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.
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. □
- Garcia-Cuerva J, Rubio de Francia JL North-Holland Mathematics Studies 16. In Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam; 1985.Google Scholar
- Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.MATHGoogle Scholar
- Torchinsky A Pure and Applied Math. 123. In Real Variable Methods in Harmonic Analysis. Academic Press, New York; 1986.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 ArticleMATHGoogle Scholar
- Pérez C: Endpoint estimate for commutators of singular integral operators. J. Funct. Anal. 1995, 128: 163–185. 10.1006/jfan.1995.1027MathSciNetView ArticleMATHGoogle Scholar
- Pérez C, Trujillo-Gonzalez R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 2002, 65: 672–692. 10.1112/S0024610702003174View ArticleMathSciNetMATHGoogle Scholar
- Chanillo S: A note on commutators. Indiana Univ. Math. J. 1982, 31: 7–16. 10.1512/iumj.1982.31.31002MathSciNetView ArticleMATHGoogle Scholar
- Lin Y: Sharp maximal function estimates for Calderón-Zygmund type operators and commutators. Acta Math. Sci., Ser. A 2011, 31: 206–215.MathSciNetMATHGoogle 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 ArticleMATHGoogle Scholar
- Bloom S: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 1985, 292: 103–122. 10.1090/S0002-9947-1985-0805955-5View ArticleMathSciNetMATHGoogle 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 ArticleMATHGoogle Scholar
- Chang DC, Li JF, Xiao J: Weighted scale estimates for Calderón-Zygmund type operators. Contemp. Math. 2007, 446: 61–70.MathSciNetView ArticleMATHGoogle Scholar
- Liu LZ: Sharp maximal function estimates and boundedness for commutators associated with general integral operator. Filomat 2011,25(4):137–151. 10.2298/FIL1104137LMathSciNetView ArticleMATHGoogle Scholar
- Garcia-Cuerva J:Weighted spaces. Diss. Math. 1979, 162: 1–63.MathSciNetMATHGoogle Scholar
- Peetre J:On convolution operators leaving -spaces invariant. Ann. Mat. Pura Appl. 1966, 72: 295–304. 10.1007/BF02414340MathSciNetView ArticleMATHGoogle Scholar
- Peetre J:On the theory of -spaces. J. Funct. Anal. 1969, 4: 71–87. 10.1016/0022-1236(69)90022-6MathSciNetView ArticleGoogle Scholar
- Di Fazio G, Ragusa MA: Commutators and Morrey spaces. Boll. Unione Mat. Ital., A 1991, 5: 323–332.MathSciNetMATHGoogle Scholar
- Di Fazio G, Ragusa MA: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 1993, 112: 241–256. 10.1006/jfan.1993.1032MathSciNetView ArticleMATHGoogle Scholar
- Liu LZ: Interior estimates in Morrey spaces for solutions of elliptic equations and weighted boundedness for commutators of singular integral operators. Acta Math. Sci., Ser. B 2005, 25: 89–94.MathSciNetMATHGoogle Scholar
- Mizuhara T: Boundedness of some classical operators on generalized Morrey spaces. Harmonic Analysis: Proceedings of a Conference Held in Sendai 1990, 183–189. JapanGoogle Scholar
- Cohen J, Gosselin J:On multilinear singular integral operators on . Stud. Math. 1982, 72: 199–223.MathSciNetMATHGoogle Scholar
- Cohen J, Gosselin J: A BMO estimate for multilinear singular integral operators. Ill. J. Math. 1986, 30: 445–465.MathSciNetMATHGoogle Scholar
- Ding Y, Lu SZ: Weighted boundedness for a class rough multilinear operators. Acta Math. Sin. 2001, 17: 517–526.