Weighted boundedness of a multilinear operator associated to a singular integral operator with general kernels
© Feng; licensee Springer. 2014
Received: 24 December 2013
Accepted: 25 April 2014
Published: 13 May 2014
In this paper, we establish the weighted sharp maximal function inequalities for a multilinear operator associated to a singular integral operator with general kernels. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.
Keywordsmultilinear operator singular integral operator sharp maximal function weighted BMO weighted Lipschitz function
1 Introduction and preliminaries
As the development of singular integral operators (see [1–3]), their commutators and multilinear operators have been well studied. In [4–6], the authors prove that the commutators generated by singular integral operators and BMO functions are bounded on for . Chanillo (see ) proves a similar result when singular integral operators are replaced by fractional integral operators. In [8, 9], the boundedness for the commutators generated by singular integral operators and Lipschitz functions on Triebel-Lizorkin and () spaces is obtained. In [10, 11], the boundedness for the commutators generated by singular integral operators and weighted BMO and Lipschitz functions on () spaces is obtained. In , some singular integral operators with general kernel are introduced, and the boundedness for the operators and their commutators generated by BMO and Lipschitz functions is obtained (see [8, 12, 13]). Motivated by these, in this paper, we study the multilinear operator generated by the singular integral operator with general kernel and the weighted Lipschitz and BMO functions.
For , let and .
We write if .
Let or and . By , we know that spaces or coincide and the norms or are equivalent with respect to different values .
In this paper, we study some singular integral operators as follows (see ).
Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 1 (see [1–3, 5, 6]) and 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 [21–23]). The main purpose of this paper is to prove the sharp maximal inequalities for the multilinear operator . As application, we obtain the weighted -norm inequality and Morrey space boundedness for the multilinear operator .
2 Theorems and lemmas
We shall prove the following theorems.
Theorem 3 Let T be a singular integral operator as in Definition 2, the sequence , , and for all α with . Then is bounded from to .
Theorem 4 Let T be a singular integral operator as in Definition 2, the sequence , , , and for all α with . Then is bounded from to .
Theorem 5 Let T be a singular integral operator as in Definition 2, the sequence , , , , and for all α with . Then is bounded from to .
Theorem 6 Let T be a singular integral operator as in Definition 2, the sequence , , , , , and for all α with . Then is bounded from to .
To prove the theorems, we need the following lemmas.
Lemma 1 (see [, p.485])
Lemma 2 (see )
Let T be a singular integral operator as in Definition 2, the sequence . Then T is bounded on for with , and weak bounded.
Lemma 4 (see )
Lemma 7 (see )
where is the cube centered at x and having side length .
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. □
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