Abstract
Let for be a homogeneous function of degree zero and be BMO functions. In this paper, we obtain some boundedness of the parametric Marcinkiewicz integral operator and its higher-order commutator on Herz spaces with variable exponent.
1. Introduction
Function spaces with variable exponent are being concerned with strong interest not only in harmonic analysis but also in applied mathematics. In the past 27 years, the theory of function spaces with variable exponent has made great progress since some elementary properties were given by Kováčik and Rákosník [1] in 1991. In [2–6], the authors proved the boundedness of some integral operators on variable spaces, respectively. Lebesgue and Sobolev spaces with integrability exponent have been widely studied; see [3, 5] and the references therein. Many applications of these spaces were given, for example, in the modeling of electrorheological fluids, in the study of image processing, and in differential equations with nonstandard growth.
On the other hand, a class of function spaces called Herz-type spaces on has attracted considerable attention in recent years because the interesting norm includes explicitly both local and global information of the function. In 2011, Izuki [7] studied the Herz spaces with variable exponent and proved the boundedness of some sublinear operators on the spaces. In addition, Wang and Liu [8] introduced a certain Herz-type Hardy spaces with variable exponent in 2012.
Suppose that denotes the unit sphere in equipped with normalized Lebesgue measure. Let for be a homogeneous function of degree zero andwhere for any . In 1958, Stein [9] introduced the Marcinkiewicz integral related to the Littlewood-Paley function on as follows: where It was shown that is of type for and of weak type .
The parametric Marcinkiewicz integral is defined by where
Let and , the higher-order commutator generated by the parametric Marcinkiewicz integral and is defined by
Motivated by [10, 11], we will study the boundedness for the parametric Marcinkiewicz integral operator and its commutator on the Herz spaces with variable exponent, where for .
Throughout this paper, we denote the Lebesgue measure and the characteristic function of a measurable set by and , respectively. The notation means that there exist constants such that . Let and for . Denote and as the sets of all positive and nonnegative integers, for , if and . In addition, and are the same as in Lemma 4.
2. Preliminaries
Firstly we give some notation and basic definitions on variable Lebesgue spaces. Given an open set , and a measurable function . is the conjugate exponent defined by
Define to be set of such that The set consists of all satisfying
By we denote the space of all measurable functions on such that for some , This is a Banach function space with respect to the Luxemburg-Nakano norm
The space is defined by
Let ; the Hardy-Littlewood maximal operator is defined by where . The set consists of satisfying the condition that is bounded on .
In variable spaces there are some important lemmas as follows.
Lemma 1 (see [2]). If and satisfiesandthen ; that is, the Hardy-Littlewood maximal operator is bounded on .
Lemma 2 (see [1] generalized Hölder inequality). Let . If and , then is integrable on and where
Lemma 3 (see [7]). Suppose . Then there exists a constant such that for all balls in ,
Lemma 4 (see [7]). Let . Then there exists a positive constant such that for all balls in and all measurable subsets ,where are constants with .
Next we recall the definition of the Herz spaces with variable exponent.
Definition 5 (see [7]). Let , and . The homogeneous Herz space consists of all such that The nonhomogeneous Herz space is defined as the set of all such that
3. Boundedness of the Parametric Marcinkiewicz Integral Operator
In this section we will prove the boundedness of the parametric Marcinkiewicz integral operators on Herz spaces with variable exponent.
A nonnegative locally integrable function on is said to belong to , if where , denotes a cube in with its sides parallel to the coordinate axes, and denotes the Lebesgue measure of .
The weighted boundedness of has been proved by Shi and Jiang [12].
Lemma 6 (see [12]). Suppose that satisfying (1). If , then for any , there is a constant , independent of , such that
Lemma 7 (see [4]). Given a family and an open set , assume that for some , and for every , Given such that satisfies (13) and (14) in Lemma 1. Then for all such that ,
Since , by Lemmas 6 and 7 it is easy to get the , -boundedness of the parametric Marcinkiewicz integral operators .
To obtain the Theorem 11, we need the following lemmas.
Lemma 8 (see [13]). If and , then
Lemma 9 (see [14]). Define a variable exponent by Then we have for all measurable functions and .
Lemma 10 (see [5]). Let satisfy conditions (13) and (14) in Lemma 1. Then for every cube (or ball) , where .
Theorem 11. Suppose that , , satisfies conditions (13) and (14) in Lemma 1; and . Then is bounded on and .
Proof. We only prove homogeneous case. The nonhomogeneous case can be proved in the same way. We suppose , since the proof of the case is easier. Let . Denote for each , then we have . Then we haveWe first estimate , by the -boundedness of the commutator ; we haveNow we estimate . We consider Note that , and . So we know that , and by mean value theorem we haveBy (31), the Minkowski inequality, and the generalized Hölder inequality we have Similarly, we consider . Noting that , by the Minkowski inequality and the generalized Hölder inequality, we have So we have Noting , we denote and . By Lemmas 8 and 9 we have When and , by Lemma 10 we have When we have So we obtain
By Lemmas 3 and 4 we have Thus we obtain If , take . Since , by the Hölder inequality we haveIf , then we haveLet us now estimate . Note that , , and , so we have . We consider Similar to the estimate for , we get Similar to the estimate for , we get So we have By Lemmas 3 and 4 we have Thus we obtain If , take . Since , by the Hölder inequality we haveIf , then we haveThus by (28), (29) and (40), (41), (48), and (49) we complete the proof of Theorem 11.
4. BMO Estimate for the Higher-Order Commutator of Parametric Marcinkiewicz Integral Operator
Let us first recall that the space consists of all locally integrable functions such that where , the supremum is taken over all cubes with sides parallel to the coordinate axes, and denotes the Lebesgue measure of .
Let . The weighted boundedness of has been proved by Shi and Jiang [12].
Lemma 12 (see [12]). Suppose that satisfying (1). If and , then for any , there is a constant , independent of , such that
By Lemmas 12 and 7 it is easy to get the -boundedness of the commutator .
Next, we will give the corresponding result about the commutator on Herz spaces with variable exponent.
Theorem 13. Suppose that satisfies conditions (13) and (14) in Lemma 1; and . Then is bounded on and .
In the proof of Theorem 13, we also need the following lemma.
Lemma 14 (see [15]). Let , be a positive integer, and be a ball in . Then we have that, for all and all with , where and .
Proof of Theorem 13. Similar to Theorem 11, we only prove homogeneous case and still suppose . Let . Denote for each , then we have . ThenNoting is bounded on , so we haveNow we estimate . We consider Note that , and , and we know that . By (31), the Minkowski inequality, and the generalized Hölder inequality we have Similarly, we consider . Noting that , by the Minkowski inequality and the generalized Hölder inequality, we have So we have Noting , we denote and . By Lemmas 8 and 9 we have When and , by Lemma 10 we have When we have So we obtain
So we haveSimilarly, by Lemma 14 we haveBy (62), (63), Lemmas 3, 4, and 14 we have Thus we obtain If , take . Since , by the Hölder inequality we haveIf , then we haveLet us now estimate . Note that , , and , so we have . We consider Similar to the estimate for , we get Similar to the estimate for , we get So we have By (62), (63), Lemmas 3, 4, and 14 we have Thus we obtain If , take . Since , by the Hölder inequality we haveIf , then we haveThus by (53), (54) and (66), (67), (74), and (75) we complete the proof of Theorem 13.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by China Postdoctoral Science Foundation Funded Project (Grant no. 2016M601105), Shandong Provincial Natural Science Foundation, China (Grant no. ZR2017MA041), a Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J18KA225), and National Natural Science Foundation of China (Grant no. 11761026).