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Commutators of Littlewood-Paley Operators on the Generalized Morrey Space
Journal of Inequalities and Applications volume 2010, Article number: 961502 (2010)
Abstract
Let ,
, and
denote the Marcinkiewicz integral, the parameterized area integral, and the parameterized Littlewood-Paley
function, respectively. In this paper, the authors give a characterization of BMO space by the boundedness of the commutators of
,
, and
on the generalized Morrey space
.
1. Introduction
Let be the unit sphere in
equipped with the Lebesgue measure
. Suppose that
satisfies the following conditions.
(a) is the homogeneous function of degree zero on
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ1_HTML.gif)
(b) has mean zero on
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ2_HTML.gif)
(c), that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ3_HTML.gif)
In 1958, Stein [1] defined the Marcinkiewicz integral of higher dimension as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ4_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ5_HTML.gif)
We refer to see [1, 2] for the properties of .
Let and
The parameterized area integral
and the parameterized Littlewood-Paley
function
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ6_HTML.gif)
where and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ7_HTML.gif)
respectively. and
play very important roles in harmonic analysis and PDE (e.g., see [3–8]).
Before stating our result, let us recall some definitions. For the commutator
formed by
and the Marcinkiewicz integral
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ8_HTML.gif)
Let and
The commutator
of
and the commutator
of
are defined, respectively, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ10_HTML.gif)
Let . It is said that
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ11_HTML.gif)
where denotes the ball in
centered at
and with radius
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ12_HTML.gif)
and .
There are some results about the boundedness of the commutators formed by BMO functions with ,
, and
(see [7, 9, 10]).
Many important operators gave a characterization of BMO space. In 1976, Coifman et al. [11] gave a characterization of BMO space by the commutator of Riesz transform; in 1982, Chanillo [12] studied the commutator formed by Riesz potential and BMO and gave another characterization of BMO space.
The purpose of this paper is to give a characterization of BMO space by the boundedness of the commutators of ,
, and
on the generalized Morrey space
.
Definition 1.1.
Let . Suppose that
be such that
is nonincreasing and
is nondecreasing. The generalized Morrey space
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ13_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ14_HTML.gif)
We refer to see [13, 14] for the known results of the generalized Morrey space for some suitable
. Noting that
, we get the Lebesque space
. For
,
coincides with the Morrey space
.
The main result in this paper is as follows.
Theorem 1.2.
Assume that is nonincreasing and
is nondecreasing. Suppose that
is defined as (1.8),
satisfies (1.1), (1.2), and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ15_HTML.gif)
If is bounded on
for some
, then
Theorem 1.3.
Let and
. Assume that
is nonincreasing and
is nondecreasing. Suppose that
is defined as (1.9),
satisfies (1.1), (1.2), and (1.15). If
is a bounded operator on
for some
, then
Theorem 1.4.
Let ,
, and
. Assume that
is nonincreasing and
is nondecreasing. Suppose that
is defined as (1.10),
satisfies (1.1), (1.2), and (1.15). If
is on
for some
, then
Remark 1.5.
It is easy to check that (see, e.g., the proof of (
) in [15, page 89]), we therefore give only the proofs of Theorem 1.2 for
and Theorem 1.3 for
.
Remark 1.6.
It is easy to see that the condition (1.15) is weaker than for
. In the proof of Theorems 1.2 and 1.3, we will use some ideas in [16]. However, because Marcinkiewicz integral and the parameterized Littlewood-Paley operators are neither the convolution operator nor the linear operators, hence, we need new ideas and nontrivial estimates in the proof.
2. Proof of Theorem 1.2
Let us begin with recalling some known conclusion.
Similar to the proof of [17], we can easily get the following.
Lemma 2.1.
If satisfies conditions (1.1), (1.2), and (1.15), let
then for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ16_HTML.gif)
Now let us return to the proof of Theorem 1.2. Suppose that is a bounded operator on
, we are going to prove that
We may assume that . We want to prove that, for any
and
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ17_HTML.gif)
holds, where Since
we may assume that
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ18_HTML.gif)
where Since
, we can easily get
Then,
has the following properties:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ19_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ21_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ23_HTML.gif)
In this proof for ,
is a positive constant depending only on
,
,
, and
. Since
satisfies (1.2), then there exists an
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ24_HTML.gif)
where is the measure on
which is induced from the Lebesgue measure on
. By the condition (1.15), it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ25_HTML.gif)
is a closed set. We claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ26_HTML.gif)
In fact, since note that
we can get
Taking
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ27_HTML.gif)
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ28_HTML.gif)
For noting that if
, then
for
. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ29_HTML.gif)
Using (2.11), we get Noting that
it follows from (2.5), (2.7), (2.8), and Hölder's inequality that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ30_HTML.gif)
For , by
, (2.4), (2.5), (2.6), the Minkowski inequality, and Lemma 2.1, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ31_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ32_HTML.gif)
Without loss of generality, we may assume that , otherwise, we get the desired result. Since
is nonincreasing, it follows that
. By (2.13), (2.15), and (2.16), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ33_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ34_HTML.gif)
Now, we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ35_HTML.gif)
where is independent of
. In fact,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ36_HTML.gif)
Now, we consider the norm of
in the following two cases.
Case 1 ().
Since is nondecreasing in
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ37_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ38_HTML.gif)
Case 2 ().
Since is nonincreasing in
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ39_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ40_HTML.gif)
Now, (2.20) is established. Then, by (2.19) and (2.20), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ41_HTML.gif)
If then Theorem 1.2 is proved. If
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ42_HTML.gif)
Let . For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ43_HTML.gif)
Noting that if and
, we get
. Applying (2.11), we have
. Since
when
and
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ44_HTML.gif)
By ,
when
and
and the Minkowski inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ45_HTML.gif)
Thus, by (2.28), (2.29), and (2.30), we get, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ46_HTML.gif)
Similar to the proof of (2.20), we can easily get . Thus, by (2.31),
, and
when
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ47_HTML.gif)
We first estimate Since
for
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ48_HTML.gif)
Now, the estimate of is divided into two cases, namely, 1:
; 2:
.
Case 1 ().
Since the function is decreasing for
and
for
by (2.27), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ49_HTML.gif)
Case 2 ().
Since the function is decreasing for
and
for
, by (2.27), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ50_HTML.gif)
From Cases 1 and 2, we know that there exists a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ51_HTML.gif)
So by (2.32), (2.33), and (2.36), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ52_HTML.gif)
Then, Theorem 1.2 is proved.
3. Proof of Theorem 1.3
Similar to the proof of Theorem 1.2, we only give the outline.
Suppose that is a bounded operator on
, we are going to prove that
We may assume that . We want to prove that, for any
and
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ53_HTML.gif)
holds, where Since
we may assume that
Let
be as (2.3), then (2.4)–(2.8) hold. In this proof for
,
is a positive constant depending only on
,
,
, and
. Since
satisfies (1.2), then there exists a
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ54_HTML.gif)
where is the measure on
which is induced from the Lebesgue measure on
. By the condition (1.15), it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ55_HTML.gif)
is a closed set. As the proof of (2.11), we can get the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ56_HTML.gif)
Taking , let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ57_HTML.gif)
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ58_HTML.gif)
For noting that if
,
, and
then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ59_HTML.gif)
Then by (3.4), we get . Since
and
we get
and
. Thus, by (2.5), (2.7), (2.8), and the Hölder inequality, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ60_HTML.gif)
By (2.5) and (2.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ61_HTML.gif)
In we have
and
. In
we get
and
It is easy to see that
Now, we estimate
by
, the Minkowski inequality, Lemma 2.1 for
, and (2.4), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ62_HTML.gif)
From (3.9) and (3.10), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ63_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ64_HTML.gif)
Without loss of generality, we may assume that , otherwise, we get the desired result. Since
is nonincreasing, we have
. Then by, (3.6), (3.8), and (3.11), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ65_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ66_HTML.gif)
Then, by (2.20) and (3.14), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ67_HTML.gif)
If then Theorem 1.3 is proved. If
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ68_HTML.gif)
Let . For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ69_HTML.gif)
For as above mentioned, we have
Since
and
, it follows the Hölder inequality that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ70_HTML.gif)
By , the Minkowski inequality, and
for
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ71_HTML.gif)
Thus, by (3.17), (3.18), and (3.19), we get, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ72_HTML.gif)
Thus, by (3.20), ,
when
and the Hölder inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ73_HTML.gif)
As the proof of (2.33) and (2.36), we can get that there exists a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ74_HTML.gif)
So, by (3.21) and (3.22), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961502/MediaObjects/13660_2010_Article_2314_Equ75_HTML.gif)
Then, Theorem 1.3 is proved.
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Acknowledgments
The authors wish to express their gratitude to the referee for his/her valuable comments and suggestions. The research was supported by NSF of China (Grant nos.: 10901017, 10931001), SRFDP of China (Grant no.: 20090003110018), and NSF of Zhenjiang (Grant no.: Y7080325).
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Chen, Y., Ding, Y. & Wang, X. Commutators of Littlewood-Paley Operators on the Generalized Morrey Space. J Inequal Appl 2010, 961502 (2010). https://doi.org/10.1155/2010/961502
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DOI: https://doi.org/10.1155/2010/961502