Weighted boundedness of multilinear singular integral operators
Journal of Inequalities and Applications volume 2014, Article number: 115 (2014)
In this paper, we establish the weighted sharp maximal function inequalities for the multilinear singular integral operators. As an application, we obtain the boundedness of the multilinear operators on weighted Lebesgue and Morrey spaces.
As the development of singular integral operators (see [1–3]), their commutators operators have been well studied. In [4–6], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on for . Chanillo (see ) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [8, 9], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and () spaces are obtained. In [10, 11], the boundedness for the commutators generated by the singular integral operators and the weighted BMO and Lipschitz functions on () spaces are obtained (also see [12, 13]). In [14–17], the authors studied some multilinear singular integral operators as follows (also see [18, 19]):
and they obtained some variant sharp function estimates and boundedness of the multilinear operators if for all α with . In this paper, we will study the multilinear operator generated by the singular integral operator and the weighted Lipschitz and BMO functions, that is, or for all α with .
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
For , let and .
For , and the non-negative weight function w, set
We write if .
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, the weighted Lipschitz space is the space of functions b such that
and the weighted BMO space is the space of functions b such that
Let or and . By , we know that spaces or coincide and the norms or are equivalent with respect to different values .
Definition 1 Let φ be a positive, increasing function on and let there exist 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
In this paper, we will study the singular integral operators as follows (see ).
Definition 2 Let be a linear operator such that T is bounded on for and weak -bounded and there exists a locally integrable function on such that
for every bounded and compactly supported function f, where K satisfies, for fixed ,
Moreover, let m be the positive integer and b be the function on . Set
The multilinear operator related to the operator T is defined by
Note that the classical Calderón-Zygmund singular integral operator satisfies the conditions of Definition 2 (see [1, 4]) and that the commutator is a particular operator of the multilinear operator if . The multilinear operator are the non-trivial generalizations of the commutator. It is well known that commutators and multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [5, 6, 19]). The main purpose of this paper is to prove the sharp maximal inequalities for the multilinear operator . As the application, we obtain the weighted -norm inequality and Morrey space boundedness for the multilinear operator .
3 Theorems and lemmas
We shall prove the following theorems.
Theorem 1 Let T be the singular integral operator as Definition 2, , , and for all α with . Then there exists a constant such that, for any and ,
Theorem 2 Let T be the singular integral operator as Definition 2, , , , and for all α with . Then there exists a constant such that, for any and ,
Theorem 3 Let T be the singular integral operator as Definition 2, , and for all α with . Then is bounded from to .
Theorem 4 Let T be the singular integral operator as Definition 2, , , and for all α with . Then is bounded from to .
Theorem 5 Let T be the singular integral operator as Definition 2, , , , and for all α with . Then is bounded from to .
Theorem 6 Let T be the singular integral operator as Definition 2, , , , , and for all α with . Then is bounded from to .
To prove the theorems, we need the following lemmas.
Lemma 1 (See [, p.485])
Let and for any function . We define, for ,
where the sup is taken for all measurable sets Q with . Then
Let , , and . Then
Lemma 3 (See )
Let and . Then, for any smooth function f for which the left-hand side is finite,
Lemma 4 (See )
Let , , and . Then, for any smooth function f for which the left-hand side is finite,
Lemma 5 (See )
Let , , , and . Then
Lemma 6 (See )
Let b be a function on and for all α with and any . Then
where is the cube centered at x and having side length .
4 Proofs of theorems
Proof of Theorem 1 It suffices to prove for and some constant , the following inequality holds:
Fix a cube and . Let and , then and for . We write, for and ,
For , noting that , w satisfies the reverse of Hölder’s inequality:
for all cube Q and some (see ). We take in Lemma 6 and have and , then by the Lemma 6 and Hölder’s inequality, we obtain
thus, by the -boundedness of T (see Lemma 2) for and , we obtain
For , by the weak boundedness of T (see Lemma 2) and Kolmogoro’s inequality (see Lemma 1), we obtain
For , note that for and , we write
For , by the formula (see ):
and Lemma 6, we have, similar to the proof of ,
thus, by ,
For , by the conditions of K, we get
Similarly, 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 . Similar to the proof of Theorem 1, we have, for and ,
For and , by using the same argument as in the proof of Theorem 1, we get
For , we have
This completes the proof of Theorem 2. □
Proof of Theorem 3 By putting in Theorem 1 and noticing that , by Lemma 3 and 4, we obtain
This completes the proof of Theorem 3. □
Proof of Theorem 4 By choosing in Theorem 1 and noticing that , by Lemmas 5 and 6, we get
This completes the proof of Theorem 4. □
Proof of Theorem 5 By setting in Theorem 2 and noting that , by Lemmas 3 and 4, we have
This completes the proof of Theorem 5. □
Proof of Theorem 6 By choosing in Theorem 2, and noticing that , by Lemmas 5 and 6, we get
This completes the proof of Theorem 6. □
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The authors declare that they have no competing interests.
The authors completed the paper together. They also read and approved the final manuscript.
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Kuang, WP., Wang, ZG. Weighted boundedness of multilinear singular integral operators. J Inequal Appl 2014, 115 (2014). https://doi.org/10.1186/1029-242X-2014-115
- multilinear operator
- singular integral operator
- sharp maximal function
- weighted BMO
- weighted Lipschitz function