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Vector-valued inequalities for the commutators of rough singular kernels
Journal of Inequalities and Applications volume 2014, Article number: 139 (2014)
Abstract
Vector-valued inequalities are considered for the commutator of the singular integral with rough kernel. The results obtained in this paper are substantial improvement and extension of some known results.
MSC: 42B20, 42B25.
1 Introduction
The homogeneous singular integral operator is defined by
when satisfies the following conditions:
-
(a)
Ω is a homogeneous function of degree zero on , i.e.,
(1.1) -
(b)
Ω has mean zero on , the unit sphere in , i.e.,
(1.2)
Using a rotation method, Calderón and Zygmund [1] proved that is bounded in for if Ω is odd or . In [2], Grafakos and Stefanov gave a nice survey, which contains a thorough discussion of the history of the operator .
For a function , let A be a linear operator on some measurable function space. Then the commutator between A and b is defined by .
In 1976, Coifman et al. [3] obtained a characterization of -boundedness of the commutators generated by the Reisz transforms () and a BMO function b. As an application of this characterization, a decomposition theorem of the real Hardy space is given in this paper. Moreover, the authors in [3] proved also that if , then the commutator for and a BMO function b is bounded on for which is defined by
In the same paper, Coifman et al. [3] outlined a different approach, which is less direct but shows a close relationship between the weighted inequalities of the operator and the weighted inequalities of the commutator . In 1993, Alvarez et al. [4] developed the idea of [3] and established a generalized boundedness criterion for the commutator of linear operators. The result of Alvarez et al. (see [4], Theorem 2.13) can be stated as follows.
Theorem A ([4])
Let . If a linear operator T is bounded on for all , (), where denotes the weight class of Muckenhoupt, then for , .
Combining Theorem A with the well-known results by Duoandikoetxea [5] on the weighted boundedness of the rough singular integral , we know that if for some , then is bounded on for . However, it is not clear up to now whether the operator with is bounded on for and all (). Hence, if , the boundedness of cannot be deduced from Theorem A. In this case, Hu [6] used the refined Fourier estimate, the Littlewood-Paley decomposition, and the properties of Young functions and got the following result.
Theorem B ([6])
Suppose that satisfying (1.1) and (1.2). Then, for and , the commutator is bounded on with bound .
Recently, Chen and Ding [7] gave a sufficient condition which contains such that the commutator of convolution operators is bounded on for . This condition was introduced by Grafakos and Stefanov in [8], and it is defined by
where is a fixed constant. Let denote the space of all integrable functions Ω on satisfying (1.3). The result in [7] can be stated as follows.
Theorem C Let Ω be a function in satisfying (1.1) and (1.2). If for some , then extends to a bounded operator from into itself for .
The condition (1.3) above has been considered by many authors in the context of rough integral operators. One can consult [9–15] among numerous references for its development and applications. The examples in [8] show that there is the following relationship between and (the Hardy space on ):
On the other hand, for all , . So, for all , .
The study of vector-valued inequalities for singular integrals with rough kernels has attracted much attention (for example, see [16]). In 2011, Tang and Wu [17] considered the vector-valued inequalities (, ), () of the commutator with the kernel satisfying (1.1) and (1.2). In this paper, we consider the vector-valued inequalities for a class of commutators of singular integrals with for some . Now we state our result as follows.
Theorem 1.1 Let Ω be a function in satisfying (1.1) and (1.2) if for some . Suppose that satisfy
-
(a)
and ; or
-
(b)
, and ; or
-
(c)
and ; or
-
(d)
, and .
Then extends to a bounded operator from into itself.
Corollary 1.2 Let Ω be a function in satisfying (1.1) and (1.2). If , then extends to a bounded operator from into itself for .
This paper is organized as follows. First, in Section 2, we give some definitions, which will be used in the proofs of the main results. In Section 3, we give some preliminary lemmas for the proof of Theorem 1.1. Then, in Section 4, we give the proof of Theorem 1.1. Throughout this paper, the letter C stands for a positive constant which is independent of the essential variables and not necessarily the same one in each occurrence. Moreover, the notations `∨’ and `∧’ denote the Fourier transform and the inverse Fourier transform, respectively. As usual, for , denotes the dual exponent of p.
We collect the notation to be used throughout this paper:
2 Definitions
Firstly, we need to recall some definitions which will be used in the proof of Theorem 1.1.
Let be a radial function which is supported in the unit ball and satisfies for . The function is supported in and satisfies the identity
We denote by and the convolution operators whose symbols are and , respectively.
The paraproduct of Bony [18] between two functions f, g is defined by
At least formally, we have the following Bony decomposition:
3 Key lemmas
Let us begin with some lemmas, which will be used in the proof of Theorem 1.1. The first one can be found in [17].
Lemma 3.1 If with and for , define the multiplier operator by and by . Then, for , for any positive integer k and , denote by (respectively ) the kth-order commutator of (respectively ). Then, for , we have
where C is independent of j and l.
Lemma 3.2 ([19])
Let , , and . Denote . Then
where C is independent of .
Lemma 3.3 ([7])
For the multiplier () defined in Section 2 and ,
where C is independent of k and δ.
Lemma 3.4 ([20])
For any and , the following properties hold:
-
(i)
if ,
-
(ii)
if .
If we replace with , the above inequalities also hold.
4 Proof of Theorem 1.1
Recall that
Let be a radial function such that , and for . Define the multiplier by . Set
for . Set
Define the operator and by
Denote by and the commutator of and , respectively. Define the operator by
Then we know
Then by the Minkowski inequality, we have, for ,
So, to prove Theorem 1.1, it suffices to prove that
It is well known that for some constant and any fixed constant (see [7] and [15]),
and
which gives that
and
If we can prove that, for any , ,
where C is independent of l and δ, we may finish the proof of Theorem 1.1. The proof of (4.4) will be postponed. Now, we will use (4.2), (4.3), and (4.4) to prove Theorem 1.1. Since
we will estimate and , respectively. We first estimate . For , taking in (4.4), then interpolating between (4.2) and (4.4), there exists a constant such that for ,
For and any fixed , interpolating between (4.4) and (4.5), there exists a constant such that for ,
Therefore we get, for ,
Next, we will estimate for (a), (b), (c), and (d), respectively. For , taking in (4.4), we get, for any ,
Taking in (4.6) gives that for any , we have
We first treat the case (a) : and . Now, for any , we take r sufficiently large such that in (4.7). Using the Riesz-Thorin interpolation theorem between (4.3) and (4.7), we have that for any ,
where . We can see that if , then θ goes to and goes to . Therefore, we get
On the other hand, fix p, for any , (4.6) also means that for any λ sufficiently large such that ,
Using the Riesz-Thorin interpolation theorem between (4.8) and (4.9), we have that
where . We can see that if , then goes to and goes to . This gives that for any fixed ,
Thus, by the inequality above, we have, for ,
Next, for the case (b) : , , and . For any , we have
Similarly, fix p, for , (4.6) also means that for any λ sufficiently small such that ,
Using the Riesz-Thorin interpolation theorem between (4.10) and (4.11), we have
where . We can see that if , then goes to and goes to . This gives that for any fixed ,
Thus, for , , and , we have
Now, for the case (c) : and . For any , we take r sufficiently small such that in (4.7). Using the Riesz-Thorin interpolation theorem between (4.3) and (4.7), we have that for any ,
where . We can see that if , then θ goes to and goes to . Therefore, we get
Then, using the previous argument, for any fixed and , we get
Thus if , then
Finally, for the case , , and , using the previous argument, we get
Therefore, we prove that
for four cases.
Now, we turn our attention to proving (4.4). Since for any , we may write
Thus,
Below we shall estimate for , respectively. As regards , by Lemma 3.1 and Lemma 3.2, we have, for ,
Similarly, we get
Hence, by (4.15), to show (4.4) it remains to give the estimate of . We will apply Bony paraproduct to do this. By (2.1),
We have
and
Then we get
Thus
-
(a)
The estimate of . Recall that . For , by Lemma 3.4(i), we know for when . Then
(4.17)
Then we get
Without loss of generality, we may assume . By Lemma 3.1, we get
Note that
By Lemma 3.3, we have, for any ,
where
and C is independent of δ and l. Then, by (4.19), (4.20) and applying Lemma 3.2 and Lemma 3.1, we have that for ,
where C is independent of l and δ.
-
(b)
The estimate of . By Lemma 3.4(i), we know for , for when . Thus
Then we get
Without loss of generality, we may assume . By the equality above and using Lemma 3.1, (see [21]) and Lemma 3.2, we have, for ,
-
(c)
The estimate of . Finally, we give the estimate of . By Lemma 3.4(ii), we know for if . We get
Thus, by Lemma 3.1, , and Lemma 3.2, we get, for ,
By (4.16), (4.21)-(4.23), we get
where C is independent of δ and l. This establishes the proof of (4.4).
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Acknowledgements
The authors would like to express their deep gratitude to the referee for giving many valuable suggestions. The research was supported by NCET of China (Grant nos.: NCET-11-0574), the Fundamental Research Funds for the Central Universities (Grant nos.: FRF-TP-12-006B), NSF of China (Grant nos.: 10901017, 11371057) and SRFDP of China (Grant nos.: 20130003110003).
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Authors’ contributions
YC carried out the vector-valued inequalities for the commutators of singular integral operator studies and drafted the manuscript. YD participated in the study of Littlewood-Paley theory. All authors read and approved the final manuscript.
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Chen, Y., Ding, Y. Vector-valued inequalities for the commutators of rough singular kernels. J Inequal Appl 2014, 139 (2014). https://doi.org/10.1186/1029-242X-2014-139
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DOI: https://doi.org/10.1186/1029-242X-2014-139