Boundedness for Commutators of Bilinear -Type Calderón-Zygmund Operators on Nonhomogeneous Metric Measure Spaces

Abstract

Let be a nonhomogeneous metric measure space. In this paper, the boundedness for commutators generated by bilinear -type Calderón-Zygmund operators and functions on is obtained.

1. Introduction

As we know, Hytönen [1] introduced nonhomogeneous metric measure spaces, which include both spaces of homogeneous type and nondoubling measure spaces as special cases. From then on, many results on nonhomogeneous metric measure spaces are obtained by many researchers. Hytönen et al. [2] and Bui and Duong [3] introduced independently the atomic Hardy space and proved that the dual space of is . The authors in [3] also proved that Calderón-Zygmund operator and commutators are bounded in for . Recently, some equivalent characterizations were established by Liu et al. [4] for the boundedness of Calderón-Zygmund operators on for . Fu et al. [5, 6] established boundedness of multilinear commutators of Calderón-Zygmund operators and generalized fractional integrals on Orlicz spaces, respectively. For more results, one can refer to [2, 4, 7–15] and the references therein.

-type Calderón-Zygmund operator was firstly introduced by Yabuta [16] in 1985. Later, many researchers further studied the properties of this operator. We [17] obtained the boundedness of -type Calderón-Zygmund operator and commutators on nondoubling measure spaces. Ri et al. [18, 19] researched the boundedness of -type Calderón-Zygmund operator on Hardy spaces with nondoubling measures and nonhomogeneous metric measure spaces, respectively. Zheng et al. [20, 21] studied the bounded properties for bilinear -type Calderón-Zygmund operator and maximal bilinear -type Calderón-Zygmund operator on nonhomogeneous metric measure spaces, respectively.

In this paper, we prepare to study the boundedness for commutators generated by bilinear -type Calderón-Zygmund operators and functions on nonhomogeneous metric measure spaces. And we obtain that these commutators are bounded on Lebesgue spaces, provided that bilinear -type Calderón-Zygmund operator is bounded from to , where and denote the Lebesgue spaces and weak Lebesgue spaces, respectively. This result includes the corresponding results on both spaces of homogeneous type and nondoubling measure spaces. It is even new in the settings of spaces of homogeneous type and nondoubling measure spaces.

Throughout this paper, denotes the set of all functions with compact support. always denotes a positive constant independent of the main parameters involved, but it may vary from line to line. And is the conjugate index of ; namely, Now, let us recall some definitions and terminologies.

Definition 1 (see [1]). A metric space is geometrically doubling if there exists some such that, for every ball , there exists a finite ball covering of such that the cardinality of this covering is at most .

Definition 2 (see [1]). A metric measure space is upper doubling if is a Borel measure on and there exists a function and a constant such that, for every , is nondecreasing, and for any , ,

Remark 3. (i) Spaces of homogeneous type are upper doubling space, if we take . Also, nondoubling measure space, which satisfies the following polynomial growth condition:for all and , is also upper doubling measure space if we take .
(ii) The authors [11] showed that there exists another function such that, for any , ,Thus, one assumes that always satisfies (3) in this paper. As the singularity of commutators is stronger than that of bilinear operators, by [22], we suppose that there exists , such that, for any , ,

Let ; a ball is -doubling if . As pointed in Lemma 2.3 of [3], there exist plenty of doubling balls with small radii and with large radii. In this paper, unless and are specified otherwise, one means doubling ball is -doubling with a fixed number , where is the geometric dimension of the space.

Definition 4 (see [3]). For two balls , let be the smallest integer such that ; we denote

Let be a nonnegative nondecreasing function on satisfying

Definition 5. A kernel is called the bilinear -type Calderón-Zygmund kernel if it satisfies the following:(i)for all with for .(ii)provided that .
A bilinear operator is called bilinear -type Calderón-Zygmund operator with the above kernel if for and ,

Remark 6. As , (ii) in Definition 5 is equivalent to (ii′) in the following statement:(ii′)provided that .

Remark 7. (i) In [20, 21], the term in (7) and (8) of this paper is substituted by . In fact, as and , Definition 5 in this paper is equivalent to Definition  1.4 in [20] or Definition  1.3 in [21]. Therefore, we can directly quote the result of Theorem  1.5 in [20] as Lemma 17 below in this paper.
(ii) Because we assume that is bounded from to , it is enough to assume that satisfies the regularity condition on the first variable, that is, (8) in this paper for getting the result of Theorem 10 below. For more details, one can refer to Remark  1.1 in [9].

Definition 8. The commutator generated by bilinear -type Calderón-Zygmund operator and is defined byAlso, and are defined as follows, respectively:

Definition 9 (see [2]). Let be some fixed constant. A function is said to belong to if there exists a constant such that, for any ball ,and for any two doubling balls ,where is the smallest -doubling ball of the form with , and is the mean value of on : namely, The minimal constant in (15) and (16) is the norm of , which is denoted by .

Theorem 10. Let , , . Assume that , with , , , if . If is bounded from to , then there exists a constant such that

Remark 11. The result of Theorem 10 is still valid for commutators of multilinear -type Calderón-Zygmund operators with functions.

2. Preliminaries

For any , the noncentered doubling maximal operator is defined by and the sharp maximal operator is denoted by where and are doubling balls.

For any , denote

Let , , and ; the noncentered maximal operator is defined by When , we simply write as .

Lemma 12 (see [3, 11]). (i) For any and – a.e. , (ii) If , then the operator is bounded on for and is bounded on for .

Lemma 13 (see [3, 5]). Assume that with if . For and , if , then there exists a constant such that

Lemma 14 (see [5, 23]). Suppose that and . Then if and only if for any ball ,and for any two doubling balls ,

Lemma 15 (see [5]). For any ,

Lemma 16 (Kolmogorov’s theorem). Let be a probability measure space and let ; then there exists a constant , such that for any measurable function .

Lemma 17 (see [20]). Let , , , and . If is bounded from to , then there exists a constant such that

3. Proof of Main Result

Lemma 18. Suppose that , , and . If   is bounded from to , then there exists a constant such that, for any , , and ,

Proof. Because is dense in for , we only consider the situation of . Also, by Lemma  3.11 in [5], we can assume that . As it has the similar method to estimate (29), (30), and (31), here we only estimate (29) for complicity.
To obtain (29), with the similar way to prove Theorem  9.1 in [24], it suffices to show thatholds for any , andfor any balls with , where is an arbitrary ball and is a doubling ball. Denote AsthusFor . Let such that . By Hölder’s inequality and Lemma 14, Let us estimate , let such that , and then Similar to estimate , Let us turn to estimate . Let , for ; then For , let and such that . Using Kolmogorov’s theorem, Hölder’s inequality, Lemma 14, and the boundedness from to of , To estimate , by Definition 5, Lemmas 14 and 15, Hölder’s inequality, and some properties of , Similarly, we obtain For , by (ii) of Definition 5 and some properties of , Let us estimate . With the help of the fact thatby Lemmas 14 and 15 and Hölder’s inequality, we haveWith the same method to estimate , Thus, taking the mean over , we have So (32) can be obtained.
Next we prove (33). Denote ; thenUsing the method to estimate ,Let us estimate . As thenLet us estimate . Note that ; write Let us estimate firstly. Since is a doubling ball, we have Thus,For , let and such that ; denote . Note that is a doubling ball. By Lemmas 16 and 14 and Hölder’s inequality, For , by (i) of Definition 5, Lemma 14, and Hölder’s inequality, Thus, Also, we have By (26) in Lemma 14, and also have similar estimate of . Therefore, Using the similar method to estimate ,Hence, (33) is proved. Thus, the result of Lemma 18 is proved.

Proof of Theorem 10. Because the proof of the result of is similar to that of , now we only prove the result of . Let , , , , , , and . By Lemmas 12–14 and 17 and Hölder’s inequality, then Thus, the proof of Theorem 10 is finished.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by Natural Science Foundation of Anhui Province (no. 1608085QA12) and Natural Science Foundation of Education Committee of Anhui Province (nos. KJ2015A117 and KJ2016A506).