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Weighted sharp maximal function inequalities and boundedness of multilinear singular integral operator satisfying a variant of Hörmander’s condition
© Wu and Zhang; licensee Springer. 2014
- Received: 9 August 2013
- Accepted: 13 January 2014
- Published: 10 February 2014
In this paper, we establish the weighted sharp maximal function inequalities for the multilinear operator associated with the singular integral operator satisfying a variant of Hörmander’s condition. As an application, we obtain the boundedness of the operator on weighted Lebesgue spaces.
- multilinear 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 , some singular integral operators satisfying a variant of Hörmander’s condition are introduced, and the boundedness for the operators and their commutators is obtained (see [16, 17]). Motivated by these results, in this paper, we will study the multilinear operator generated by the singular integral operator satisfying a variant of Hörmander’s condition and the weighted Lipschitz and BMO functions, that is, or for all α with .
First, let us introduce some notation. Throughout this paper, Q will denote a cube of with sides parallel to the axes. For a non-negative integrable function ω, let and .
For , let and .
Let and . By , we know that spaces coincide and the norms are equivalent with respect to different values .
Remark We note that if and .
In this paper, we will study some singular integral operators as follows (see ).
Note 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 [12–14]). The main purpose of this paper is to prove the sharp maximal inequalities for the multilinear operator . As the application, we obtain the weighted -boundedness for the multilinear operator .
We give some preliminary lemmas.
Lemma 1 (see [, p.485])
Lemma 2 (see )
Let T be the singular integral operator as Definition 2. Then T is bounded on for with , and weak bounded.
Lemma 3 (see )
Lemma 4 (see )
Let , . Then there exists such that for any .
Lemma 5 (see )
Lemma 6 (see )
Let , . Then there exists such that for any , where .
Lemma 7 (see )
Lemma 10 (see )
where is the cube centered at x and having side length .
We shall prove the following theorems.
Theorem 3 Let T be the singular integral operator as Definition 3, , , and for all α with . Then is bounded from to .
Theorem 4 Let T be the singular integral operator as Definition 3, , , , and for all α with . Then is bounded from to .
Corollary Let be the commutator generated by the singular integral operator T as Definition 2 and b. Then Theorems 1-4 hold for .
These results 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. □
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