- Open Access
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.
- multi-linear operator
- singular integral operator with non-smooth kernel
- sharp inequality
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.
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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