Abstract
In this paper, we obtain the weighted endpoint estimates for the commutators of the singular integral operators with the functions and the associated maximal operators on Orlicz-Morrey Spaces. We also get the similar results for the commutators of the fractional integral operators with the functions and the associated maximal operators.
1. Introduction and Main Results
The Morrey spaces were introduced by Morrey in [1] to investigate the local behavior of solutions to second-order elliptic partial differential equations. Chiarenza and Frasca [2] showed the boundedness of the Hardy-Littlewood maximal operator, singular integral operators, and the fractional integral operators on the Morrey spaces. Komori and Shirai [3] introduced the weighted Morrey spaces and proved that, for and , and are bounded on , and if and , then for all and any cube ,
In this paper, we obtain the weighted endpoint estimates for the commutators of the singular integral operators with functions and associated maximal operators. We also obtain the similar results for the commutators of the fractional integral operators with functions and associated maximal operators.
Let be a measurable function on and , , for two weights and , and the weighted Morrey space is defined by whereand the supremum is taken over all cubes in . When , we write as .
We say that is a singular integral operator if there exists a function which satisfies the following conditions:
The space is defined by where .
For the singular integral operator and , the commutator is defined by
In order to state our results, we need to recall some notations and facts about the Young functions and Orlicz spaces; for further information, see [4]. A function is a Young function if it is convex and increasing, and if and as .
Let be a Young function, and two weights and , and the weighted Orlicz-Morrey Class is defined as where When , we write as .
Given a locally integrable function and a Young function , define the mean Luxemburg norm of on a cube by
For , and a Young function , we define Orlicz maximal operator
If , we write simply as . If and , is the Hardy-Littlewood maximal operator . If , we write simply as .
If and , is the fractional maximal operator of order and we write it as . If , we write simply as .
Take , which means for a.e. .
Given , for an appropriate function on , the fractional integral operator (or the Riesz potential) of order is defined by
For , we define the commutators of the operator and by
The following theorems are our main results.
Theorem 1. Let and , then there exists a positive constant such that, for any cube and any ,
Theorem 2. Let be any singular integral operator, , , and . Then there exists a positive constant such that, for any cube and any ,
Theorem 3. Let , , , , , , and . Then there exists a positive constant such that, for any cube and any ,
Theorem 4. Let , , , , , , , and . Then there exists a positive constant such that, for any cube and any ,
2. Proof of Theorems 1 and 2
Lemma 5 (see [5]). Let , then there exists a positive constant such that, for any weight and all , for every locally integrable function .
Lemma 6 (see [6]). Let , then there exist a constant and such that, for any cube and a measurable subset ,
Proof of Theorem 1. Fix a cube centered at . By Lemma 5, we have To estimate term I, since , we have For term II, observe that, for , and . We have Therefore we obtain Since , using Lemma 6, we get This ends the proof.
Lemma 7 (see [7]). Let be any Calderón-Zygmund singular integral operator, , , and . Then there exists a positive constant such that, for all weights ,
Lemma 8 (see [6]). Let , then there exist a constant and such that, for any cube ,
Proof of Theorem 2. Fix a cube centered at . By Lemma 7, we have To estimate term I, since , it is easy to prove that , , and we have For term II, observe that, for , , is a cube, and , by Lemma 8, for any , we have Noticing the definition of the maximal function , we obtain By Lemma 6, we get in which we take small enough such that . This ends the proof.
3. Proof of Theorems 3 and 4
Given an increasing function , as in [8], we define the function by
If is submultiplicative, then . Also, for all , .
In this section, we set , it is submultiplicative, and so . Let , and be a number . Denote SoThe function is invertible with
Lemma 9 (see [8]). If is decreasing, then, for any positive sequence ,
Lemma 10. Let , . Then there exists a constant such that, for any , for any weight , we have
Proof. By homogeneity, we may assume that . Define the set It is easy to see that is open and we may assume that it is not empty. To estimate the size of , it is enough to estimate the size of every compact set contained in . We can cover by a finite family of cubes for which Using Vitali’s covering lemma, we can extract a subfamily of disjoint cubes such that For each , by homogeneity and the properties of the norm , we have For each , we have It is easy to see that is decreasing; by Lemma 9, we have This ends the proof.
Proof of Theorem 3. Fix a cube centered at . By Lemma 10, we have Now we estimate term I. Noticing that, for , we have Since , we get For term II, observe that, for , and . As in the proof of Theorem 1, we have Since , is submultiplicative, and using Lemma 6, we get This ends the proof.
Lemma 11 (see [9]). Let , , , and . Then there exists a constant such that, for any ,
Lemma 12 (see [6]). Let , , and , then .
Proof of Theorem 4. Fix a cube centered at , for any and , and by Lemma 12, we have . By Lemma 11, we obtain Now we estimate term I. Noticing that , Lemma 8, we have . Then For term II, as the proof of Theorem 2, for , By Lemma 6, we get in which we take small enough such that and . This ends the proof.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This paper is supported by the Natural Science Foundation of Education Department of Hebei Province (No. Z2014031).