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Inequalities and boundedness for commutators related to integral operator with general kernel
Journal of Inequalities and Applications volume 2014, Article number: 172 (2014)
Abstract
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.
MSC:42B20, 42B25.
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 [8]) 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 [12], 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.
First, let us introduce some notations. 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
here, and in what follows, . It is well known that (see [1, 2])
We say that f belongs to if belongs to and define . It has been known that (see [14])
Let
For , let .
For and , set
The weight is defined by (see [1])
and
For and , let be the weighted homogeneous Triebel-Lizorkin space (see [11]).
For , the Lipschitz space is the space of functions f such that
In this paper, we will study some integral operators as follows (see [12]).
Definition 1 Let be defined on and b be a locally integrable function on , set
and
for every bounded and compactly supported function f.
Let H be the Banach space . For each fixed , we view and as the mappings from to H. Set
which T is bounded on . The commutator related to is defined by
and for we find that there is a sequence of positive constant numbers such that for any ,
and
where and .
Definition 2 Let φ be a positive, increasing function on and there exists a constant such that
Let f be a locally integrable function on . Set, for and ,
where . The generalized fractional Morrey space is defined by
We write if , which is the generalized Morrey space. If , , then , which is the classical Morrey spaces (see [14, 15]). If , then , which is the Lebesgue spaces.
As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of the operator on the Morrey spaces (see [7, 16–19]).
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 [6], 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.
2 Theorems
We shall prove the following theorems.
Theorem 1 Let T be the integral operator as Definition 1, the sequence , , and . Then there exists a constant such that, for any and ,
Theorem 2 Let T be the integral operator as Definition 1, the sequence , , and . Then there exists a constant such that, for any and ,
Theorem 3 Let T be the integral operator as Definition 1, the sequence , and . Then there exists a constant such that, for any and ,
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 [12])
Let T be the integral operator as Definition 1, the sequence . Then T is bounded on for .
Lemma 2 (see [11])
For and , we have
Lemma 3 (see [1])
Let and . Then, for any smooth function f for which the left-hand side is finite,
Lemma 4 (see [8])
Suppose that , and . Then
Lemma 5 Let , . Then, for any smooth function f for which the left-hand side is finite,
Proof For any cube in , we know for any cube by [20]. Noticing that and if , by Lemma 3, we have, for ,
thus
and
This finishes the proof. □
Lemma 6 Let T be the integral operator as Definition 1, , and . Then
Lemma 7 Let , , and . Then
The proofs of the two lemmas are similar to that of Lemma 5 by Lemmas 1 and 4, we omit the details.
Proof of Theorem 1 It suffices to prove for and some constant , the following inequality holds:
Fix a cube and . Write, for and ,
Then
For , by Hölder’s inequality and Lemma 2, we obtain
For , by the boundedness of T, we get
For , recalling that , we have
This completes 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 ,
Then
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 It suffices to prove for and some constant , the following inequality holds:
Fix a cube and . Write, for and ,
Then
For , by Hölder’s inequality, we get
For , choose , by Hölder’s inequality and the boundedness of T, we obtain
For , recalling that , taking , with , by Hölder’s inequality, we obtain
This completes the proof of Theorem 3. □
Proof of Theorem 4 Choose in Theorem 1, we have, by Lemmas 1, 4, and 5,
This completes the proof of Theorem 4. □
Proof of Theorem 5 Choose in Theorem 1, we have, by Lemmas 5-7,
This completes the proof of Theorem 5. □
Proof Theorem 6 Choose in Theorem 2. By using Lemma 3, we obtain
This completes the proof of the theorem. □
Proof of Theorem 7 Choose in Theorem 3, we have
This completes the proof of the theorem. □
4 Applications
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.
Fixed and . Let ψ be a fixed function which satisfies:
-
(1)
,
-
(2)
,
-
(3)
when .
We denote and the characteristic function of by . The Littlewood-Paley commutators are defined by
and
where
and for . Set . We also define
and
which are the Littlewood-Paley operators (see [3]). Let H be the space
or
then, for each fixed , and may be viewed as the mapping from to H, and it is clear that
and
It is easily to see that , , and satisfy the conditions of Theorems 1-6 (see [21–23]), thus Theorems 1-6 hold for , , and .
Application 2 Marcinkiewicz operators.
Fixed and . Let Ω be homogeneous of degree zero on with . Assume that . The Marcinkiewicz commutators are defined by
and
where
and
Set
We also define
and
which are the Marcinkiewicz operators (see [24]). Let H be the space
or
Then it is clear that
and
It is easy to see that , , and satisfy the conditions of Theorems 1-6 (see [21–24]), thus Theorems 1-6 hold for , , and .
Application 3 Bochner-Riesz operator.
Let , and for . Set
The maximal Bochner-Riesz commutator is defined by
We also define
which is the maximal Bochner-Riesz operator (see [25]). Let H be the space , then
It is easy to see that satisfies the conditions of Theorems 1-6 (see [21]), thus Theorems 1-6 hold for .
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Zeng, J. Inequalities and boundedness for commutators related to integral operator with general kernel. J Inequal Appl 2014, 172 (2014). https://doi.org/10.1186/1029-242X-2014-172
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DOI: https://doi.org/10.1186/1029-242X-2014-172