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Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space
Journal of Inequalities and Applications volume 2011, Article number: 479576 (2011)
Abstract
Let , the authors introduce in this paper a class of the hypersingular Marcinkiewicz integrals along surface with variable kernels defined by
, where
with
. The authors prove that the operator
is bounded from Sobolev space
to
space for
, and from Hardy-Sobolev space
to
space for
. As corollaries of the result, they also prove the
boundedness of the Littlewood-Paley type operators
and
which relate to the Lusin area integral and the Littlewood-Paley
function.
1. Introduction
Let be the
-dimensional Euclidean space and
be the unit sphere in
equipped with the normalized Lebesgue measure
. For
, let
.
Before stating our theorems, we first introduce some definitions about the variable kernel . A function
defined on
is said to be in
,
, if
satisfies the following two conditions:
(1), for any
and any
;
(2).
In 1955, Calderón and Zygmund [1] investigated the boundedness of the singular integrals
with variable kernel. They found that these operators connect closely with the problem about the second-order linear elliptic equations with variable coefficients. In 2002, Tang and Yang [2] gave
boundedness of the singular integrals with variable kernels associated to surfaces of the form
, where
for any
. That is, they considered the variable Calderón-Zygmund singular integral operator
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ1_HTML.gif)
On the other hand, as a related vector-valued singular integral with variable kernel, the Marcinkiewicz singular with rough variable kernel associated with surfaces of the form is considered. It is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ2_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ3_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ4_HTML.gif)
If , we put
. Historically, the higher dimension Marcinkiewicz integral operator
with convolution kernel, that is
, was first defined and studied by Stein [3] in 1958. See also [4–6] for some further works on
with convolution kernel. Recently, Xue and Yabuta [7] studied the
boundedness of the operator
with variable kernel.
Theorem 1.1 (see [7]).
Suppose that is positively homogeneous in
of degree 0, and satisfies (1.4) and
(2′), for some
. Let
be a positive and monotonic (or negative and monotonic)
function on
and let it satisfy the following conditions:
(i) for some
;
(ii) is monotonic on
.
Then there is a constant C such that , where constant
is independent of
.
Since the condition (2) implies (2′), so the boundedness of
holds if
with
.
Our aim of this paper is to study the hypersingular Marcinkiewicz integral along surfaces with variable kernel
, and with index
, on the homogeneous Sobolev space
for
and the homogeneous Hardy-Sobolev space
for some
. Let
be as above, we then define the operators
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ5_HTML.gif)
Our main results are as follows.
Theorem 1.2.
Suppose that ,
satisfies (1.4) and
with
. Let
be a positive and increasing
function on
and let it satisfy the following conditions:
(i);
(ii) on
.
Then there is a constant C such that , where constant
is independent of
.
Theorem 1.3.
Suppose , and that
, with
, and satisfies (1.4). Let
be a positive and increasing
function on
and let it satisfy the following conditions:
(i);
(ii),
.
Then, for , there is a constant C such that
, where constant
is independent of any
.
Furthermore, our result can be extended to the Littlewood-Paley type operators and
with variable kernels and index
, which relate to the Lusin area integral and the Littlewood-Paley
function, respectively. Let
be as above, we then define the operators
and
for
, respectively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ6_HTML.gif)
with , where
. As an application of Theorem 1.2, we have the following conclusion.
Theorem 1.4.
Under the assumption of Theorem 1.2, then Theorem 1.2 still holds for and
.
By Theorems 1.2 and 1.3 and applying the interpolation theorem of sublinear operator, we obtain the boundedness of
.
Corollary 1.5.
Suppose , and that
,
, and satisfies (1.4). Let
be given as in Theorem 1.3. Then, for
, there exists an absolute positive constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ7_HTML.gif)
for all .
Remark 1.6.
It is obvious that the conclusions of Theorem 1.2 are the substantial improvements and extensions of Stein's results in [3] about the Marcinkiewicz integral with convolution kernel, and of Ding's results in [8] about the Marcinkiewicz integral
with variable kernels.
Remark 1.7.
Recently, the authors in [9] proved the boundedness of hypersingular Marcinkiewicz integral with variable kernels on homogeneous Sobolev space for
and
without any smoothness on
. So Corollary 1.5 extended the results in [9, Theorem  5].
Throughout this paper, the letter always remains to denote a positive constant not necessarily the same at each occurrence.
2. The Bounedness on Sobolev Spaces
Before giving the definition of the Sobolev space, let us first recall the Triebel-Lizorkin space.
Fix a radial function satisfying
and
, and
if
. Let
. Define the function
by
, such that
.
For ,
, and
, the homogeneous Triebel-Lizorkin space
is the set of all distributions
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ8_HTML.gif)
For , the homogeneous Sobolev spaces
is defined by
, namely
. From [10] we know that for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ9_HTML.gif)
and if is a nonnegative integer, then for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ10_HTML.gif)
For , we define the homogeneous Hardy-Sobolev space
by
. It is well known that
for
, one can refer [10] for the details.
Next, let us give the main lemmas we will use in proving theorems.
Lemma 2.1 (see [11]).
Suppose that and
has the form
where
is a solid spherical harmonic polynomial of degree
. Then the Fourier transform of
has the form
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ11_HTML.gif)
and ,
is the Bessel function.
Lemma 2.2 (see [12]).
For , and
, there exists
such that for any
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ12_HTML.gif)
Lemma 2.3.
Let ,
,
is a
function on
and let it satisfy the conditions (i) and (ii) in Theorem 1.2.
Denote , if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ13_HTML.gif)
Then there exists a constant independent of
, such that
for every integer
.
Proof.
Let , then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ14_HTML.gif)
So it suffices to show .
Decompose this integral into two parts .
For , by using Lemma 2.2 and
, we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ15_HTML.gif)
For the other part , applying Stirling's formula, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ16_HTML.gif)
Also in [13], the authors proved the following inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ17_HTML.gif)
So by (2.9) and (2.10), , and noting that
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ18_HTML.gif)
So far we can deduce the desired conclusion of Lemma 2.3.
Proof of Theorem 1.2.
The basic idea of proof can go back to [14], for recently papers, one see [8, 15]. By the same argument as in [1], let denote the complete system of normalized surface spherical harmonics. See [14] for instance, we can decompose
as following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ19_HTML.gif)
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ20_HTML.gif)
then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ21_HTML.gif)
Then, applying Hölder inequality twice, we have for any that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ22_HTML.gif)
By [14, page 230, equation (4.4)], we can observe that the series in the first parenthesis on the right-hand side of the inequality above, for each fixed, is equal to
, where
is the Sobolev space on
with
for any
. So if we take
sufficiently close to 1, then by the Sobolev imbedding theorem
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ23_HTML.gif)
with .
By Fourier transform and (2.16), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ24_HTML.gif)
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ25_HTML.gif)
For the integral on the right hand side of the above inequality, by changing of variable, we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ26_HTML.gif)
So we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ27_HTML.gif)
Put and
, we can deduce from Lemma 2.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ28_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ29_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ30_HTML.gif)
By [14], we know that .
So we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ31_HTML.gif)
Set ,
and note that
, we can deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ32_HTML.gif)
Noting that is increasing, by using the second mean-value theorem, we get, for some
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ33_HTML.gif)
From (2.26), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ34_HTML.gif)
Thus using Lemma 2.3, we can deduce the desired conclusion of Theorem 1.2.
Proof of Theorem 1.4.
First, we know that . On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ35_HTML.gif)
Thus, using Theorem 1.2, we can finish Theorem 1.4.
3. The Bounedness on Hardy-Sobolev Spaces
In order to prove the boundedness for operator on Hardy-Sobolev spaces and prove Theorem 1.3, we first introduce a new kind of atomic decomposition for Hardy-Sobolev space as following which will be used next.
Definition 3.1 (see [16]).
For , the function
is called a
atom if it satisfies the following three conditions:
(1) with a ball
;
(2);
(3), for any polynomial
of degree
.
By [16], we have that every can be written as a sum of
atoms
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ36_HTML.gif)
Proof of Theorem 1.3.
Similar to the argument of Lemma  3.3 in [17] and using above atomic decomposition, it suffices to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ37_HTML.gif)
with the constant independent of any
atom
.
Assume . We first note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ38_HTML.gif)
For , using Theorem 1.2, it is not difficult to deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ39_HTML.gif)
For , we first consider the case
, according to [15, Lemma  5.5], for
and
atom
with support
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ40_HTML.gif)
Using Minkowski inequality and Hölder inequality for integrals, and (3.5), we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ41_HTML.gif)
For the integral on the right hand side of the above inequality, by changing of variable and noting that ,
, we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ42_HTML.gif)
By (3.7), we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ43_HTML.gif)
Thus by (3.5) and the condition ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F479576/MediaObjects/13660_2010_Article_2342_Equ44_HTML.gif)
As for , similar to the argument of
, we can easily get
. So far the proof of Theorem 1.3 has been finished.
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Acknowledgments
This project supported by the National Natural Science Foundation of China under Grant no. 10747141, Zhejiang Provincial National Natural Science Foundation of China under Grant no. Y604056, and Science Foundation of Shaoguan University under Grant no. 200915001.
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Ruiying, W., Yin, L. Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space. J Inequal Appl 2011, 479576 (2011). https://doi.org/10.1155/2011/479576
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DOI: https://doi.org/10.1155/2011/479576