Weighted boundedness of multilinear operators associated to singular integral operators with non-smooth kernels
© Lu; licensee Springer. 2014
Received: 18 January 2014
Accepted: 24 May 2014
Published: 24 July 2014
In this paper, we establish the weighted sharp maximal function inequalities for a multilinear operator associated to a singular integral operator with non-smooth kernel. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.
1 Introduction and preliminaries
As the development of singular integral operators (see [1, 2]), their commutators and multilinear operators have been well studied. In [3–5], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on for . Chanillo (see ) proves a similar result when the singular integral operators are replaced by the fractional integral operators. In [7, 8], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and () spaces is obtained. In [9, 10], the boundedness for the commutators generated by the singular integral operators and the weighted BMO and Lipschitz functions on () spaces is obtained (also see ). In [12, 13], some singular integral operators with non-smooth kernels are introduced, and the boundedness for the operators and their commutators is obtained (see [14–17]). Motivated by these, in this paper, we study multilinear operators generated by singular integral operators with non-smooth kernels and the weighted Lipschitz and BMO functions.
In this paper, we study some singular integral operators as follows (see ).
for some .
- (1)There exists an ‘approximation to the identity’ such that has the associated kernel and there exist so that
- (2)There exists an ‘approximation to the identity’ such that has the associated kernel which satisfies
Note that the commutator is a particular operator of the multilinear operator if . The multilinear operator is a non-trivial generalization of the commutator. It is well known that commutators and multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [18–20]). The main purpose of this paper is to prove sharp maximal inequalities for the multilinear operator . As an application, we obtain the weighted -norm inequality and Morrey space boundedness for the multilinear operator .
For , let and .
We write if .
where and denotes the side length of Q. For , let .
Let or and . By , we know that spaces or coincide and the norms or are equivalent with respect to different values .
2 Theorems and lemmas
We shall prove the following theorems.
Theorem 3 Let T be a singular integral operator with non-smooth kernel as given in Definition 3, , and for all α with . Then is bounded from to .
Theorem 4 Let T be a singular integral operator with non-smooth kernel as given in Definition 3, , , and for all α with . Then is bounded from to .
Theorem 5 Let T be a singular integral operator with non-smooth kernel as given in Definition 3, , , , and for all α with . Then is bounded from to .
Theorem 6 Let T be a singular integral operator with non-smooth kernel as given in Definition 3, , , , , and for all α with . Then is bounded from to .
To prove the theorems, we need the following lemmas.
Lemma 1 (see [, p.485])
Let T be a singular integral operator with non-smooth kernel as given in Definition 2. Then T is bounded on for with , and weak bounded.
Lemma 7 (see )
where is the cube centered at x and having side length .
where and denotes the side length of Q.
for some . This completes the proof. □
The same argument as in the proof of Lemma 8 will give the proof of Lemma 9, we omit the details.
3 Proofs of theorems
These complete the proof of Theorem 1. □
This completes the proof of Theorem 2. □
This completes the proof of Theorem 3. □
This completes the proof of Theorem 4. □
This completes the proof of Theorem 5. □
This completes the proof of Theorem 6. □
for some and for all .
Now, let L be a linear operator on with so that generates a holomorphic semigroup , . Applying Theorem 6 of  and Theorems 1-6, 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
Then Theorems 1-6 hold for the multilinear operator associated to and b.
The author completed the paper, and read and approved the final manuscript.
Project was supported by Scientific Research Fund of Hunan Provincial Education Departments (13C1007).
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