Boundedness for Commutators of Calderón-Zygmund Operator on Herz-Type Hardy Space with Variable Exponent ()
Received 25 May 2016; accepted 26 June 2016; published 29 June 2016

1. Introduction
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,
(1.1)
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
(1.2)
and
(1.3)
provided that 
(1.4)
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 Lp to Lp 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
(1.5)
The space
is defined by
![]()
The Lebesgue space
is a Banach space with the norm defined by
(1.6)
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
(1.7)
(1.8)
then, we have
.
Proposition 1.2. [6] Suppose that
, if
then
(1.9)
for all balls
with
.
Definition 1.3. [7] Let
,
and
. The homogeneous Herz space with variable exponent
is defined by
(1.10)
where
(1.11)
The non-homogeneous Herz space with variable exponent
is defined by
(1.12)
where
(1.13)
Definition 1.4. [1] Let
,
and
and
. Suppose that
is maximal function of f. Homogeneous variable exponent Herz-tybe Hardy spaces
is defined by
(1.14)
with norm
(1.15)
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
,
(1.16)
Definition 1.6. [7]
the Lipschiz space is defined by
(1.17)
Definition 1.7. For
, the bounded mean oscillation space
is defined by
![]()
2. Main Result and Proof
In order to prove result, we need recall some lemma.
Lemma 2.1. ( [3] ) Let
, T be Calderón-Zygmund operator,
,
Then,
(2.1)
Lemma 2.2. ( [8] ) Let
; if
and
, then
(2.2)
where ![]()
Lemma 2.3. ( [2] ) Let
. Then for all ball B in
,
(2.3)
Lemma 2.4. ( [2] ) Let
then for all measurable subsets
, and all ball B in ![]()
(2.4)
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
(2.5)
(2.6)
By virtue of Lemma 2.1, we can easily see that
![]()
First we estimate F1. For each
and we shall get
(2.7)
![]()
Thus by Lemma 2.3, Lemma 2.4 and Proposition 1.2, we get
(2.8)
When
and
, by Hölder’s inequality and (2.8), we calculations
(2.9)
where
by
, we get
(2.10)
Now we estimate F3. For each
, we shall get
(2.11)
![]()
Using the Lemma 2.3 and Lemma 2.4 and Proposition 1.2, we obtain
(2.12)
When
and
, by Hölder’s inequality and (2.12), we have
(2.13)
When
by
, we have
(2.14)
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
(2.16)
Thus by Lemma 2.3, Lemma 2.4 and noting that
we get
(2.17)
When
and
, by Hölder’s inequality and (2.17), we calculations
(2.18)
when
by
, we get
(2.19)
Finally we consider the estimates of F3. Note that for each
,
, and
, by generalized Hölder’s inequality and Lemma 2.2. we have
(2.20)
Thus by Proposition 1.2, and Lemma 2.5, we get
(2.21)
Thus by Lemma 2.3, Lemma 2.4 and noting that
we get
(2.22)
When
and
, by Hölder’s inequality and (2.22),we calculations
(2.23)
when
by
, we get
(2.24)
combining (2.14)-(2.24) the prove is completed.
Acknowledgements
This paper is supported by National Natural Foundation of China (Grant No. 11561062).
NOTES
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*Corresponding author.