Norm inequalities for higher-order commutators of one-sided oscillatory singular integrals
- Shaoguang Shi^{1}Email author and
- Lei Zhang^{1}
https://doi.org/10.1186/s13660-016-1025-0
© Shi and Zhang 2016
Received: 20 September 2015
Accepted: 18 February 2016
Published: 3 March 2016
Abstract
In the present paper, we study the weighted norm inequalities for higher-order commutators formed by a class of one-sided oscillatory singular integrals and \(BMO\) functions. We obtain that the boundedness of these commutators can be deduced by that of one-sided Calderón-Zygmund singular integral operators.
Keywords
MSC
1 Introduction
- (1)there exists a finite constant \(C_{1}\) such that$$\bigl|K(x-y)-K(x)\bigr|\leq\frac{C_{1}|y|}{|x|^{2}} \quad\mbox{for all } |x|>2|y|; $$
- (2)there exists a finite constant \(C_{2}\) such that$$\biggl\vert \int_{\varepsilon< |x|< N}K(x)\,dx\biggr\vert \leq C_{2} \quad\mbox{for all } \varepsilon \mbox{ and } N \mbox{ such that } 0< \varepsilon< N; $$
- (3)there exists a finite constant \(C_{3}\) such that$$\bigl|K(x)\bigr|\leq\frac{C_{3}}{|x|} \quad\mbox{for all } x\neq0. $$
This paper is devoted to the weighted norm inequalities for \(T_{b}^{k,+}\) and \(T_{b}^{k,-}\) with \(k>1\). Due to their similarities, we further consider only the operator \(T_{b}^{k,+}\).
Now, we can formulate our results.
Theorem 1.1
Let \(1< p<\infty\), \(k\in\mathbb{Z}^{+}\), \(w\in A_{p}^{+}\), and let \(b\in BMO(\mathbb{R})\). Then the operator \(T_{b}^{k,+}\) is bounded on \(L^{p}(w)\).
Theorem 1.2
We end this section with the outline of this paper. In Section 2, some lemmas are collected for the proofs of our main results. Section 3 contains the proofs of Theorem 1.1 and Theorem 1.2. Throughout this paper, the letter C will denote a positive constant that may vary from line to line but will remain independent of the relevant quantities.
2 Basic lemmas
We provide in this section some lemmas that are crucial for the proofs in Section 3. Together with the characterizations of the weighted inequalities for one-sided operators, we can obtain some properties of the classes \(A_{p}^{+}\) and \(A_{p}^{-}\).
Lemma 2.1
- (1)
If \(w\in A_{p}^{+}\), then \(w^{1+\varepsilon}\in A_{p}^{+}\) for some \(\varepsilon>0\) with \(1\leq p<\infty\).
- (2)
Let \(1< p<\infty\). Then \(w\in A_{p}^{+}\) if and only if there exist \(w_{1}\in A_{1}^{+}\) and \(w_{2}\in A_{1}^{-}\) such that \(w=w_{1}(w_{2})^{1-p}\).
- (3)
Let \(1< p<\infty\). Then \(w\in A_{p}^{+}\) if and only if \(w^{1-p'}\in A_{p'}^{-}\), where \(\frac{1}{p}+\frac{1}{p'}=1\).
- (4)
Let \(1< p<\infty\) and \(w\in A_{p}^{+}\). Then \(A_{p}^{+}(\delta^{\lambda}(w))=A_{p}^{+}(w)\), where \(\delta^{\lambda}(w)(x)=w(\lambda x)\) for all \(\lambda>0\).
In [30], the authors obtained some weighted norm estimates for \(M_{b}^{k,+}\).
Lemma 2.2
- (1)
\(M_{b}^{k,+}\) is bounded on \(L^{p}(w)\) for every \(1< p<\infty\) with \(w\in A_{p}^{+}\).
- (2)
\(M_{b}^{k,+}\) is bounded on \(L^{p}(dx)\) for some \(1< p<\infty\).
- (3)
\(b\in BMO(\mathbb{R})\).
Lemma 2.3
[33]
- (1)
Let \(1< p<\infty\), and let \(b\in BMO(\mathbb {R})\). Then there exists \(\lambda>0\) such that \(e^{\lambda b}\in A_{p}^{+}\).
- (2)
Let \(1< p<\infty\) and \(\lambda>0\). Then there exists \(\eta=\eta(\lambda,p)>0\) such that for \(b\in BMO(\mathbb{R})\) and \(\|b\|_{BMO(\mathbb{R})}<\eta\), we have \(e^{\lambda b}\in A_{p}^{+}\).
To prove Theorem 1.1, we still need a celebrated interpolation theorem of operators with change of measures.
Lemma 2.4
[34]
3 Proofs of the main results
The proof of Theorem 1.1 is a by-product of the following two lemmas.
Lemma 3.1
Lemma 3.2
3.1 Proof of Lemma 3.1
3.2 Proof of Lemma 3.2
3.3 Proof of Theorem 1.2
Declarations
Acknowledgements
This work was partially supported by National Natural Science Foundation of China (Grant Nos. 11271175, 11301249) and the Applied Mathematics Enhancement Program of Linyi University (No. LYDX2013BS059). The authors thank the anonymous referees cordially for their valuable suggestions on this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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