- Open Access
Sharp boundedness for vector-valued multilinear integral operators
© Pan and Tong; licensee Springer. 2014
Received: 15 January 2014
Accepted: 11 July 2014
Published: 21 August 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 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.
Let M be the Hardy-Littlewood maximal operator, that is, . For , we denote by the operator M iterated k times, i.e., and for .
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.
- (1)If and , then
- (2)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.
Example 2 Marcinkiewicz operators.
Example 3 Bochner-Riesz operators.
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 ()
where is the cube centered at x and having side length .
Lemma 2 ([, p.485])
Lemma 3 ()
3 Proof of the theorem
It is only to prove Theorem 1.
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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