Applied Mathematics
Vol.4 No.11C(2013), Article ID:38699,7 pages DOI:10.4236/am.2013.411A3005
Boundedness of Hyper-Singular Parametric Marcinkiewicz Integrals with Variable Kernels
1Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, China
2College of Mathematics and Computer Science, Hunan Normal University, Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), Hunan Normal University, Changsha, China
Email: *fendui@yahoo.com
Copyright © 2013 Qiquan Fang, Xianliang Shi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received September 13, 2013; revised October 13, 2013; accepted October 20, 2013
Keywords: Hyper-Singular Marcinkiewicz Integral; Variable Kernel; Multilinear Commutator; Hardy Type Space
ABSTRACT
In this article, we consider the boundedness of on Hardy type space Where,
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 haveby 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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NOTES
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