent. M. Luzki [2] introduced the Herz space with variable exponent and proved the boundedness of some sublinear operator on these spaces. Li’na Ma, Shuhai Li and Huo Tang [3] proved the boundedness of commutators of a class of generalized Calderón-Zygmund operators on Labesgue space with variable exponent by Lipschitz function. Mitsuo Izuki [4] proved the boundedness of commutators on Herz spaces with variable exponent. Lijuan Wang and S. P. Tao [5] proved the boundedness of Littlewood- Paley operators and their commutators on Herz-Morrey space with variable exponent. In this paper we prove the boundedness of commutators of singular integrals with Lipschitz function or BMO function on Herz-type Hardy space with variable exponent.

In this section, we will recall some definitions.

Definition 1.1. Let T be a singular integral operator which is initially defined on the Schwartz space. Its values are taken in the space of tempered distributions such that for x not in the support of f,

where f is in, the space of compactly bounded function.

Let Here the kernel k is function in away from the diagonal and satisfies the standard estimate

and

provided that

provided that such that is called standard kernel and the class of all kernels that

satisfy (1.2), (1.3), (1.4) is denoted by. Let T be as in (1.1) with kernel. If T is bounded from L^p to L^p with, then we say that T is Calderón-Zygmund operator.

Let Ω be a measurable set in with. We first defined Lebesgue spaces with variable exponent.

Definition 1.2. [4] Let be a measurable function. The Lebesgue space with variable exponent is defined by

The space is defined by

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

We denote

.

Then consists of all satisfying and.

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

Let

Proposition 1.1. See [1] . If satisfies

then, we have.

Proposition 1.2. [6] Suppose that, if then

for all balls with.

Definition 1.3. [7] Let, and. The homogeneous Herz space with variable exponent is defined by

where

The non-homogeneous Herz space with variable exponent is defined by

where

Definition 1.4. [1] Let, and and. Suppose that is maximal function of f. Homogeneous variable exponent Herz-tybe Hardy spaces is defined by

with norm

Definition 1.5. [1] Let, , and non negative integer

A function g on is said to be a central, if satisfies

1);

2);

3).

What’s more, when,

Definition 1.6. [7] the Lipschiz space is defined by

Definition 1.7. For, the bounded mean oscillation space is defined by

In order to prove result, we need recall some lemma.

Lemma 2.1. ( [3] ) Let, T be Calderón-Zygmund operator, ,

Then,

Lemma 2.2. ( [8] ) Let; if and, then

where

Lemma 2.3. ( [2] ) Let. Then for all ball B in,

Lemma 2.4. ( [2] ) Let then for all measurable subsets, and all ball B in

where, are constants with

Lemma 2.5. ( [4] ) Let, and with then

Lemma 2.6. ( [9] ) Let function and T be a Calderón-Zygmund operator. Then

Theorem 2.1. Let, , , , and

where are a constants, then are bounded from to.

Proof: we suffices to prove homogeneous case. Let, in the sense, where each is a central -atom with supp. Write

We have

By virtue of Lemma 2.1, we can easily see that

First we estimate F₁. For each and we shall get

Thus by Lemma 2.3, Lemma 2.4 and Proposition 1.2, we get

When and, by Hölder’s inequality and (2.8), we calculations

where by, we get

Now we estimate F₃. For each, we shall get

Using the Lemma 2.3 and Lemma 2.4 and Proposition 1.2, we obtain

When and, by Hölder’s inequality and (2.12), we have

When by, we have

Combining (2.10)-(2.14), we get

Theorem 2.2. Let, , , and where are a

constants, then are bounded from to.

Proof: we suffices to prove homogeneous case. Let, in the sense, where each is a central -atom with supp. Write

We have

By inequality (2.5)we have

Firstly we estimate F₂ by Lemma 2.6 we can see

Now we consider the estimates of F₁. Note that for each, , and, by generalized Hölder’s inequality and Lemma 2.2, we have

Thus by Lemma 2.5 we get

Thus by Lemma 2.3, Lemma 2.4 and noting that we get

When and, by Hölder’s inequality and (2.17), we calculations

when by, we get

Finally we consider the estimates of F₃. Note that for each, , and, by generalized Hölder’s inequality and Lemma 2.2. we have

Thus by Proposition 1.2, and Lemma 2.5, we get

Thus by Lemma 2.3, Lemma 2.4 and noting that we get

When and, by Hölder’s inequality and (2.22),we calculations

when by, we get

combining (2.14)-(2.24) the prove is completed.

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

Omer Abdalrhman,Afif Abdalmonem,Shuangping Tao, (2016) Boundedness for Commutators of Calderón-Zygmund Operator on Herz-Type Hardy Space with Variable Exponent. Journal of Applied Mathematics and Physics,04,1157-1167. doi: 10.4236/jamp.2016.46120