Sharp function estimates and boundedness for commutators associated with general singular integral operator
© Zeng; licensee Springer. 2014
Received: 28 September 2013
Accepted: 14 February 2014
Published: 4 March 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.
For , let .
For , , and the non-negative weight function w, let be the weighted homogeneous Triebel-Lizorkin space (see ).
- (1)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 ).
where and .
Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 1 with (see ).
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 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 .
Lemma 3 (see )
Lemma 4 (see )
If , then for and any cube Q.
This finishes the proof. □
The proofs of the two lemmas are similar to that of Lemma 7 by Lemmas 1 and 5, we omit the details.
These complete the proof of Theorem 1. □
This completes the proof of Theorem 2. □
This completes the proof of Theorem 3. □
This completes the proof of Theorem 4. □
This completes the proof of the theorem. □
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