BMO Functions Generated by Weights on Ball Banach Function Spaces

Abstract

Let be a ball Banach function space on . We introduce the class of weights . Assuming that the Hardy-Littlewood maximal function is bounded on and , we obtain that . As a consequence, we have , where is the variable exponent Lebesgue space. As an application, if a linear operator is bounded on the weighted ball Banach function space for any , then the commutator is bounded on with .

1. Introduction

It is well known that there is a relation between weights and , i.e., for any ,

See, for instance, [1] (p. 409). The purpose of this note is to reveal the relation between and weights over the ball Banach function space .

To state our results, we begin with the definition of the ball Banach function space. Denote by the symbol the set of all measurable functions on . For any and , let and

Definition 1. A Banach space is called a ball Banach function space if it satisfies that (i) implies that almost everywhere(ii) almost everywhere implies that (iii) almost everywhere implies that (iv) implies that , where is as in (2);(v)for any , there exists a positive constant , depending on , such that, for any ,

For any ball Banach function space , the associate space (Köthe dual) is defined by setting where is called the associate norm of (see, for instance, [2] (Chapter 1, Definitions 2.1 and 2.3)).

Remark 2. By [3] (Proposition 2.3), we know that, ifis a ball Banach function space, then its associate spaceis also a ball Banach function space.

Now, we introduce the class of weights and recall the function space BMO. A weight is a locally integrable function such that almost everywhere .

Definition 3. Letbe a ball Banach function space. We say that a weightbelongs toifhere and hereafter is the characteristic function for .

Remark 4. (1)There is an immediate consequence. Let be a ball Banach function space. If , then (2)We recall that the definition of . Let . A weight belongs to if

By the definition of and , if and only if for any .

The classical function space is the collection of all locally integrable functions such that where the supremum is taking all balls in and is the mean value of the function on , namely,

By the well-known John-Nirenberg inequality, John and Nirenberg [4] proved that there exists a positive constant such that where and

We also recall that the Hardy-Littlewood maximal function is defined by setting, for any locally integrable function and ,

Now, we state our result as the following theorem.

Theorem 5. Letbe ball Banach function spaces. If the Hardy-Littlewood maximal functionis bounded onand, then

Remark 6. Let, Theorem5goes back to the classical result for.

As an example, let be the collection of all measurable functions . Then, the variable Lebesgue space is defined to be the set of all measurable functions on such that

Denote and . A measurable function is said to be globally log-Hölder continuous if there exists a such that, for any , where the implicit positive constants are independent of and .

Definition 7 ([5], Definition 1.4.). Given an exponent function and a weight, we say thatif there exists a constantsuch that for every ball, where for almost everywhere .

Remark 8. Letbe a globally log-Hölder continuous function satisfying. By [3] (Lemma 2.5 and Proposition 3.8.), for any ball, . This shows that for

Let be a globally log-Hölder continuous function satisfying . We know that is bounded on and its duality ; see, for instance, [6, 7] and their references.

Corollary 9. Letbe a globally log-Hölder continuous function satisfying. Then,.

2. Proof of Theorem 5

The following lemmas give two elementary properties of ball Banach function spaces, whose proof is similar to the one corresponding to Banach function spaces; see [2].

Lemma 10 (Holder’s inequality). Letbe a ball Banach function space with the associate space. Ifand, thenis integrable and

Lemma 11 (G. G. Lorentz, W. A. J. Luxembourg). Every ball Banach function spacecoincides with its second associate space. In other words, a functionbelongs toif and only if it belongs toand, in that case,

Under weak boundedness of the Hardy-Littlewood maximal function on , the norm enjoys the following property; see [8] (Lemma 2.2).

Lemma 12. Letbe a ball Banach function space and suppose that the Hardy-Littlewood maximal operatoris weakly bounded onor, that is, there exists a positive constantsuch thator holds for all and all . Then, there exists a positive constant such that for all balls , .

Remark 13. By Lemma10, we havefor any ball.

Lemma 14. Letandbe a ball Banach function space. Suppose that the Hardy-Littlewood maximal operatoris weakly bounded on. Then,if and only if there exists a positive constantsuch that for any ball

Proof. We first prove the sufficiency. In fact, by the definition of , we have Conversely, suppose that . Then by Lemmas 10 and 12Also,

The John-Nirenberg inequality for ball Banach function spaces was established by Izuki et al. ([9], Theorem 3.1).

Lemma 15. Letbe a ball Banach function space such thatis bounded onand write. Then, there exists a positive constantsuch that for all balls, and,

As a consequence of Lemma 15, we have the following inequality.

Lemma 16. Letbe a ball Banach function space. Suppose thatis bounded on. Suppose that. Then for anyand ball, we havewhere is as in Lemma 15.

Proof. By Lemma 15, we have

Lemma 17. Letbe a ball Banach function space. If, then.

Proof. Let . Then, . By Lemmas 10, 12, and 14, we obtain that

Proof of Theorem 18. By Lemma 17, for any and , . Conversely, suppose that . Since is bounded on , by Lemma 16, we know that there exist and such that, for any ball ,

Similarly, since is bounded on , by Lemmas 11 and 16, we know that there exist and such that, for any ball ,

Taking and and applying Lemma 14, we get the desired result.

3. Applications

In this section, we will show that the boundedness of the commutator of a linear operator on with the BMO function can be derived from the weighted boundedness of on . We first establish the following Minkowski-type inequality.

Lemma 19. Letbe a Banach function space anda measurable function on. If, for almost every, and, for almost every, , then

Proof. By Lemma 11, we have From the Fubini theorem and Lemma 10, it follows that which implies the desired conclusion. This finishes the proof of Lemma 19.

Let be a linear operator defined by

Given a symbol , we define the commutator .

Let be a weight. Define and . We say that is bounded on if there exists a positive constant such that for all ,

Theorem 20. Letbe a ball Banach function space. Suppose thatis bounded onand. If, for any, is bounded onthen, for all, is bounded on, i.e.,where is independent of .

Proof. We adapt the idea from [10, 11]. Without loss of generality, we assume that in . By Theorem 5, there exists a such that . As well known, for every , and . Thus, For any , define . Then, is analytic on and the Cauchy integral formula implies that For any , set . Since , we have By this and (38), we conclude that Applying Lemma 19 and the weighted boundedness of , we have We complete the proof of Theorem 20.

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 project is supported by the Scientific Research Fund of Young Teachers in Longqiao College (Grant No. LQKJ2020-01).