A sharp inequality for multilinear singular integral operators with non-smooth kernels
© Gu and Cai; licensee Springer. 2013
Received: 17 March 2013
Accepted: 12 August 2013
Published: 16 September 2013
In this paper, we establish a sharp inequality for some multilinear singular integral operators with non-smooth kernels. As an application, we obtain the weighted -norm inequality and -type inequality for the multilinear operators.
1 Definitions and results
As the development of singular integral operators and their commutators, multilinear singular integral operators have been well studied (see [1–6]). In this paper, we study some multilinear operator associated to the singular integral operators with non-smooth kernels as follows.
for some .
- (1)There exists an ‘approximation to the identity’ such that has an associated kernel and there exist so that
- (2)There exists an ‘approximation to the identity’ such that has an associated kernel which satisfies
for some , .
Note that when , is just the multilinear commutator of T and (see ). However, 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 widely studied by many authors (see [1–4]). Hu and Yang (see ) proved a variant sharp estimate for the multilinear singular integral operators. In , Pérez and Trujillo-Gonzalez proved a sharp estimate for the multilinear commutator when and noted that . The main purpose of this paper is to prove a sharp function inequality for the multilinear singular integral operator with non-smooth kernel when for all α with . As an application, we obtain an () norm inequality and an -type inequality for the multilinear operators. In [9–12], the boundedness of a singular integral operator with non-smooth kernel is obtained. In , the boundedness of the commutator associated to the singular integral operator with non-smooth kernel is obtained. Our works are motivated by these papers.
We denote the Muckenhoupt weights by for (see ).
We shall prove the following theorems.
2 Proof of the theorem
To prove the theorems, we need the following lemma.
Lemma 1 (see )
where is the cube centered at x and having side length .
Lemma 2 ([, p.485])
Lemma 3 (see )
Let T be a singular integral operator with non-smooth kernel as given in Definition 2. Then T is bounded on for every and bounded from to .
for every , .
where and denotes the side length of Q.
for some . This completes the proof. □
This completes the proof of Theorem 1. □
for some and for all .
Now, let L be a linear operator on with so that generates a holomorphic semigroup , . Applying Theorem 6 of , we get the following.
- (i)The holomorphic semigroup , is represented by the kernels which satisfy, for all , an upper bound
- (ii)The operator L has a bounded holomorphic functional calculus in , that is, for all and , the operator satisfies
- (a)For and ,
- (b)is bounded on for any and , that is,
- (c)There exists a constant such that for all and ,
Project supported by Hunan Provincial Natural Science Foundation of China (12JJ6003) and Scientific Research Fund of Hunan Provincial Education Departments (12K017).
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