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Weighted sharp maximal function inequalities and boundedness of multilinear singular integral operator satisfying a variant of Hörmander’s condition
Journal of Inequalities and Applications volume 2014, Article number: 57 (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.
As the development of singular integral operators (see [1, 2]), their commutators and multilinear operators have been well studied. In [3–5], the authors proved that the commutators generated by the singular integral operators and BMO functions are bounded on for . Chanillo (see ) proved a similar result when 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 are obtained. In [9, 10], the boundedness for the commutators generated by the singular integral operators and the weighted BMO and Lipschitz functions on () spaces are obtained (also see ). In [12, 13], the authors studied some multilinear singular integral operators as follows (also see ):
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 any locally integrable function f, the sharp maximal function of f is defined by
It is well known that (see )
For , let and .
For , and the non-negative weight function ω, set
The weight is defined by (see )
Given a non-negative weight function ω. For , the weighted Lebesgue space is the space of functions f such that
Given the non-negative weight function ω, the weighted BMO space is the space of functions b such that
For , the weighted Lipschitz space is the space of functions b such that
Remark (1) It has been known that (see ), for , and ,
Let and . By , we know that spaces coincide and the norms are equivalent with respect to different values .
Definition 1 Let be a finite family of bounded functions in . For any locally integrable function f, the Φ sharp maximal function of f is defined by
where the infimum is taken over all m-tuples of complex numbers and is the center of Q. For , let
Remark We note that if and .
Definition 2 Given a positive and locally integrable function f in , we say that f satisfies the reverse Hölder’s condition (write this as ), if for any cube Q centered at the origin we have
In this paper, we will study some singular integral operators as follows (see ).
Definition 3 Let and satisfy
there exist functions and such that , and for a fixed and any ,
For , we define the singular integral operator related to the kernel K by
Moreover, let m be the positive integer and b be the function on . Set
The multilinear operator related to the operator T is defined by
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])
Let and for any function . We define, for ,
where the sup is taken for all measurable sets Q with . Then
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 )
Let . Then
Lemma 4 (see )
Let , . Then there exists such that for any .
Lemma 5 (see )
Let , , and . Then there exists such that for ,
Lemma 6 (see )
Let , . Then there exists such that for any , where .
Lemma 7 (see )
Let , , . Then there exists such that
Let , , and . Then
Let , , and such that . Then, for any smooth function f for which the left-hand side is finite,
Lemma 10 (see )
Let b be a function on and for all α with and any . Then
where is the cube centered at x and having side length .
3 Theorems and proofs
We shall prove the following theorems.
Theorem 1 Let T be the singular integral operator as Definition 3, , , , and for all α with . Then there exist a constant , , , and such that, for any and ,
Theorem 2 Let T be the singular integral operator as Definition 3, , , , and for all α with . Then there exists a constant such that, for any and ,
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 .
Proof of Theorem 1 It suffices to prove for and some constant , the following inequality holds:
where Q is any a cube centered at , and . Fix a cube and . Let and , then and for . We write, for and ,
For , noting that , w satisfies the reverse of Hölder’s inequality:
for all cubes Q and some (see ). We take in Lemma 10 and have and , then by Lemma 10 and Hölder’s inequality, we obtain
thus, by Lemma 7, we obtain
For , we know by Lemma 4, thus
then, by the weak boundedness of T (see Lemma 2) and Kolmogorov’s inequality (see Lemma 1), we obtain, by Lemma 5,
For , note that for and , we write
For , by the formula (see ):
and Lemma 10, we have, similar to the proof of and for ,
For , we get
Similarly, we have
These results complete the proof of Theorem 1. □
Proof of Theorem 2 It suffices to prove for and some constant that the following inequality holds:
where Q is any cube centered at , and . Fix a cube and . Similar to the proof of Theorem 1, we have, for and ,
For and , by using the same argument as in the proof of Theorem 1, we get
For , we have
This completes the proof of Theorem 2. □
Proof of Theorem 3 Notice that and by Lemma 6, thus, by Theorem 1, Lemmas 2 and 9,
This completes the proof of Theorem 3. □
Proof of Theorem 4 Choose in Theorem 2 and notice , then we have, by Lemmas 8 and 9,
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.
Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.
Coifman RR, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. 1976, 103: 611–635. 10.2307/1970954
Pérez C: Endpoint estimate for commutators of singular integral operators. J. Funct. Anal. 1995, 128: 163–185. 10.1006/jfan.1995.1027
Pérez C, Trujillo-Gonzalez R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 2002, 65: 672–692. 10.1112/S0024610702003174
Chanillo S: A note on commutators. Indiana Univ. Math. J. 1982, 31: 7–16. 10.1512/iumj.1982.31.31002
Janson S: Mean oscillation and commutators of singular integral operators. Ark. Mat. 1978, 16: 263–270. 10.1007/BF02386000
Paluszynski M: Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J. 1995, 44: 1–17.
Bloom S: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 1985, 292: 103–122. 10.1090/S0002-9947-1985-0805955-5
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.2007.08.034
He YX, Wang YS: Commutators of Marcinkiewicz integrals and weighted BMO. Acta Math. Sin. Chin. Ser. 2011, 54: 513–520.
Cohen J, Gosselin J:On multilinear singular integral operators on . Stud. Math. 1982, 72: 199–223.
Cohen J, Gosselin J: A BMO estimate for multilinear singular integral operators. Ill. J. Math. 1986, 30: 445–465.
Ding Y, Lu SZ: Weighted boundedness for a class of rough multilinear operators. Acta Math. Sin. 2001, 17: 517–526.
Grubb DJ, Moore CN: A variant of Hörmander’s condition for singular integrals. Colloq. Math. 1997, 73: 165–172.
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.
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.
Garcia-Cuerva J Dissert. Math. 162. Weighted Hp Spaces 1979.
The authors declare that they have no competing interests.
CW carried out all of the paper, MZ participated in the proof of Theorem 2. All authors read and approved the final manuscript.
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Wu, C., Zhang, M. Weighted sharp maximal function inequalities and boundedness of multilinear singular integral operator satisfying a variant of Hörmander’s condition. J Inequal Appl 2014, 57 (2014). https://doi.org/10.1186/1029-242X-2014-57
- multilinear operator
- singular integral operator
- sharp maximal function
- weighted BMO
- weighted Lipschitz function