Inequalities and boundedness for commutators related to integral operator with general kernel
© Zeng; licensee Springer. 2014
Received: 9 March 2014
Accepted: 25 April 2014
Published: 10 May 2014
In this paper, we establish the sharp maximal function inequalities for the commutators related to some integral operator with general kernel and the and Lipschitz functions. As an application, we obtain the boundedness of the commutators on Lebesgue, Morrey and Triebel-Lizorkin space. The operator includes Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.
Keywordscommutator Littlewood-Paley operator Marcinkiewicz operator Bochner-Riesz operator sharp maximal function Morrey space Triebel-Lizorkin space Lipschitz function
1 Introduction and preliminaries
As the development of singular integral operators (see [1–3]), their commutators have been well studied (see [4–6]). In [5–7], the authors prove that the commutators generated by the singular integral operators and functions are bounded on for . Chanillo (see ) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [9–11], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and () spaces are obtained. In , some singular integral operators with general kernel are introduced, and the boundedness for the operators and their commutators generated by and Lipschitz functions are obtained (see [12, 13]). The purpose of this paper is to prove the sharp maximal function inequalities for the commutator associated with some integral operator with general kernel and the and Lipschitz functions. As an application, we obtain the boundedness of the commutator on Lebesgue, Morrey and Triebel-Lizorkin space. The operator includes Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.
For , let .
For and , let be the weighted homogeneous Triebel-Lizorkin space (see ).
In this paper, we will study some integral operators as follows (see ).
for every bounded and compactly supported function f.
where and .
It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [5, 6]). In , Pérez and Trujillo-Gonzalez prove a sharp estimate for the multilinear commutator. The main purpose of this paper is to prove the sharp maximal inequalities for the commutator. As the application, we obtain the -norm inequality, Morrey and Triebel-Lizorkin spaces boundedness for the commutator.
We shall prove the following theorems.
Theorem 4 Let T be the integral operator as Definition 1, the sequence , , , and . Then is bounded from to .
Theorem 5 Let T be the integral operator as Definition 1, the sequence , , , , and . Then is bounded from to .
Theorem 6 Let T be the integral operator as Definition 1, the sequence , , , and . Then is bounded from to .
Theorem 7 Let T be the integral operator as Definition 1, the sequence and . Then is bounded on for .
3 Proofs of theorems
To prove the theorems, we need the following lemma.
Lemma 1 (see )
Let T be the integral operator as Definition 1, the sequence . Then T is bounded on for .
Lemma 2 (see )
Lemma 3 (see )
Lemma 4 (see )
This finishes the proof. □
The proofs of the two lemmas are similar to that of Lemma 5 by Lemmas 1 and 4, we omit the details.
This completes 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 the theorem. □
This completes the proof of the theorem. □
In this section we shall apply Theorems 1-6 of the paper to some particular operators such as the Littlewood-Paley operators, Marcinkiewicz operator and Bochner-Riesz operator.
Application 1 Littlewood-Paley operators.
Application 2 Marcinkiewicz operators.
Application 3 Bochner-Riesz operator.
It is easy to see that satisfies the conditions of Theorems 1-6 (see ), thus Theorems 1-6 hold for .
- 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
- 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
- Chanillo S: A note on commutators. Indiana Univ. Math. J. 1982, 31: 7–16. 10.1512/iumj.1982.31.31002MathSciNetView ArticleMATHGoogle Scholar
- Chen WG: Besov estimates for a class of multilinear singular integrals. Acta Math. Sin. 2000, 16: 613–626. 10.1007/s101140000059View ArticleMathSciNetMATHGoogle Scholar
- Janson S: Mean oscillation and commutators of singular integral operators. Ark. Math. 1978, 16: 263–270. 10.1007/BF02386000MathSciNetView ArticleMATHGoogle 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
- Chang DC, Li JF, Xiao J: Weighted scale estimates for Calderón-Zygmund type operators. Contemp. Math. 2007, 446: 61–70.MathSciNetView ArticleMATHGoogle Scholar
- Lin Y: Sharp maximal function estimates for Calderón-Zygmund type operators and commutators. Acta Math. Sci. Ser. A Chin. Ed. 2011, 31: 206–215.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
- Chiarenza F, Frasca M: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. 1987, 7: 273–279.MathSciNetMATHGoogle Scholar
- Di Fazio G, Ragusa MA: Commutators and Morrey spaces. Boll. Unione Mat. Ital., A (7) 1991, 5: 323–332.MathSciNetMATHGoogle 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. 2005,25(1):89–94.MathSciNetMATHGoogle Scholar
- Mizuhara T: Boundedness of some classical operators on generalized Morrey spaces. In: Harmonic Analysis ICM-90 Satellite Conference Proceedings (Sendai, 1990), 1991, 183–189.Google Scholar
- Coifman RR, Rochberg R: Another characterization of BMO. Proc. Am. Math. Soc. 1980, 79: 249–254. 10.1090/S0002-9939-1980-0565349-8MathSciNetView ArticleMATHGoogle Scholar
- Liu LZ: The continuity of commutators on Triebel-Lizorkin spaces. Integral Equ. Oper. Theory 2004, 49: 65–76. 10.1007/s00020-002-1186-8View ArticleMathSciNetMATHGoogle Scholar
- Liu LZ: Triebel-Lizorkin space estimates for multilinear operators of sublinear operators. Proc. Indian Acad. Sci. Math. Sci. 2003, 113: 379–393. 10.1007/BF02829632MathSciNetView ArticleMATHGoogle Scholar
- Liu LZ: Boundedness for multilinear Littlewood-Paley operators on Triebel-Lizorkin spaces. Methods Appl. Anal. 2004,10(4):603–614.MathSciNetMATHGoogle Scholar
- Torchinsky A, Wang S: A note on the Marcinkiewicz integral. Colloq. Math. 1990,60(61):235–243.MathSciNetMATHGoogle Scholar
- Lu SZ: Four Lectures on Real Hp Spaces. World Scientific, River Edge; 1995.View ArticleGoogle 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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.