Boundedness of Hyper-Singular Parametric Marcinkiewicz Integrals with Variable Kernels ()
1. Introduction
A function 
defined on 
is said to belong to
, if it satisfies the following three conditions:
1) 
for any 
and any
;
2) 
3)
, for any
.
In [1], the authors considered the hyper-singular parametric Marcinkiewicz integral with variable kernel as follows:
where
When
, we set
, which is the parametric Marcinkiewicz integral with variable kernels considered in [2].
For 
the homogenous Lipschitz space 
is the space of function 
such that
where 
denotes 
-th difference operator (see [3]).
In 2006, Lu and Xu studied the boundedness of the commutator of 
in [4]. They proved that:
Theorem A [4]. Suppose 
for 
If 
and
, then 
maps 
continuously into
. Here 
is defined as follows:
(1)
where
Let 
In this article, we mainly consider the commutator 
defined by
(2)
where
Given any positive integer
, for all
, we denote by 
the family of all finite subsets 
of 
of 
different elements. For any
, we associate the complementary sequence 
given by
, (see [5]).
For any
, we will denote 
and the product 
When
, we have
by definition, we have
. Similarly, when
, we have 
and
. With this notation, if
we write 
When
, we write 
Definition1.1.Let
, 
be defined as above such that
.
A function 
on 
is called a 
-atom if 1)
, for some 
and
;
2) 
3) 
for any
and 
Definition 1.2. Let
, we say that a distribution 
on 
belongs to 
if and only if 
can be written as 
in the distributional sensewhere each 
is a 
-atom and
Moreover,
with the infimum taken over all the above decompositions of 
as above Definition 1.3. A function 
is said to satisfy the 
-Dini condition, if
(3)
where 
denotes the integral modulus of continuity of order 
of 
defined by
We will denote simply 
-Dini condition for 
- Dini condition when
.
2. Main Theorem
Now let us formulate our main results as follows.
Theorem 2.1. Suppose that 
is the commutator
(2), and let
, then 
is bounded from 
into
. That is,
Theorem 2.2. Suppose that 
is the commutator
(2), and let 
If
satisfies the following two conditions:
1) 
satisfies 
-Dini condition (3);
2) there exists
such that 
then 
is bounded from 
into
. That is
Remark Obviously, 
is the commutator of the operator 
in [1]. At the same time, we change the course of the statement in [4].
In order to prove our Theorems, we need several preliminary lemmas.
Lemma 2.1. [6] Let 
and suppose 
If there exists a constant 
such that
, then for any
,
where the constant 
is independent of 
and
.
lemma 2.2. [7] Let
, 
and 
be defined as 
If there exists
, such that 
then
is bounded from 
into
. That is
3. Proofs
3.1. Proof of Theorem 2.1.
Applying the Minkowski’ inequality, we can get
By Lemma 2.2 , we have
This completes the proof of Theorem 2.1.
3.2. Proof of Theorem 2.2.
Noting that
, we can choose 
such that
. It is easy to see that
. Next , we choose 
such that 
It follows from Theorem 
that 
is bounded from 
into
. That is
(4)
By the atomic decomposition theory on Hardy type space, it suffices to prove that there is a constant 
such that for all 
-atom the following holds
Without loss of generality we may assume that
. We write 
We split
into two parts as follows:
We can easily see that
. By (4) and the size condition of atom
, we have
Next we estimate
. Let us consider
:
for
. By the mean value theorem, we have
Thus, by the Minkowski’s inequality for integrals,
Applying the Hölder inequality and the size condition of
, we have
So we can get
Noting that 
we have
For
, we write
So 
is dominated by
Now let us estimate
. By the vanishing condition of
, we have
where 
Since 
we get from Hölder’s inequality and Lemma
,
Now we estimate
. Applying Minkowski's inequality, the size condition of
, we obtain
So we have
Thus
So when
, we have
Combining the estimates for 
and
, we have
This completes the proof of Theorem 2.2.
4. Acknowledgements
The authors would like to thank anonymous reviewers for their comments and suggestions. The authors are partially supported by project 11226108, 11071065, 11171306 funded by NSF of China, project 20094306110004 funded by RFDP of high education of China.
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