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A sharp inequality for multilinear singular integral operators with non-smooth kernels
Journal of Inequalities and Applications volume 2013, Article number: 439 (2013)
Abstract
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.
MSC:42B20, 42B25.
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.
Definition 1 A family of operators , , is said to be ‘approximations to the identity’ if, for every , can be represented by the kernel in the following sense:
for every with , and satisfies
where s is a positive, bounded and decreasing function satisfying
for some .
Definition 2 A linear operator T is called a singular integral operator with non-smooth kernel if T is bounded on and associated with a kernel such that
for every continuous function f with compact support, and for almost all x not in the support of f.
-
(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
and
for some , .
Let be positive integers (), , and let be functions on (). Set, for ,
The multilinear operator associated to T is defined by
Note that when , is just the multilinear commutator of T and (see [7]). 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 [8]) proved a variant sharp estimate for the multilinear singular integral operators. In [7], 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 [13], the boundedness of the commutator associated to the singular integral operator with non-smooth kernel is obtained. Our works are motivated by these papers.
First, let us introduce some notations. Throughout this paper, Q denotes 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 [14, 15])
We say that f belongs to if belongs to and . Let M be a Hardy-Littlewood maximal operator defined by
For , we denote by the operator M iterated k times, i.e., and
The sharp maximal function associated with the ‘approximations to the identity’ is defined by
where and denotes the side length of Q. For , we denote by
Let Φ be a Young function and be the complementary associated to Φ. For a function f, we denote the Φ-average by
and the maximal function associated to Φ by
The Young functions used in this paper are and , the corresponding average and maximal functions are denoted by , and , . Following [11, 12, 16], we know the generalized Hölder inequality
and the following inequality, for , with , and any , ,
We denote the Muckenhoupt weights by for (see [14]).
We shall prove the following theorems.
Theorem 1 If T is a singular integral operator with non-smooth kernel as given in Definition 2, let for all α with and . Then there exists a constant such that for any , and ,
Theorem 2 If T is a singular integral operator with non-smooth kernel as given in Definition 2, let for all α with and . Then is bounded on for any and , that is,
Theorem 3 If T is a singular integral operator with non-smooth kernel as given in Definition 2, let , for all α with and . Then there exists a constant such that for all ,
2 Proof of the theorem
To prove the theorems, we need the following lemma.
Lemma 1 (see [1])
Let b 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 ([[14], 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 (see [17])
Let for , we denote that . Then
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 any , there exists a constant independent of γ such that
for , where D is a fixed constant which only depends on n. Thus
for every , .
Lemma 6 Let be an ‘approximation to the identity’ and . Then, for every , and ,
where and denotes the side length of Q.
Proof We write, for any cube Q with ,
We have, by the generalized Hölder inequality,
For , notice for and , then and , then
where the last inequality follows from
for some . This completes the proof. □
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 write, for ,
and
then
Now, let us estimate , , , , , , , and , respectively. First, for and , by Lemma 1, we get
by Lemma 2 and the weak type of T (Lemma 4), we obtain
For , we get, by Lemma 2 and the generalized Hölder inequality,
For , similar to the proof of , we get
Similarly, for , taking such that , we obtain, by Lemma 3 and the generalized Hölder inequality,
For , , and , by Lemma 6, we get
For , we write
By Lemma 1 and the following inequality (see [15])
we know that for and ,
Note that and for and . By the conditions on K and , we obtain
For , we get, by the generalized Hölder inequality,
Similarly,
For , taking such that , by Lemma 3 and the generalized Hölder inequality, we get
Thus
This completes the proof of Theorem 1. □
By Theorem 1 and the -boundedness of , we may obtain the conclusions of Theorem 2. By Theorem 1 and [16, 17], we may obtain the conclusions of Theorem 3.
3 Applications
In this section we shall apply Theorems 1, 2 and 3 of the paper to the holomorphic functional calculus of linear elliptic operators. First, we review some definitions regarding the holomorphic functional calculus (see [9]). Given , define
and its interior by . Set . A closed operator L on some Banach space E is said to be of type θ if its spectrum and if for every , there exists a constant such that
For , let
where . Set
If L is of type θ and , we define by
where Γ is the contour parameterized clockwise around with . If, in addition, L is one-to-one and has a dense range, then, for ,
where . L is said to have a bounded holomorphic functional calculus on the sector if
for some and for all .
Now, let L be a linear operator on with so that generates a holomorphic semigroup , . Applying Theorem 6 of [9], we get the following.
Theorem 4 Assume the following conditions are satisfied:
-
(i)
The holomorphic semigroup , is represented by the kernels which satisfy, for all , an upper bound
for , and , where and s is a positive, bounded and decreasing function satisfying
-
(ii)
The operator L has a bounded holomorphic functional calculus in , that is, for all and , the operator satisfies
Then, for for all α with and , the multilinear operator associated to and satisfies:
-
(a)
For and ,
-
(b)
is bounded on for any and , that is,
-
(c)
There exists a constant such that for all and ,
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Acknowledgements
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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Gu, G., Cai, M. A sharp inequality for multilinear singular integral operators with non-smooth kernels. J Inequal Appl 2013, 439 (2013). https://doi.org/10.1186/1029-242X-2013-439
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DOI: https://doi.org/10.1186/1029-242X-2013-439