-Type sharp estimates and boundedness on a Morrey space for Toeplitz-type operators associated to singular integral operators satisfying a variant of Hörmander’s condition
© Feng; licensee Springer. 2013
Received: 4 June 2013
Accepted: 15 November 2013
Published: 17 December 2013
In this paper, we prove the -type sharp maximal function estimates for the Toeplitz-type operators associated to certain singular integral operators satisfying a variant of Hörmander’s condition. As an application, we obtain the weighted boundedness of the operators on the Lebesgue and Morrey spaces.
As the development of singular integral operators (see [1, 2]), their commutators have been well studied. In [3, 4], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on for . Chanillo (see ) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [6, 7], some Toeplitz-type operators related to the singular integral operators and strongly singular integral operators are introduced, and the boundedness for the operators generated by BMO and Lipschitz functions is obtained. In , some singular integral operators satisfying a variant of Hörmander’s condition are introduced, and the boundedness for the operators is obtained (see , ). In this paper, we prove the sharp maximal function inequalities for the Toeplitz-type operator related to some singular integral operators satisfying a variant of Hörmander’s condition. As an application, we obtain the weighted boundedness of the Toeplitz-type operator on Lebesgue and Morrey spaces.
Remark We note that if and .
In this paper, we study some singular integral operators as follows (see ).
where are T or ±I (the identity operator), are the bounded linear operators on for and , , .
Remark Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 4 (see , ). Also note that the commutator is a particular operator of the Toeplitz-type operators . The Toeplitz-type operators are the non-trivial generalizations of the commutator. It is well known that commutators 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 Toeplitz-type operator . As the application, we obtain the weighted -norm inequality and Morrey space boundedness for the Toeplitz-type operators .
3 Theorems and lemmas
We shall prove the following theorems.
Theorem 2 Let T be the singular integral operator as Definition 4, , and . If for any (), then is bounded on .
Theorem 3 Let T be the singular integral operator as Definition 4, , , and . If for any (), then is bounded on .
To prove the theorems, we need the following lemmas.
Lemma 1 ([, p.485])
Lemma 2 (see )
Lemma 3 (see )
Let T be the singular integral operator as Definition 4. Then T is bounded on for , and weak bounded.
Lemma 4 (see )
This finishes the proof. □
The proof of the lemma is similar to that of Lemma 6 by Lemma 3, we omit the details.
4 Proofs of theorems
This completes the proof of Theorem 1. □
This completes the proof. □
This completes the proof. □
- 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 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
- Coifman R, Rochberg R: Another characterization of BMO. Proc. Am. Math. Soc. 1980, 79: 249–254. 10.1090/S0002-9939-1980-0565349-8MathSciNetView ArticleGoogle 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.MathSciNetGoogle Scholar
- Garcia-Cuerva J, Rubio de Francia JL North-Holland Math. Stud. 16. In Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam; 1985.Google Scholar
- Grubb DJ, Moore CN: A variant of Hörmander’s condition for singular integrals. Colloq. Math. 1997, 73: 165–172.MathSciNetGoogle 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.MathSciNetGoogle Scholar
- Mizuhara T: Boundedness of some classical operators on generalized Morrey spaces. ICM-90 Satell. Conf. Proc. Harmonic Analysis 1990, 183–189. Sendai, 1990Google 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
- Di Fazio G, Ragusa MA: Commutators and Morrey spaces. Boll. Unione Mat. Ital., A 1991, 5(7):323–332.MathSciNetGoogle Scholar
- Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.Google 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
- Torchinsky A Pure and Applied Math. 123. In Real Variable Methods in Harmonic Analysis. Academic Press, New York; 1986.Google Scholar
- Chanillo S: A note on commutators. Indiana Univ. Math. J. 1982, 31: 7–16. 10.1512/iumj.1982.31.31002MathSciNetView 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/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.