- Open Access
Vector-valued inequalities for the commutators of rough singular kernels
© Chen and Ding; licensee Springer. 2014
- Received: 25 September 2013
- Accepted: 20 March 2014
- Published: 4 April 2014
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.
- singular integral
- rough kernel
- vector-valued inequalities
- (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  proved that is bounded in for if Ω is odd or . In , 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 the same paper, Coifman et al.  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.  developed the idea of  and established a generalized boundedness criterion for the commutator of linear operators. The result of Alvarez et al. (see , Theorem 2.13) can be stated as follows.
Theorem A ()
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  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  used the refined Fourier estimate, the Littlewood-Paley decomposition, and the properties of Young functions and got the following result.
Theorem B ()
Suppose that satisfying (1.1) and (1.2). Then, for and , the commutator is bounded on with bound .
where is a fixed constant. Let denote the space of all integrable functions Ω on satisfying (1.3). The result in  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 .
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 ). In 2011, Tang and Wu  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.
and ; or
, and ; or
and ; or
, 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.
Firstly, we need to recall some definitions which will be used in the proof of Theorem 1.1.
We denote by and the convolution operators whose symbols are and , respectively.
Let us begin with some lemmas, which will be used in the proof of Theorem 1.1. The first one can be found in .
where C is independent of j and l.
Lemma 3.2 ()
where C is independent of .
Lemma 3.3 ()
where C is independent of k and δ.
Lemma 3.4 ()
If we replace with , the above inequalities also hold.
for four cases.
- (a)The estimate of . Recall that . For , by Lemma 3.4(i), we know for when . Then(4.17)
- (b)The estimate of . By Lemma 3.4(i), we know for , for when . Thus
- (c)The estimate of . Finally, we give the estimate of . By Lemma 3.4(ii), we know for if . We get
where C is independent of δ and l. This establishes the proof of (4.4).
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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