Sharp maximal function inequalities and boundedness for commutators related to generalized fractional singular integral operators
© Gu and Cai; licensee Springer. 2014
Received: 9 March 2014
Accepted: 25 April 2014
Published: 23 May 2014
In this paper, some sharp maximal function inequalities for the commutators related to certain generalized fractional singular integral operators are proved. As an application, we obtain the boundedness of the commutators on Lebesgue, Morrey and Triebel-Lizorkin spaces.
1 Introduction and preliminaries
By using a classical result of Coifman et al. (see ), we know that the commutator is bounded on (). Now, as the development of singular integral operators and their commutators, some new integral operators and their commutators are studied (see [2–7]). In [5, 8, 9], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and () 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 [10, 11]). Motivated by these, in this paper, we will prove some sharp maximal function inequalities for the commutator associated with certain generalized fractional singular integral operators and the and Lipschitz functions. As an application, we obtain the boundedness of the commutator on Lebesgue, Morrey and Triebel-Lizorkin space.
For , let .
For and , let be the homogeneous Triebel-Lizorkin space (see ).
In this paper, we will study some singular integral operators as follows (see ).
where and . We write .
where Ω is homogeneous of degree zero on , and for some , that is, there exists a constant such that for any , . When , is the Riesz potential (fractional integral operator).
It is well known that commutators are of great interest in harmonic analysis, and they have been widely studied by many authors (see [2, 7]). In , Pérez and Trujillo-Gonzalez prove a sharp estimate for the commutator. The main purpose of this paper is to prove some sharp maximal inequalities for the commutator . As an application, we obtain the -norm inequality, and for Morrey and Triebel-Lizorkin spaces boundedness for the commutator.
We shall prove the following theorems.
Theorem 4 Let there be a sequence , , , and . Suppose is a bounded linear operator from to and has a kernel K satisfying (1). Then is bounded from to .
Theorem 5 Let there be a sequence , , , , and . Suppose is a bounded linear operator from to and has a kernel K satisfying (1). Then is bounded from to .
Theorem 6 Let there be a sequence , , , and . Suppose is a bounded linear operator from to and has a kernel K satisfying (1). Then is bounded from to .
Theorem 7 Let there be a sequence , , and . Suppose is a bounded linear operator from to and has a kernel K satisfying (1). Then is bounded from to .
Theorem 8 Let there be a sequence , , , and . Suppose is a bounded linear operator from to and has a kernel K satisfying (1). Then is bounded from to .
Corollary Let , that is, is the singular integral operator as Definition 1. Then Theorems 1-8 hold for .
3 Proofs of theorems
To prove the theorems, we need the following lemma.
Lemma 1 (see )
Let be the singular integral operator as Definition 1. Then is bounded from to for and .
Lemma 2 (see )
Lemma 3 (see )
This finishes the proof. □
The proofs of two Lemmas are similar to that of Lemma 5 by Lemmas 1 and 4, 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 Theorem 5. □
This completes the proof of Theorem 6. □
This completes the proof of the theorem. □
This completes the proof of the theorem. □
Project supported by Hunan Provincial Natural Science Foundation of China (12JJ6003) and Scientific Research Fund of Hunan Provincial Education Departments (12K017).
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