Commutator of Marcinkiewicz Integral Operators on Herz-Morrey-Hardy Spaces with Variable Exponents ()
1. Introduction
Firstly in 1938, Marcinkiewicz [1] introduced the Marcinkiewicz integral. Next, the Marcinkiewicz integral operator has been studied extensively by many mathematicians in various fields. For example, Stain in [2] introduced the Marcinkiewicz integral operator related to the littlewood-Paley
function on
and proved that
is of type
for
and of week type
. In [3], Ding, Fan and Pan improved the above result and obtained the
and weighted
boundedness of the Marcinkiewicz cussed the boundedness for the commutator generated by the Marcinkiintegral
under some weak conditions. Torchinsky and Wang in [4] discussed integral
and
function on Lebesgue spaces
.
On the other hand, a class of functional spaces called Herz-Morrey-Hardy spaces with variable exponent has attracted great interest in recent years. We find that in successive studies in this field, in [5] [6] Xu, Yang introduced Herz-Morrey-Hardy spaces with variable exponents and their some applications. He obtained that certain singular integral operators are bounded from Herz-Morrey-Hardy spaces with variable exponents into Herz-Morrey spaces with variable exponents as an application of the atomic characterization. Also, he established their molecular decomposition, and by using their atomic and molecular decompositions, he gave the boundedness of a convolution type singular integral on Herz-Morrey-Hardy spaces with variable exponents. Omer in [7] proved the boundedness of commutators generated by the Calderón-Zygmund and used properties of variable exponent, BMO(Rn) function and Lipschitz function to prove this boundedness. Also, Yang in [8] established some boundedness for
and
on the homogeneous Morrey-Herz-type Hardy spaces with variable exponents and studied Boundedness of Calderón-Zygmund operator on these spaces.
Suppose
denotes the unit sphere in
equipped with the normalized measure
. Let
be homogenous function of degree zero and satisfies
(1.1)
where
for any
.
Then the Marcinkiewicz integral operator
is defined by
(1.2)
where
(1.3)
Let
and
be a locally integrable function on
, the commutator generated by the Marcinkiewicz integral
and b is defined by
(1.4)
Motivated by [6] and [7], the aim of this paper is to study the boundedness for the commutator of Marcinkiewicz integral operator
on the Herz-Morrey-Hardy space with variable exponent where
for
, with BMO function and Lipschitz function, we will define The definitions of the Morrey-Herz spaces with variable exponents, the Morrey-Herz-Hardy spaces with variable exponents (which will be defined in the next section), and the preliminary lemmas are presented in Section 2. In Section 3, we will prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent with
. Lastly, in Section 4 we will prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent with function
.
A given open set
and a measurable function
,
denotes the set of measurable function f on
such that for some
,
(1.5)
the space
is defined by
(1.6)
The Lebesgue spaces
is Banach spaces with the norm defined by
(1.7)
where
,
.
Denotes
. Let M be the Hardy-Littlewood maximal operator. We denote
to be the set of all functions
satisfying the M is bounded on
.
Definition 1.1. [6]
Let
,
,
. Let
be a bounded real-valued measurable function on
The nonhomogeneous Morrey-Herz space
and homogeneous Morrey-Herz space with variable exponents
are respectively defined by
(1.8)
and
(1.9)
where
(1.10)
(1.11)
Definition 1.2. [9]
For all
, the Lipschitz space
is defined by
(1.12)
Definition 1.3. [5]
Let
and
. The nonhomogeneous Herz-Morrey-Hardy space with variable exponent
and homogeneous Herz-Morrey-Hardy space with variable exponents
are respectively defined by
(1.13)
(1.14)
Definition 1.4. [10] (Hölder’s inequality) Let
and
. Then the discrete and integral forms of Hölder’s inequality are given as
(1.15)
for continuous function f and g on
.
Definition 1.5. [10] (Minkowski’s inequality) Let
. Then the discrete and integral forms of Minkowski’s inequality are given as
(1.16)
for continuous function f and g on
. for more general functions can be obtained naturally. A further generalization is: If
, then
(1.17)
2. Preliminaries
In this section, we give some preliminaries which we used to prove theorems.
Lemma 2.1. [11] Let
. Then for any
and
, we have
where
.
This inequality is called the generalized Hölder inequality with respect to the variable
spaces.
Lemma 2.2. [12] Given
, for any
, when
, we get
where
.
Proposition 2.3. [13] If
satisfies
then
.
Lemma 2.4. [14] Let k be a positive integer and B be a ball in
. Then we have that for all
and
with
, we have
1)
,
2)
,
where
and
.
Lemma 2.5. [15] Let
, then there exist positive constants
, such that for all balls
and all measurable subset
,
where
are constants with
.
Lemma 2.6. [16] If
, then there exists a constant
such that for any balls B in
,
Lemma 2.7. [6] Let
, and
be log-Hölder continuous both at the origin and infinity,
,
as in lemma 2.4. Then
(or
) if and only if
(or
), in the sense of
, where each
is a central
atom with support contained in
and
or
,
moreover
or
,
where infimum is taken over all above decomposition of f.
Lemma 2.8. [17] Let
and
. If
, then
Lemma 2.9. [18] Let
satisfies
-Dini condition with
. If there exist constants
and
such that
, then for every
, we have
Lemma 2.10. [15] Given E, let
be a measurable function (with respect to product measure) such that for almost every
. Then
Lemma 2.11. [19] If
and
, then
Lemma 2.12. [19] Let
satisfies Proposition 2.3. Then
for every cube (or ball)
, where
.
3. Lipschitz Boundedness for the Commutator of Marcikiewicz Integrals Operator
In this section, we prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent so when
under some conditions.
Theorem 3.1.
Suppose that
with
. If
satisfies proposition 2.3 with
,
with
and satisfies
let
and
or (
). Then the commutator
is bounded from
(or
) to
(or
).
To the proof the above theorem, we will recall the following lemma.
Lemma 3.1. [15]
Suppose that
with
. If
satisfies Proposition 2.3 with
with
. Then the commutator
is bounded from
to
.
Next, we will give the Lipschitz estimate about the commutator
on Herz-Morrey-Hardy spaces with variable exponent.
Proof Theorem 3.1:
To prove this theorem, we only prove the homogeneous case. Let
. By lemma 2.6 we have
converged in
, where each
is a central
atom with support contained in
and
Here we denote
. By lemma 2.8 we have
In beginning, we examine a function which we will use in proving
When
and
with
, it follows from
that
. We have
(3.1)
Then by (3.1), the Minkowski’s inequality, the generalized Hölder’s inequality and the vanishing of the moment of
we have
Similarly, we consider
. Noting that
. By the Minkowski’s inequality, the generalized Hölder’s inequality and the vanishing moments of
we have
So we have
From lemma 2.10 and the Minkowski’s inequality we have
For
, noting
, we denote
and
. By lemma 2.2 we have
When
and
, by Lemma 2.12 we have
When
we have
So we obtain
By lemma 2.9 we have
Now, by using the generalized Hölder’s inequality we get:
(3.2)
For
similar to the method of
we have
Now, by using the generalized Hölder’s inequality we get:
(3.3)
Now by (3.3), (3.4), and lemmas 2.5 and 2.6, we have
Firstly we estimate I. We need to show that there exists a positive constant C, such that
, we consider
By the
, bounbedness of the commutator
on
(see [15] ), we have the following. Therefore, when
(3.4)
When
, let
we have
(3.5)
We estimate
by lemma 2.1 when
by
, we get
(3.6)
When
, let
. Since
, by Hölder’s inequality, we have
(3.7)
Secondly we estimate
. We need to show that there exists a positive constant C, such that
, we consider
When
, we get
(3.8)
When
, let
we have
(3.9)
For
, when
, by
we get
(3.10)
When
, let
. Since
, by Hölder’s inequality, we have
(3.11)
Thirdly, we estimate
, we need to show that there exists a positive constant C, such that
When
, by the boundedness of
in
( [20] ), we have
(3.12)
When
, by
and the boundedness of
in
( [20] ) and Hölder’s inequality, we get
(3.13)
When
, by
we get
(3.14)
When
, let
. Since
, and by Hölder’s inequality, we have
(3.15)
Joint the estimates for I, II and III, we obtain
Then we complete the proof of Theorem 3.1.
4. BMO Boundedness for the Commutator of Marcikiewicz Integrals Operator
In this section, we prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent with function
.
Theorem 4.1.
Suppose that
with
. If
satisfies proposition 2.3 and
. Let
and
(or
). Then
is bounded from
(or
) to
(or
).
proof:
In a way similar to theorem (3.2) we only prove the homogeneous case. Let
and
. Let us write
Then we have
From the Hölder’s inequality, we have
Noting
, we denote
and
. By lemmas 3.2, 3.10 we have
By lemma (2.12), when
and when
respectively we have
,
and
we obtain
.
So we have
(4.1)
Similarly by lemma 2.4 we have
(4.2)
Now, by (4.1), (4.2), lemmas 2.4, 2.5 and 2.3, we have
(4.3)
By the boundedness of
in
see [7], we have
So we have
Firstly we estimate H. We need to show that there exists a positive constant C, such that
Consider
By boundedness of
in
, see ( [20] ), when
we have
(4.4)
When
and
, and let
, we have