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Sharp function estimates and boundedness for commutators associated with general singular integral operator
Journal of Inequalities and Applications volume 2014, Article number: 108 (2014)
In this paper, we establish the sharp maximal function estimates for the commutator associated with the singular integral operator with general kernel. As an application, we obtain the boundedness of the commutator on weighted Lebesgue, Morrey, and Triebel-Lizorkin spaces.
1 Introduction and preliminaries
As the development of singular integral operators (see [1–3]), their commutators have been well studied. In [4–6], the authors prove that the commutators generated by the singular integral operators and functions are bounded on for . Chanillo (see ) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [8–10], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and () spaces are obtained. In [11, 12], the boundedness for the commutators generated by the singular integral operators and the weighted and Lipschitz functions on () spaces are obtained. In , some singular integral operators with general kernel are introduced, and the boundedness for the operators and their commutators generated by and Lipschitz functions are obtained (see [13, 14]). In this paper, we will study the commutators generated by the singular integral operators with general kernel and the weighted Lipschitz functions.
First, let us introduce some notation. Throughout this paper, Q will denote a cube of with sides parallel to the axes. For any locally integrable function f, the sharp maximal function of f is defined by
For , let .
For and , set
The weight is defined by (see )
The weight is defined by (see ), for ,
Given a non-negative weight function w. For , the weighted Lebesgue space is the space of functions f such that
For , , and the non-negative weight function w, let be the weighted homogeneous Triebel-Lizorkin space (see ).
For and the non-negative weight function w, the weighted Lipschitz space is the space of functions b such that
It is well known that, for , , and ,
Let and . By , we know that spaces coincide and the norms are equivalent with respect to different values .
In this paper, we will study some singular integral operators as follows (see ).
Definition 1 Let be a linear operator such that T is bounded on and there exists a locally integrable function on such that
for every bounded and compactly supported function f, where K satisfies the following: there is a sequence of positive constant numbers such that for any ,
where and .
Let b be a locally integrable function on . The commutator related to T is defined by
Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 1 with (see ).
Definition 2 Let φ be a positive, increasing function on and there exists a constant such that
Let w be a non-negative weight function on and f be a locally integrable function on . Set, for ,
where . The generalized weighted Morrey space is defined by
It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [5, 6]). In , Pérez and Trujillo-Gonzalez prove a sharp estimate for the multilinear commutator. The main purpose of this paper is to prove the sharp maximal inequalities for the commutator. As the application, we obtain the weighted -norm inequality, and Morrey and Triebel-Lizorkin spaces’ boundedness for the commutator.
We shall prove the following theorems.
Theorem 1 Let T be the singular integral operator as Definition 1, the sequence , , , , and . Then there exists a constant such that, for any and ,
Theorem 2 Let T be the singular integral operator as Definition 1, the sequence , , , , and . Then there exists a constant such that, for any and ,
Theorem 3 Let T be the singular integral operator as Definition 1, the sequence , , , , , and . Then is bounded from to .
Theorem 4 Let T be the singular integral operator as Definition 1, the sequence , , , , , , and . Then is bounded from to .
Theorem 5 Let T be the singular integral operator as Definition 1, the sequence , , , , , and . Then is bounded from to .
3 Proofs of theorems
To prove the theorems, we need the following lemmas.
Lemma 1 (see )
Let T be the singular integral operator as Definition 1, the sequence . Then T is bounded on for with .
For any cube Q, , , and , we have
Lemma 3 (see )
For , , and , we have
Lemma 4 (see )
Let and . Then, for any smooth function f for which the left-hand side is finite,
Suppose that , , , and . Then
If , then for and any cube Q.
Lemma 7 Let , , , , and be the weighted Morrey space as Definition 2. Then, for any smooth function f for which the left-hand side is finite,
Proof Notice that for any cube by  and Lemma 6; thus, for and any cube Q, we have, by Lemma 4,
This finishes the proof. □
Lemma 8 Let , , , T be the singular integral operator as Definition 1 and be the weighted Morrey space as Definition 2. Then
Lemma 9 Let , , , and be the weighted Morrey space as Definition 2. Then
The proofs of the two lemmas are similar to that of Lemma 7 by Lemmas 1 and 5, we omit the details.
Proof of Theorem 1 It suffices to prove, for and some constant , that the following inequality holds:
Fix a cube and . Write, for and ,
For , by Hölder’s inequality and Lemma 2, we obtain
For , by the boundedness of T, we get
For , recalling that , we have
These complete the proof of Theorem 1. □
Proof of Theorem 2 It suffices to prove for and some constant , the following inequality holds:
Fix a cube and . Write, for and ,
By using the same argument as in the proof of Theorem 1, we get
This completes the proof of Theorem 2. □
Proof of Theorem 3 Choose in Theorem 1, notice that and ; we have, by Lemmas 1, 4, and 5,
This completes the proof of Theorem 3. □
Proof of Theorem 4 Choose in Theorem 1, notice that and ; we have, by Lemmas 7-9,
This completes the proof of Theorem 4. □
Proof Theorem 5 Choose in Theorem 2, notice that and . By using Lemma 3, we obtain
This completes the proof of the theorem. □
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The author declares that they have no competing interests.
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Cite this article
Zeng, J. Sharp function estimates and boundedness for commutators associated with general singular integral operator. J Inequal Appl 2014, 108 (2014). https://doi.org/10.1186/1029-242X-2014-108
- singular integral operator
- sharp maximal function
- Morrey space
- Triebel-Lizorkin space