- Research
- Open access
- Published:
Boundedness of Littlewood-Paley operators and their commutators on Herz-Morrey spaces with variable exponent
Journal of Inequalities and Applications volume 2014, Article number: 227 (2014)
Abstract
The aim of this paper is to establish the vector-valued inequalities for Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley g-functions and -functions, and their commutators on the Herz-Morrey spaces with variable exponent . By applying the properties of spaces and the vector-valued inequalities for Littlewood-Paley operators and their commutators generated by BMO function on , the boundedness of the vector-valued Littlewood-Paley operators and their commutators is obtained on .
MSC:42B20, 42B25, 42B35.
1 Introduction
Let and satisfies
-
(i)
,
-
(ii)
,
-
(iii)
, ,
where C, ε, γ are all positive constants. Denote with and . Given a function , the Lusin area integral of f is defined by
where denote the usual cone of aperture one
As , we denote as .
Now let us turn to introduce the other two Littlewood-Paley operators. It is well known that the Littlewood-Paley operators include also the Littlewood-Paley g-functions and the Littlewood-Paley -functions besides the Lusin area integrals. The Littlewood-Paley g-functions, which can be viewed as a ‘zero-aperture’ version of , and -functions, which can be viewed as a ‘infinite-aperture’ version of , are defined, respectively, by
If we take ψ to be the poisson kernel, then the functions defined above are the classical Littlewood-Paley operators.
Let , , the corresponding m-order commutators of Littlewood-Paley operators above generated by a function b are defined by
and
The Littlewood-Paley operators are very important objects in the study of harmonic analysis. They play very important roles in harmonic analysis and PDE (see [1–3]), so it is natural and meaningful to consider the boundedness of Littlewood-Paley operators and their commutators. Lu and Yang investigated the behavior of Littlewood-Paley operators in the space in [4]. In 2005, Zhang and Liu proved the commutator is bounded on (see [5]). In 2009, Xue and Ding gave the weighted estimate for Littlewood-Paley operators and their commutators (see [6]). There are some other results about Littlewood-Paley operators in [7–9].
On the other hand, Lebesgue spaces with variable exponent become one of the important function spaces due to the fundamental paper [10] by Kováčik and Rákosník. In the past 20 years, the theory of these spaces has made progress rapidly, and the study of it has many applications in fluid dynamics, elasticity, calculus of variations and differential equations with non-standard growth conditions (see [11–15]). In [16], Cruz-Uribe et al. proved the extrapolation theorem which leads the boundedness of some classical operators including the commutator on . Karlovich and Lerner also independently obtained the boundedness of the singular integral commutators in [17]. Recently, Izuki considered the boundedness of vector-valued sub-linear operators and fractional integrals on Herz-Morrey spaces with variable exponent in [18] and [19], respectively.
Inspired by the above works, in this paper we will consider the vector-valued inequalities of the Littlewood-Paley operators and their m-order commutators on Herz-Morrey spaces with variable exponent. To do this, we need recall some definitions about the spaces with variable exponent.
Let E be a Lebesgue measurable set in with measure .
Definition 1.1 [10]
Let be a measurable function.
The Lebesgue space with variable exponent is defined by
The space is defined by
The Lebesgue space is a Banach space with the norm defined by
We denote
Then consists of all satisfying and .
Let M be the Hardy-Littlewood maximal operator. We denote to be the set of all functions satisfying the condition that M is bounded on .
Let , , , .
Definition 1.2 [18]
Let , , , and . The homogeneous Herz-Morrey spaces with variable exponent is defined by
where
Remark 1.1 Comparing the homogeneous Herz-Morrey spaces with variable exponent with the homogeneous Herz spaces with variable exponent (see [20]), where is defined by
Obviously, .
We now make some conventions. Throughout this paper, given a function f, we denote the mean value of f on E by . means the conjugate exponent of , namely holds. C always means a positive constant independent of the main parameters and may change from one occurrence to another.
2 Preliminary lemmas
In this section, we need some conclusions which will be used in the proofs of our main results.
Lemma 2.1 [10] (Generalized Hölder’s Inequality)
Let .
-
(1)
For any , ,
where .
-
(2)
For any , , when , we have
where .
Lemma 2.2 [17]
If , then there exist constants , such that for all balls and all measurable subsets ,
Lemma 2.3 [18]
If , then there exist constants , such that for all balls ,
Lemma 2.4 [20]
Let , m is a positive integer, there exist constants , such that for any with ,
-
(1)
;
-
(2)
.
Lemma 2.5 [21]
Given an open set , suppose that satisfies
Then , that is, the Hardy-Littlewood maximal operator is bounded on .
Let be the classical Muckenhoupt weighted class.
Lemma 2.6 [16]
-
(1)
Given a family ℱ and open set , assume that for some , and for every ,
Let be such that there exists , with . Then for every and sequence ,
-
(2)
If , then for all ,
Lemma 2.7 [6]
Let satisfies (i)-(iii), , , there exists a constant C and for all bounded functions f with compact support:
-
(1)
If , , then
-
(2)
If , , , then
Remark 2.1
-
(1)
If we replace with , then the results of Lemma 2.7 also hold. That is due to for any , , the inequalities and hold.
-
(2)
According to the argument in [6], we may deduce that Lemma 2.7 is also suit to .
Combining Lemmas 2.6-2.7 and Remark 2.1, we get the following conclusions.
Lemma 2.8 Let satisfies (i)-(iii). If , , then for all bounded compactly support functions such that , that is , the below vector-valued inequalities hold:
-
(1)
,
-
(2)
,
-
(3)
,
where , .
Lemma 2.9 Let , satisfies (i)-(iii), . If , , then for all bounded compactly support functions such that , that is , the following vector-valued inequalities hold:
-
(1)
,
-
(2)
,
-
(3)
,
where , .
3 Main results and their proofs
In this section, we will establish the vector-valued inequalities of the Littlewood-Paley operators, , , , and their m-order commutators on Herz-Morrey spaces with variable exponent, respectively.
We begin with the study of the vector-valued inequalities of the Littlewood-Paley operators on .
Theorem 3.1 Suppose that satisfies (i)-(iii), . Let , , , where , is the constant in Lemma 2.2. Then for all function sequences , the following vector-valued inequalities hold:
-
(1)
,
-
(2)
,
-
(3)
,
where , , and the constant C is independent of .
Proof Firstly, we consider the inequality (1). For any function sequence satisfies , we write
Thus
For the term , noting that , we can easily obtain, by Lemma 2.8,
We now turn to estimate . Let , . We write
For I, observe that as , , , , then we have , and . Hence, it follows from the condition (ii) that
For the estimate of II, denote
Noting that , by the condition (iii), we have
Since , then we obtain
and
Thus, we have
On the other hand, by the condition (ii) and , we deduce
Thus, similar to , we get
Combining the estimates of I, II, we obtain
Therefore, by using the above inequality and Minkowski’s inequality, we have
It follows from Hölder’s inequality and Lemmas 2.2-2.3 that
Noting that , thus, when , we get
Finally, let us to estimate . By using Minkowski’s inequality, we obtain
Let , , , . Then we have . Hence, it follows from the condition (ii) that
If , then . Applying the condition (ii), we obtain
Thus
Therefore, similar to , as , by applying Minkowski’s inequality, Lemmas 2.1-2.3, and , we have
Adding up the results of , , , we have
That is, the inequality (1) in Theorem 3.1 holds.
Next we show the other two vector-valued inequalities also hold.
We consider first. Similar to , via calculation, we shall get the following conclusions (also see [22]):
-
(1)
If , , , then ;
-
(2)
If , , , then .
Thus, with a similar argument in the proof of the inequality (1) in Theorem 3.1, by Lemma 2.8 and the above two conclusions, we shall get the following inequality immediately:
For , by the definition of and , we have
By the above estimate of , we know that, as , , ,
Thus, when , we have
On the other hand, as , , , we have
Furthermore, when , we obtain
Hence, similar to the proof of the inequality (1) too, by Lemma 2.8 and the above two conclusions about , we get the following inequality immediately:
This completes the proof of Theorem 3.1. □
Now, let us to establish the vector-valued inequalities of the commutators generated by the Littlewood-Paley operators with BMO functions on .
Theorem 3.2 Suppose that satisfies (i)-(iii), , , . Let , , , where , is the constant in Lemma 2.2. Then for all function sequence , the following vector-valued inequalities hold:
-
(1)
,
-
(2)
,
-
(3)
,
where , , and the constant C is independent of .
Proof Firstly, we consider the inequality (1). Let . For any function sequence satisfies . We write
Thus,
We are now going to estimate each term, respectively. For the term , noting that , we can easily obtain by Lemma 2.9
For the term , observe that as , , . With the same argument as in the estimate of , we have
Thus,
Hence, by Minkowski’s inequality, we get
Furthermore, by Hölder’s inequality and Lemmas 2.2-2.4, we obtain
Therefore, as , noting that
then we have
Now we turn to estimate . Observe that as , , . With the same argument as in the estimate of , then we have
Thus,
Hence, by Minkowski’s inequality, we get
Furthermore, by Hölder’s inequality and Lemmas 2.2-2.4, we obtain
Therefore, similar to , as , noting that
then we have
Adding up the results of , , , we have
That is, the inequality (1) in Theorem 3.2 holds.
For the proofs about the vector-valued inequalities of and , with a similar arguments in the proof of the vector-valued inequality (1) in Theorem 3.2, by Lemma 2.9 and the estimates of , and in Theorem 3.1, it is not difficult to deduce
and, if and ,
We complete the proof of Theorem 3.2. □
Remark 3.1 By Remark 1.1, we can easily to see that the results in Theorems 3.1-3.2 are also suitable for Herz spaces with variable exponent .
References
Stein E: The development of square functions in the work of A. Zygmund. Bull. Am. Math. Soc. (N.S.) 1982, 7: 359-376. 10.1090/S0273-0979-1982-15040-6
Kenig C: Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. Am. Math. Soc., Providence; 1994.
Chang S, Wilson J, Wolff T: Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv. 1985, 60: 217-246. 10.1007/BF02567411
Lu SZ, Yang DC: The central BMO spaces and Littlewood-Paley operators. Approx. Theory Appl. 1995, 11: 72-94.
Zhang MJ, Liu LZ: Sharp weighted inequality for multilinear commutator of the Littlewood-Paley operator. Acta Math. Vietnam. 2005,30(2):181-189.
Xue QY, Ding Y: Weighted estimates for multilinear commutators of the Littlewood-Paley operators. Sci. China Ser. A 2009,39(3):315-332.
Wei XM, Tao SP: Boundedness for parametrized Littlewood-Paley operators with rough kernels on weighted weak Hardy spaces. Abstr. Appl. Anal. 2013. Article ID 651941, 2013: Article ID 651941
Wei XM, Tao SP:The boundedness of Littlewood-Paley operators with rough kernels on weighted spaces. Anal. Theory Appl. 2013,29(2):135-148.
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
Kováčik O, Rákosník J:On spaces and . Czechoslov. Math. J. 1991,41(116):592-618.
Acerbi E, Mingione G: Gradient estimates for a class of parabolic systems. Duke Math. J. 2007, 136: 285-320. 10.1215/S0012-7094-07-13623-8
Acerbi E, Mingione G: Regularity results for stationary electrorheological fluids. Arch. Ration. Mech. Anal. 2002, 164: 213-259. 10.1007/s00205-002-0208-7
Diening L, Ružička M:Calderón-Zygmund operators on generalized Lebesgue spaces and problems related to fluid dynamics. J. Reine Angew. Math. 2003, 563: 197-220.
Ružička M Lecture Notes in Math. 1748. In Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin; 2000.
Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR, Ser. Mat. 1986 (in Russian), 50: 675-710. (in Russian)
Cruz-Uribe D, Fiorenza A, Martell JM, Pérez C:The boundedness of classical operators on variable spaces. Ann. Acad. Sci. Fenn., Math. 2006, 31: 239-264.
Karlovich AY, Lerner AK:Commutators of singular integrals on generalized spaces with variable exponent. Publ. Mat. 2005, 49: 111-125.
Izuki M: Fractional integrals on Herz-Morrey spaces with variable exponent. Hiroshima Math. J. 2010, 40: 343-355.
Izuki M: Boundedness of vector-valued sublinear operators on Herz-Morrey spaces with variable exponent. Math. Sci. Res. J. 2009, 13: 243-253.
Izuki M: Boundedness of commutators on Herz spaces with variable exponent. Rend. Circ. Mat. Palermo 2010,59(2):199-213. 10.1007/s12215-010-0015-1
Nekvinda A:Hardy-Littlewood maximal operator on . Math. Inequal. Appl. 2004, 7: 255-265.
Lu SZ, Yang DC, Hu GE: Herz Type Spaces and Their Applications. Science Press, Beijing; 2008:131-132.
Acknowledgements
Shuangping Tao is supported by National Natural Foundation of China (Grant No. 11161042 and 11071250).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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
Wang, L., Tao, S. Boundedness of Littlewood-Paley operators and their commutators on Herz-Morrey spaces with variable exponent. J Inequal Appl 2014, 227 (2014). https://doi.org/10.1186/1029-242X-2014-227
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-227