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.
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 .
3 Theorems and proofs
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. □
- Garcia-Cuerva J, Rubio de Francia JL North-Holland Math. 16. In Weighted Norm Inequalities and Related Topics. Elsevier, Amsterdam; 1985.Google Scholar
- Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.Google Scholar
- Coifman RR, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. 1976, 103: 611–635. 10.2307/1970954MathSciNetView ArticleGoogle Scholar
- Pérez C: Endpoint estimate for commutators of singular integral operators. J. Funct. Anal. 1995, 128: 163–185. 10.1006/jfan.1995.1027MathSciNetView ArticleGoogle Scholar
- Pérez C, Trujillo-Gonzalez R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 2002, 65: 672–692. 10.1112/S0024610702003174View ArticleGoogle Scholar
- Chanillo S: A note on commutators. Indiana Univ. Math. J. 1982, 31: 7–16. 10.1512/iumj.1982.31.31002MathSciNetView ArticleGoogle Scholar
- Janson S: Mean oscillation and commutators of singular integral operators. Ark. Mat. 1978, 16: 263–270. 10.1007/BF02386000MathSciNetView ArticleGoogle Scholar
- Paluszynski M: Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J. 1995, 44: 1–17.MathSciNetView ArticleGoogle Scholar
- Bloom S: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 1985, 292: 103–122. 10.1090/S0002-9947-1985-0805955-5View ArticleGoogle Scholar
- Hu B, Gu J: Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz spaces. J. Math. Anal. Appl. 2008, 340: 598–605. 10.1016/j.jmaa.2007.08.034MathSciNetView ArticleGoogle Scholar
- He YX, Wang YS: Commutators of Marcinkiewicz integrals and weighted BMO. Acta Math. Sin. Chin. Ser. 2011, 54: 513–520.Google Scholar
- Cohen J, Gosselin J:On multilinear singular integral operators on . Stud. Math. 1982, 72: 199–223.MathSciNetGoogle Scholar
- Cohen J, Gosselin J: A BMO estimate for multilinear singular integral operators. Ill. J. Math. 1986, 30: 445–465.MathSciNetGoogle Scholar
- Ding Y, Lu SZ: Weighted boundedness for a class of rough multilinear operators. Acta Math. Sin. 2001, 17: 517–526.MathSciNetView ArticleGoogle Scholar
- Grubb DJ, Moore CN: A variant of Hörmander’s condition for singular integrals. Colloq. Math. 1997, 73: 165–172.MathSciNetGoogle Scholar
- Lu DQ, Liu LZ:-Type sharp estimates and weighted boundedness for commutators related to singular integral operators satisfying a variant of Hörmander’s condition. Bol. Soc. Parana. Mat. 2013, 31: 99–114.MathSciNetGoogle Scholar
- Trujillo-Gonzalez R: Weighted norm inequalities for singular integral operators satisfying a variant of Hörmander’s condition. Comment. Math. Univ. Carol. 2003, 44: 137–152.MathSciNetGoogle Scholar
- Garcia-Cuerva J Dissert. Math. 162. Weighted Hp Spaces 1979.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.