Journal of Applied Mathematics and Physics
Vol.08 No.01(2020), Article ID:97568,17 pages
10.4236/jamp.2020.81005
Iterated Commutators for Multilinear Singular Integral Operators on Morrey Space with Non-Doubling Measures
Tie Li, Yinsheng Jiang, Yaoming Niu
Baotou Teacher’s College of Inner Mongolia University of Science and Technology, Baotou, China
Copyright © 2020 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: November 26, 2019; Accepted: December 28, 2019; Published: December 31, 2019
ABSTRACT
Let be a non-negative Radon measure on which only satisfies the following growth condition that there exists a positive constant C such that for all , and some fixed . This paper is interested in the properties of the iterated commutators of multilinear singular integral operators on Morrey spaces . Precisely speaking, we show that the iterated commutators generated by multilinear singular integrals operators are bounded from to where (Regular Bounded Mean Oscillation space) and with and .
Keywords:
Non-Doubling Measures, Morrey Space, Multilinear Singular Integral Operators, RBMO, Commutator
1. Introduction
Let be a positive Radon measures on satisfying only the growth condition, that is, there exists a constant and such that
(1)
for any cube with sides parallel to the coordinate axes. will be the cube centered at x with side length . For , rQ will denote the cube with the same center as Q and with . The set of all cubes , satisfying is denoted by . In this note, we do not assume that is doubling.
Nazarov, Treil and Volberg developed the theory of the singular integrals for the measures with growth condition to investigate the analytic capacity on the complex plane [1] [2]. Tolsa showed that the analytic capacity is subadditive and that it is bi-Lipschitz invariant [3] [4] and defined for the growth measures RBMO (regular bounded mean oscillation) space, the Hardy space and the Littlewood-Paley decomposition [5] [6]. He also gave his space in terms of the grand maximal operator [7]. Recently many people paid attention to the measure with growth condition because of recovering the Calderón-Zygmund theory and solving the long-standing open Painlevé problem.
The boundedness of fractional integral operators on the classical Morrey spaces was studied by Adams [8], Chiarenza et al. [9]. In [9], by establishing a pointwise estimate of fractional integrals in terms of the Hardy-Littlewood maximal function, they showed the boundedness of fractional integral operators on the Morrey spaces. In 2005, Sawano and Tanaka [10] gave a natural definition of Morrey spaces for Radon measures which might be non-doubling but satisfied the growth condition, and they investigated the boundedness in these spaces of some classical operators in harmonic analysis. Later on, Sawano [11] defined the generalized Morrey spaces on for non-doubling measure and showed the properties of maximal operators, fractional integral operators and singular operators in this space.
A classical result of commutator is due to Coifman, Rochberg and Weiss [12], if and T is a Calderón-Zygmund operator, then the commutator is bounded on spaces for . The same result for the multilinear commutator was obtained by Pérez and Trujillo-Gonzalez [13]. Tolsa [5] developed the theory of Calderón-Zygmund operators and their commutators with RBMO functions in the setting of non-doubling measures. Hu, Meng and Yang [14] considered the multilinear commutator on Lebesgue spaces with non-doubling measures. Chen and Sawyer [15] modified the definition of RBMO to investigate the commutators of the potential operators and RBMO functions.
In the last decade, multilinear singular integrals of Calderón-Zygmund type have attracted great attentions. Some interesting results refer to [16] [17] [18] [19] [20] in the text of Lebesgue measures. It points out that Perez and Pradolini [21] introduced a said iterated commutators generated by the multilinear singular integral operators with Calderón-Zygmund type and vector function and obtained the boundedness from to with for (in fact, they considered the weighted case). Xu [22] extended the result to the case of the non-doubling measures. Very recently, Tao and He [23] obtained the boundedness of the multilinear Calderón-Zygmund operators on the generalized Morrey spaces over the quasi-metric space of non-homogeneoustype. The aim of this paper is to study the iterated commutators of multilinear singular integral operators on Morrey spaces with non-doubling measures.
Before stating our result, we recall some definitions and notation. Given large enough but depending only on the dimension d, we say that a cube is doubling if . For any fixed cube , let N be the smallest nonnegative integer such that is doubling. We denote this cube by .
For two cubes in , we suppose
(2)
where is the first positive integer k such that . This was introduced by Tolsa in [5].
Let be the mean value of f on Q, namely, . The regularity bounded mean oscillations function spaces were introduced by Tolsa [5].
Definition 1.1. Let be a fixed constant. We say that is in RBMO if there exists a constant such that
(3)
for any cube Q, and
(4)
for any two doubling cubes . The mininal constant is the norm of f, and it will be denoted by .
The definition of the Morrey space with non-doubling measure is given in the following [10].
Definition 1.2. Let and , the Morrey space is defined as
where
(5)
It is easy to observe that , and Hölder’s inequality tells us for all , then we have . The space is a Banach space with its norm and the parameter appearing in the definition does not affect it. The Morrey space norm reflects local regularity of f more precisely than the Lebesgue space norm. See [10] [11] [24] for details. We will denote by .
Denoting by , we consider the multilinear singular integral operator as follows,
(6)
whenever are -functions with compact support and . Moreover,
(7)
and, for some ,
(8)
provided that and .
Let for and let , then the iterated commutator is formally defined as
(9)
Suppose the main result in this paper can be stated as follow.
Theorem 1.1. Let as in (9) and satisfying conditions (7) and (8). Let with and . Suppose for . If maps to , then the commutators are bounded from to , that is,
(10)
More generally, denote by the family of all subsets of i different elements of , and let and . For any , we define
(11)
In case , one sees that is just the commutator . So we have a more generalization version of the theorem as following.
Theorem 1.2. Let as in (11) and satisfying conditions (7) and (8). Let with and . Suppose for . If maps to , then for all , the commutators are bounded from to , that is,
(12)
2. Proof of Main Results
Before proving our theorem, we recall the following maximal operator,
(13)
we will use the sharp maximal estimates. Let f be a function in , the sharp maximal function of f is defined by
(14)
The non-centered doubling maximal operator is defined by
(15)
By the Lebesgue differential theorem, it is easy to see that for any and . Define the non-centered maximal operator,
(16)
for and , where the supremum is taking over all the cubes Q containing the point x.
To prove Theorem 1.2 is reduced to the following lemmas.
Lemma 2.1. Let and and . If maps to and satisfying conditions (7) and (8). Then we have
(17)
We postpone the proof of Lemma 2.1 after of Theorem 1.2.
Lemma 2.2. [10] Let , and , then the operator is bounded on and
with the constant C independent of f.
Lemma 2.3. [24] Suppose that , and there exists an increasing sequence of concentric doubling cubes, , such that
Then there exist a constant independent on f such that
Lemma 2.4. [25] Let and . If maps to and satisfying conditions (7) and (8). Then there exists a constant C independent of such that
Remark 2.1. If maps to and satisfying conditions (7) and (8), with , and . From Corollary 1.8 in [23], we can easily get that is bounded from to .
Proof of Theorem 1.2. Using Lemma 2.1, Lemma 2.2, Lemma 2.3 and Lemma 2.4, we get that
Applying the inequality (17) in Lemma 2.1, for , we have
where and are two nonempty subsets of and . Hence, we can make use of induction on to get that
(18)
This completes the proof of Theorem 1.2.☐
For simplicity of the notation, we only show the special case of Lemma 2.1. The similar process with minor modification will be to able prove Lemma 2.1 for the general case.
Lemma 2.5. Let and . If maps to and satisfying conditions (7) and (8), then there exists a constant independent of and such that
(19)
In order to prove Lemma 2.5, we have the following decomposition for the commutators . For any , writing , thus it is clear that . Moreover,
(20)
By expanding and
hence we can obtain from the equality (20) that
(21)
where are constants depending only on m and i.
Proof of Lemma 2.5. For simplicity, we denote by the quantities on the right hand side of the inequality (19). Recall the definition of the sharp maximal operator , and use the standard technique, see [15] for example, we only need to prove that
(22)
and
(23)
with the absolute constant C independent of and R, where R is any doubling cube with . In fact, we take
(24)
and clearly
(25)
Recall the equality (21), for any , we have that
(26)
In order to show the inequality (22), we will calculate the integrals for the three functions above, respectively. Firstly, for , by the Hölder inequality one sees that
(27)
where we have choose such that .
Similarly, for , by the Hölder inequality, we also deduce that
(28)
To estimate the integral related to the function , we split as , where and , this yields
(29)
where each term in satisfies that , for some and some . So we can decompose the function further into three parts as follows
(30)
For ,we can take such that and . Let for each , then . Using Lemma 2.4, we know that the is bounded from to . Hence, by this boundedness and Hölder inequality, we have
(31)
In order to estimate the integral of terms and over Q, we will give their point-wise estimates. In fact, for , since we observe that
(32)
where we have used the fact (see [5] ) that, there is an absolute constant C such that, for any , integer and cubes Q,
(33)
On the other hand, for , we note for any that
(34)
where we have use the inequation (33) again.
Taking the mean over , we can obtain that
(35)
Combing the inequalities (26) (27), (28), (30), (31) and (33), we see from the estimates of and that the desired inequality (22) holds.
Next we turn to estimate the inequality (23). For any cubes with , where R is doubling. We denote by N, then and . We recall the equality (29) and let and , and let . Then we can write
(36)
For the term , noting
(37)
and the similar argument as that for the estimate of , we can obtain that
(38)
To estimate , we recall the notations and note that, for any sequences and ,
(39)
Using this equality and expanding , we observe that
(40)
Similarly,
(41)
Thus
(42)
To estimate the integrals above, we recall that and let , then we can write that and , and thus we have
(43)
where are constant independent of and Q. From the equality (43), we can deduce that
(44)
where
and
Along the same lines as that of the pointwise estimates of ,we can obtain that, for and if ,
(45)
Let if ; and
thus for ,
Hence we get from (45) that, for ,
(46)
where we have used the fact that the cubes R and are comparable, which implies and . Using the inequality (46) above and the identity (44), we obtain that, for ,
The estimates of and is very similar to the one used in the estimate of . In fact, repeating the similar procedures used in (45) and (46) for , and noting that since by the definition of N, we can deduce that
(47)
It is left to estimate the term in of the case . A small modification is needed to estimate this term. For and , one sees
This and the inequality (47) follows
Moreover, combing the estimates of and , we obtain the desired inequality (23).
Finally, let us show how to acquire the inequality (19) from the two inequalities (22) and (23). Fix the point x and let Q be any cube that . notice , hence we see from the inequalities (22) and (23) that
(48)
On the other hand for all doubling cubes with such that , where is the constant in Lemma 6 in [15], using (23), we have
(49)
and moreover the inequality (49) holds for any doubling cubes Q, R with . Therefore,
(50)
According to the estimates (48) (50) and the definition of the sharp maximal function, we deduce the inequality (19) and so finish the proof of the Lemma 2.5.☐
3. Conclusions
The proof of Lemma 2.5 can be slightly modified to prove the conclusion of Lemma 2.1. Therefore we show that the iterated commutators generated by multilinear singular integrals operators are bounded from
to . Suppose , the detailed conclusion can be described as follows: Let as in (9) and satisfying conditions (7)
and (8). Let with and . Suppose for . If maps to , then the commutators are bounded from to , that is,
More generally, let as in (11) and satisfying conditions (7) and (8). Let with and . Suppose for . If maps to , then for all , the commutators are bounded from to , that is,
Funding
Supported by the National Natural Science Foundation of China (Grant No. 11161044 and Grant No. 11661061) and Natural science foundation of Inner Mongolia (No. 2019MS01003).
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Li, T., Jiang, Y.S. and Niu, Y.M. (2020) Iterated Commutators for Multilinear Singular Integral Operators on Morrey Space with Non-Doubling Measures. Journal of Applied Mathematics and Physics, 8, 53-69. https://doi.org/10.4236/jamp.2020.81005
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