Inequalities and boundedness for commutators related to integral operator with general kernel
© Zeng; licensee Springer. 2014
Received: 9 March 2014
Accepted: 25 April 2014
Published: 10 May 2014
In this paper, we establish the sharp maximal function inequalities for the commutators related to some integral operator with general kernel and the and Lipschitz functions. As an application, we obtain the boundedness of the commutators on Lebesgue, Morrey and Triebel-Lizorkin space. The operator includes Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.
1 Introduction and preliminaries
As the development of singular integral operators (see [1–3]), their commutators have been well studied (see [4–6]). In [5–7], 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 [9–11], 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 [12, 13]). The purpose of this paper is to prove the sharp maximal function inequalities for the commutator associated with some integral operator with general kernel and the and Lipschitz functions. As an application, we obtain the boundedness of the commutator on Lebesgue, Morrey and Triebel-Lizorkin space. The operator includes Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.
For , let .
For and , let be the weighted homogeneous Triebel-Lizorkin space (see ).
In this paper, we will study some integral operators as follows (see ).
for every bounded and compactly supported function f.
where and .
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 -norm inequality, Morrey and Triebel-Lizorkin spaces boundedness for the commutator.
We shall prove the following theorems.
Theorem 4 Let T be the integral operator as Definition 1, the sequence , , , and . Then is bounded from to .
Theorem 5 Let T be the integral operator as Definition 1, the sequence , , , , and . Then is bounded from to .
Theorem 6 Let T be the integral operator as Definition 1, the sequence , , , and . Then is bounded from to .
Theorem 7 Let T be the integral operator as Definition 1, the sequence and . Then is bounded on for .
3 Proofs of theorems
To prove the theorems, we need the following lemma.
Lemma 1 (see )
Let T be the integral operator as Definition 1, the sequence . Then T is bounded on for .
Lemma 2 (see )
Lemma 3 (see )
Lemma 4 (see )
This finishes the proof. □
The proofs of the two lemmas are similar to that of Lemma 5 by Lemmas 1 and 4, we omit the details.
This completes 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 the theorem. □
This completes the proof of the theorem. □
In this section we shall apply Theorems 1-6 of the paper to some particular operators such as the Littlewood-Paley operators, Marcinkiewicz operator and Bochner-Riesz operator.
Application 1 Littlewood-Paley operators.
Application 2 Marcinkiewicz operators.
Application 3 Bochner-Riesz operator.
It is easy to see that satisfies the conditions of Theorems 1-6 (see ), thus Theorems 1-6 hold for .
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