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Sharp boundedness for vector-valued multilinear integral operators
Journal of Inequalities and Applications volume 2014, Article number: 315 (2014)
In this paper, the sharp inequalities for some vector-valued multilinear integral operators are obtained. As applications, we get the weighted () norm inequalities and an -type estimate for the vector-valued multilinear operators.
1 Preliminaries and theorems
As the development of singular integral operators and their commutators, multilinear singular integral operators have been well studied (see [1–5]). In this paper, we study some vector-valued multilinear integral operators as follows.
Suppose that are positive integers (), and are functions on (). Let be defined on . Set
for every bounded and compactly supported function f, where
Let H be the Banach space such that, for each fixed , and may be viewed as a mapping from to H. For , the vector-valued multilinear operator related to is defined by
and satisfies: for fixed ,
if . Set
Suppose that is bounded on for and weak -bounded.
Note that when , is just a vector-valued multilinear commutator of T and A (see ). While when , is a non-trivial generalization of the commutator. It is well known that multilinear operators are of great interest in harmonic analysis and have been studied by many authors (see [1–5]). In , Hu and Yang proved a variant sharp estimate for the multilinear singular integral operators. In , Pérez and Trujillo-Gonzalez proved a sharp estimate for some multilinear commutator. The main purpose of this paper is to prove a sharp inequality for the vector-valued multilinear integral operators. As applications, we obtain the weighted () norm inequalities and an -type estimate for the vector-valued multilinear operators.
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 function of f is defined by
where, and in what follows, . It is well known that (see )
We say that f belongs to if belongs to and . For , we denote by
Let M be the Hardy-Littlewood maximal operator, that is, . For , we denote by the operator M iterated k times, i.e., and for .
Let Φ be a Young function and be the complementary associated to Φ, we denote the Φ-average by, for a function f,
and the maximal function associated to Φ by
The Young functions to be used in this paper are and , the corresponding Φ-average and maximal functions are denoted by , and , . We have the following inequality, for any and (see )
For , we denote that
the space is defined by
It has been known that (see )
It is obvious that coincides with the space if , and if . We denote the Muckenhoupt weights by for (see ).
Now we state our main results as follows.
Theorem 1 Let , and for all α with and . Define . Then, for any , there exists a constant such that for any and ,
Theorem 2 Let , and for all α with and .
If and , then
If . Define and . Then there exists a constant such that for all ,
Remark The conditions in Theorems 1 and 2 are satisfied by many operators.
Now we give some examples.
Example 1 Littlewood-Paley operators.
Fix and . Let ψ be a fixed function which satisfies the following properties:
We denote that and the characteristic function of by . The Littlewood-Paley multilinear operators are defined by
and for . Set . We also define that
which are the Littlewood-Paley operators (see ). Let H be the space
Then, for each fixed , and may be viewed as the mapping from to H, and it is clear that
Example 2 Marcinkiewicz operators.
Fix and . Let Ω be homogeneous of degree zero on with . Assume that . The Marcinkiewicz multilinear operators are defined by
We also define that
which are the Marcinkiewicz operators (see ). Let H be the space
Then it is clear that
Example 3 Bochner-Riesz operators.
Let , and for . Set
The maximal Bochner-Riesz multilinear operators are defined by
We also define that
which is the maximal Bochner-Riesz operator (see ). Let H be the space , then
It is easy to see that satisfies the conditions of Theorems 1 and 2 (see ), thus Theorems 1 and 2 hold for .
2 Some lemmas
We give some preliminary lemmas.
Lemma 1 ()
Let A be a function on and for all α with and some . Then
where is the cube centered at x and having side length .
Lemma 2 ([, p.485])
Let and for any function , we define that, for ,
where the sup is taken for all measurable sets E with . Then
Lemma 3 ()
Let for , we denote that . Then
3 Proof of the theorem
It is only to prove Theorem 1.
Proof of Theorem 1 It suffices to prove for and some constant that the following inequality holds:
Without loss of generality, we may assume . Fix a cube and . Let and , then and for . We split for and . Write
Then, by Minkowski’s inequality, we have
Now, let us estimate , , , and , respectively. First, for and , by Lemma 1, we get
Thus, by Lemma 2 and the weak type of , we obtain
For , note that , similar to the proof of and by using Lemma 3, we get
For , similar to the proof of , we get
Similarly, for , by using Lemma 3, we get
For , we write
By Lemma 1, we know that, for and ,
Note that for and , we obtain, by the condition of ,
Thus, by Minkowski’s inequality, we get
For , by the formula (see )
and Lemma 1, we have
For , similar to the proof of , and , we get
For , by using Lemma 3, we obtain
This completes the proof of Theorem 1. □
By Theorem 1 and the -boundedness of , we may obtain the conclusions (1), (2) of Theorem 2.
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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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Pan, J., Tong, L. Sharp boundedness for vector-valued multilinear integral operators. J Inequal Appl 2014, 315 (2014). https://doi.org/10.1186/1029-242X-2014-315
- vector-valued multilinear operator
- Littlewood-Paley operator
- Marcinkiewicz operator
- Bochner-Riesz operator
- sharp estimate