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Weighted boundedness for Toeplitz type operators related to strongly singular integral operators
Journal of Inequalities and Applications volume 2014, Article number: 42 (2014)
Abstract
In this paper, we show the sharp maximal function estimates for the Toeplitz type operators related to the strongly singular integral operators. As an application, we obtain the boundedness of the operators on weighted Lebesgue and Triebel-Lizorkin spaces.
MSC:42B20, 42B25.
1 Introduction and Preliminaries
As a development of singular integral operators [1, 2], their commutators have been well studied. In [3–5], the authors prove that the commutators generated by the singular integral operators and functions are bounded on for . Chanillo [6] proves a similar result when singular integral operators are replaced by the fractional integral operators. In [7–9], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and () spaces are obtained. In [10, 11], the boundedness for the commutators generated by the singular integral operators and the weighted and Lipschitz functions on () spaces are obtained. In [12, 13], 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 and Lipschitz functions is obtained. In this paper, we will study the Toeplitz type operators related to the strongly singular integral operator and the weighted Lipschitz functions.
First, let us introduce some notation. Throughout this paper, Q will denote a cube of with sides parallel to the axes. For any locally integrable function f, the sharp maximal function of f is defined by
where we write . It is well known that [1, 2]
Let
For , set .
For and , set
The weight is defined by [1]
and
The weight is defined by [14], for ,
Given a non-negative weight function w, for , the weighted Lebesgue space is the space of functions f such that
For , and the non-negative weight function w, let be the weighted homogeneous Triebel-Lizorkin space [9].
For and the non-negative weight function w, the weighted Lipschitz space is the space of functions b such that
Remark (1) For , and , it is well known that
(2) Let and . By [15], we know that the spaces coincide and the norms are equivalent with respect to different values of .
Definition Let be a bounded linear operator. T is called a strongly singular integral operator if it satisfies the following conditions:
-
(i)
T extends to a bounded operator on ;
-
(ii)
there exists a function continuous away from the diagonal on such that
if for some , , and for with disjoint support;
-
(iii)
for some , T and extend to a bounded operator from into , where .
Let b be a locally integrable function on . The Toeplitz type operator related to T is defined by
where are strongly singular integral operators or ±I (the identity operator), are bounded linear operators on for , , .
Note that the commutator is a particular case of the Toeplitz type operators . The Toeplitz type operators are 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 [4, 5]. In [16–19], the boundedness of the strongly singular integral operator is obtained. In [20], a sharp function estimate of the strongly singular integral operator is obtained. In [21], the boundedness of the strongly singular integral operators and their commutators is obtained. In [13], the Toeplitz type operators related to the strongly singular integral operators are introduced, and the boundedness for the operators generated by and Lipschitz functions is obtained. Our works are motivated by these papers. The main purpose of this paper is to prove sharp maximal inequalities for the Toeplitz type operators . As applications, we obtain the weighted -norm inequality and the Triebel-Lizorkin space boundedness for the Toeplitz type operators .
We need the following preliminary lemmas.
Lemma 1 ([16])
Let T be a strongly singular integral operator. Then T is bounded on for with , and when , , T is bounded from into .
Lemma 2 ([15])
For any cube , , and , we have
Lemma 3 ([9])
For , , and , we have
Lemma 4 ([1])
Let and . Then, for any smooth function f for which the left-hand side is finite,
Lemma 5 ([14])
Suppose that , , , and . Then
2 Theorems and proofs
We shall prove the following theorems.
Theorem 1 Let , , , and . If () and , then there exists a constant such that, for any and ,
Theorem 2 Let , , , and . If () and , then there exists a constant such that, for any and ,
Theorem 3 Let , , , and . If () and , then is bounded from to .
Theorem 4 Let , , , , and . If () and , then is bounded from to .
Proof of Theorem 1 It suffices to prove for and some constant that the following inequality holds:
Without loss of generality, we may assume are T (). Fix a cube and . We have the following two cases.
Case 1. . Write
Then
For , by Hölder’s inequality, boundedness of T, and Lemma 2, we obtain
thus
For , by and for and , we obtain, for ,
thus
Case 2. . Set and write
Then
For , since , there exists q such that , , and T is bounded from into . By using the same argument as in the proof of , we get
thus
For , by using the same argument as in the proof of , we get, for ,
thus
These complete the proof of Theorem 1. □
Proof of Theorem 2 It suffices to prove for and some constant that the following inequality holds:
Without loss of generality, we may assume are T (). Fix a cube and . We have the following two cases.
Case 1. . Similar to the proof of Theorem 1, we have
and
By using the same argument as in the proof of Theorem 1, we get
Case 2. . Set , where , and write
and
By using the same argument as in the proof of Theorem 1, for , there exists q such that , , and T is bounded from into , and we get
This completes the proof of Theorem 2. □
Proof of Theorem 3 Choose in Theorem 1, notice that and , and we have, by Lemmas 1, 4, and 5,
This completes the proof of Theorem 3. □
Proof of Theorem 4 Choose in Theorem 2. By using Lemma 3, we obtain
This completes the proof of the theorem. □
Remark A typical example of strongly singular integral operators is a class of multiplier operators whose symbol is given by for and [18–20, 22].
References
Garcia-Cuerva J, Rubio de Francia JL North-Holland Math. 16. In Weighted Norm Inequalities and Related Topics. North-Holland, 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
Chen WG: Besov estimates for a class of multilinear singular integrals. Acta Math. Sin. 2000, 16: 613-626. 10.1007/s101140000059
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 . 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
Krantz S, Li S: Boundedness and compactness of integral operators on spaces of homogeneous type and applications. J. Math. Anal. Appl. 2001, 258: 629-641. 10.1006/jmaa.2000.7402
Lin Y, Lu SZ: Toeplitz type operators associated to strongly singular integral operator. Sci. China Ser. A 2006, 36: 615-630.
Muckenhoupt B, Wheeden RL: Weighted norm inequalities for fractional integral. Trans. Am. Math. Soc. 1974, 192: 261-274.
Garcia-Cuerva, J: Weighted spaces. Diss. Math. 162 (1979)
Alvarez J, Milman M: continuity properties of Calderón-Zygmund type operators. J. Math. Anal. Appl. 1986, 118: 63-79. 10.1016/0022-247X(86)90290-8
Alvarez J, Milman M: Vector-valued inequalities for strongly singular Calderón-Zygmund operators. Rev. Mat. Iberoam. 1986, 2: 405-426.
Fefferman C: Inequalities for strongly singular convolution operators. Acta Math. 1970, 124: 9-36. 10.1007/BF02394567
Fefferman C, Stein EM: spaces of several variables. Acta Math. 1972, 129: 137-193. 10.1007/BF02392215
Chanillo S: Weighted norm inequalities for strongly singular convolution operators. Trans. Am. Math. Soc. 1984, 281: 77-107. 10.1090/S0002-9947-1984-0719660-6
Garcia-Cuerva J, Harboure E, Segovia C, Torrea JL: Weighted norm inequalities for commutators of strongly singular integrals. Indiana Univ. Math. J. 1991, 40: 1397-1420. 10.1512/iumj.1991.40.40063
Sjölin P:An inequality for strongly singular integrals. Math. Z. 1979, 165: 231-238. 10.1007/BF01437558
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Chen, D. Weighted boundedness for Toeplitz type operators related to strongly singular integral operators. J Inequal Appl 2014, 42 (2014). https://doi.org/10.1186/1029-242X-2014-42
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DOI: https://doi.org/10.1186/1029-242X-2014-42