- Research
- Open access
- Published:
Weighted boundedness of multilinear operators associated to singular integral operators with non-smooth kernels
Journal of Inequalities and Applications volume 2014, Article number: 276 (2014)
Abstract
In this paper, we establish the weighted sharp maximal function inequalities for a multilinear operator associated to a singular integral operator with non-smooth kernel. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.
MSC:42B20, 42B25.
1 Introduction and preliminaries
As the development of singular integral operators (see [1, 2]), their commutators and multilinear operators have been well studied. In [3–5], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on for . Chanillo (see [6]) proves a similar result when the 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 is obtained. In [9, 10], the boundedness for the commutators generated by the singular integral operators and the weighted BMO and Lipschitz functions on () spaces is obtained (also see [11]). In [12, 13], some singular integral operators with non-smooth kernels are introduced, and the boundedness for the operators and their commutators is obtained (see [14–17]). Motivated by these, in this paper, we study multilinear operators generated by singular integral operators with non-smooth kernels and the weighted Lipschitz and BMO functions.
In this paper, we study some singular integral operators as follows (see [13]).
Definition 1 A family of operators , , is said to be an ‘approximation to the identity’ if, for every , can be represented by a kernel in the following sense:
for every with , and satisfies
where ρ is a positive, bounded and decreasing function satisfying
for some .
Definition 2 A linear operator T is called a singular integral operator with non-smooth kernel if T is bounded on and associated with the kernel so that
for every continuous function f with compact support, and for almost all x not in the support of f.
-
(1)
There exists an ‘approximation to the identity’ such that has the associated kernel and there exist so that
-
(2)
There exists an ‘approximation to the identity’ such that has the associated kernel which satisfies
and
for some , . Moreover, let m be a positive integer and b be a 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 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 [18–20]). The main purpose of this paper is to prove sharp maximal inequalities for the multilinear operator . As an application, we obtain the weighted -norm inequality and Morrey space boundedness for the multilinear operator .
Now, let us introduce some notations. 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, and in what follows, . It is well known that (see [1, 2])
Let
For , let and .
For , and the non-negative weight function w, set
We write if .
The sharp maximal function associated with the ‘approximation to the identity’ is defined by
where and denotes the side length of Q. For , let .
The weight is defined by (see [1]), for ,
and
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, the weighted Lipschitz space is the space of functions b such that
and the weighted BMO space is the space of functions b such that
Remark (1) It has been known that (see [9, 21]), for , and ,
-
(2)
It has been known that (see [1, 21]), for , and ,
-
(3)
Let or and . By [22], we know that spaces or coincide and the norms or are equivalent with respect to different values .
Definition 3 Let φ be a positive, increasing function on , and let there exist a constant such that
Let w be a non-negative weight function on and f be a locally integrable function on . Set, for and ,
where . The generalized fractional weighted Morrey space is defined by
We write if , which is the generalized weighted Morrey space. If , , then , which is the classical Morrey space (see [23, 24]). If , then , which is the weighted Lebesgue space (see [1]).
As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of the operator on the Morrey spaces (see [22, 25–27]).
2 Theorems and lemmas
We shall prove the following theorems.
Theorem 1 Let T be a singular integral operator with non-smooth kernel as given in Definition 2, , , and for all α with . Then there exists a constant such that, for any and ,
Theorem 2 Let T be a singular integral operator with non-smooth kernel as given in Definition 2, , , , and for all α with . Then there exists a constant such that, for any and ,
Theorem 3 Let T be a singular integral operator with non-smooth kernel as given in Definition 3, , and for all α with . Then is bounded from to .
Theorem 4 Let T be a singular integral operator with non-smooth kernel as given in Definition 3, , , and for all α with . Then is bounded from to .
Theorem 5 Let T be a singular integral operator with non-smooth kernel as given in Definition 3, , , , and for all α with . Then is bounded from to .
Theorem 6 Let T be a singular integral operator with non-smooth kernel as given in Definition 3, , , , , and for all α with . Then is bounded from to .
To prove the theorems, we need the following lemmas.
Lemma 1 (see [[1], p.485])
Let , and for any function , we define that, for ,
where the sup is taken for all measurable sets Q with . Then
Let T be a singular integral operator with non-smooth kernel as given in Definition 2. Then T is bounded on for with , and weak bounded.
Let be an ‘approximation to the identity’. For any , there exists a constant independent of γ such that
for , where D is a fixed constant which only depends on n. Thus, for , , and ,
Let , , and . Then
Let be an ‘approximation to the identity’, , , , and . Then
Let , , , and . Then
Lemma 7 (see [19])
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 .
Lemma 8 Let be an ‘approximation to the identity’, and . Then, for every , , and ,
where and denotes the side length of Q.
Proof We write, for any cube Q with ,
We have, by Hölder’s inequality,
For II, notice for and , then and , then
where the last inequality follows from
for some . This completes the proof. □
Lemma 9 Let be an ‘approximation to the identity’, , , and . Then, for every , and ,
The same argument as in the proof of Lemma 8 will give the proof of Lemma 9, we omit the details.
3 Proofs of theorems
Proof of Theorem 1 It suffices to prove for and some constant that the following inequality holds:
where and d denotes the side length of Q. Fix a cube and . Let and , then and for . We write, for and ,
and
then
For , noting that , w satisfies the reverse of Hölder’s inequality
for all cube Q and some (see [1]). We take in Lemma 7 and have and , then by Lemma 7 and Hölder’s inequality, we get
Thus, by the -boundedness of T (see Lemma 2) for and , we obtain
For , by the weak boundedness of T (see Lemma 2) and Kolmogorov’s inequality (see Lemma 1), we obtain
For and , by Lemma 8 and similar to the proof of and , we get
For , note that for and . We have, by Lemma 7 and similar to the proof of ,
Thus, by the conditions on K and , and ,
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:
where and d denotes the side length of Q. 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
Thus
For and , by Lemma 9 and similar to the proof of and , we get
For , by Lemma 7 and similar to the proof of , for , we have
Thus
This completes the proof of Theorem 2. □
Proof of Theorem 3 Choose in Theorem 1 and notice , then we have, by Lemmas 3 and 4,
This completes the proof of Theorem 3. □
Proof of Theorem 4 Choose in Theorem 1 and notice , then we have, by Lemmas 5 and 6,
This completes the proof of Theorem 4. □
Proof of Theorem 5 Choose in Theorem 2 and notice , then we have, by Lemmas 3 and 4,
This completes the proof of Theorem 5. □
Proof of Theorem 6 Choose in Theorem 2 and notice , then we have, by Lemmas 5 and 6,
This completes the proof of Theorem 6. □
4 Applications
In this section we shall apply the theorems of the paper to the holomorphic functional calculus of linear elliptic operators. First, we review some definitions regarding the holomorphic functional calculus (see [13]). Given . Define
and its interior by . Set . A closed operator L on some Banach space E is said to be of type θ if its spectrum and for every , there exists a constant such that
For , let
where . Set
If L is of type θ and , we define by
where Γ is the contour parameterized clockwise around with . If, in addition, L is one-to-one and has a dense range, then, for ,
where . L is said to have a bounded holomorphic functional calculus on the sector if
for some and for all .
Now, let L be a linear operator on with so that generates a holomorphic semigroup , . Applying Theorem 6 of [12] and Theorems 1-6, we get the following.
Corollary Assume that the following conditions are satisfied:
-
(i)
The holomorphic semigroup , is represented by the kernels which satisfy, for all , an upper bound
for , and , where and s is a positive, bounded and decreasing function satisfying
-
(ii)
The operator L has a bounded holomorphic functional calculus in ; that is, for all and , the operator satisfies
Then Theorems 1-6 hold for the multilinear operator associated to and b.
Author’s contributions
The author completed the paper, and read and approved the final manuscript.
References
Garcia-Cuerva J, Rubio de Francia JL North-Holland Mathematics Studies 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
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.
Duong XT, McIntosh A: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoam. 1999, 15: 233–265.
Martell JM: Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications. Stud. Math. 2004, 161: 113–145. 10.4064/sm161-2-2
Deng DG, Yan LX: Commutators of singular integral operators with non-smooth kernels. Acta Math. Sci. 2005, 25: 137–144.
Liu LZ: Sharp function boundedness for vector-valued multilinear singular integral operators with non-smooth kernels. J. Contemp. Math. Anal. 2010, 45: 185–196. 10.3103/S1068362310040011
Liu LZ: Multilinear singular integral operators on Triebel-Lizorkin and Lebesgue spaces. Bull. Malays. Math. Sci. Soc. 2012, 35: 1075–1086.
Zhou XS, Liu LZ: Weighted boundedness for multilinear singular integral operators with non-smooth kernels on Morrey space. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2010, 104: 115–127. 10.5052/RACSAM.2010.11
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 rough multilinear operators. Acta Math. Sin. 2001, 17: 517–526.
Garcia-Cuerva J Dissert. Math. 162. Weighted Hp Spaces 1979.
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.1993.1032
Peetre J:On convolution operators leaving -spaces invariant. Ann. Mat. Pura Appl. 1966, 72: 295–304. 10.1007/BF02414340
Peetre J:On the theory of -spaces. J. Funct. Anal. 1969, 4: 71–87. 10.1016/0022-1236(69)90022-6
Di Fazio G, Ragusa MA: Commutators and Morrey spaces. Boll. Unione Mat. Ital., A 1991, 5: 323–332.
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.
Mizuhara T: Boundedness of some classical operators on generalized Morrey spaces. Harmonic Analysis: Proceedings of a Conference Held in Sendai, Japan 1990, 183–189.
Acknowledgements
Project was supported by Scientific Research Fund of Hunan Provincial Education Departments (13C1007).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Lu, D. Weighted boundedness of multilinear operators associated to singular integral operators with non-smooth kernels. J Inequal Appl 2014, 276 (2014). https://doi.org/10.1186/1029-242X-2014-276
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-276