Weighted boundedness of multilinear singular integral operators
© Kuang and Wang; licensee Springer. 2014
Received: 9 January 2014
Accepted: 28 February 2014
Published: 18 March 2014
In this paper, we establish the weighted sharp maximal function inequalities for the multilinear singular integral operators. As an application, we obtain the boundedness of the multilinear operators on weighted Lebesgue and Morrey spaces.
Keywordsmultilinear operator singular integral operator sharp maximal function weighted BMO weighted Lipschitz function
and they obtained some variant sharp function estimates and boundedness of the multilinear operators if for all α with . In this paper, we will study the multilinear operator generated by the singular integral operator and the weighted Lipschitz and BMO functions, that is, or for all α with .
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 will study the singular integral operators as follows (see ).
Note that the classical Calderón-Zygmund singular integral operator satisfies the conditions of Definition 2 (see [1, 4]) and that the commutator is a particular operator of the multilinear operator if . The multilinear operator are the non-trivial generalizations 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 [5, 6, 19]). The main purpose of this paper is to prove the sharp maximal inequalities for the multilinear operator . As the application, we obtain the weighted -norm inequality and Morrey space boundedness for the multilinear operator .
3 Theorems and lemmas
We shall prove the following theorems.
Theorem 3 Let T be the singular integral operator as Definition 2, , and for all α with . Then is bounded from to .
Theorem 4 Let T be the singular integral operator as Definition 2, , , and for all α with . Then is bounded from to .
Theorem 5 Let T be the singular integral operator as Definition 2, , , , and for all α with . Then is bounded from to .
Theorem 6 Let T be the singular integral operator as Definition 2, , , , , 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 3 (See )
Lemma 4 (See )
Lemma 5 (See )
Lemma 6 (See )
where is the cube centered at x and having side length .
4 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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