Boundedness of Marcinkiewicz Integrals on RBMO Spaces over Nonhomogeneous Metric Measure Spaces

Abstract

Let be a metric measures space satisfying the upper doubling conditions and the geometrically doubling conditions in the sense of Hytönen. Under the assumption that the dominating function satisfies the weak reverse doubling condition, the authors prove that Marcinkiewicz integral with kernel satisfying certain stronger Hörmander-type condition is bounded on RBMO space.

1. Introduction

We all know that the Littlewood-Paley function has been playing an important role in harmonic analysis and the Marcinkiewicz integral is an essential Littlewood-Paley function. As an analogy to Littlewood-Paley function without going into the interior of the unit disk, in 1938, Marcinkiewicz introduced the integral on one dimensional Euclidean space , which now is called the Marcinkiewicz integral, and conjectured that it is bounded on for (see [1]). In 1944, Zygmund proved the Marcinkiewicz conjecture by using the complex variable method in [2]. Particularly, in 1958, Stein introduced the higher dimensional Marcinkiewicz integral (see [3]). Let be homogeneous of degree zero in for and integrable and have mean value zero on the unit sphere . The higher dimensional Marcinkiewicz integral is defined by Recently, many papers focus on the boundedness of this operator on various function spaces. We refer the reader to see [4, 5].

Many results from real analysis and harmonic analysis on the classical Euclidean spaces have been extended to the space of homogeneous type by Coifman and Weiss in [6]. Recall that a metric space equipped with a Borel measure is called a space of homogeneous type, if satisfies the following doubling measure condition that there exists a positive constant such that, for all balls with and ,

Meanwhile, many classical results concerning the theory of Calderón-Zygmund operators and function spaces have been proved still valid for nondoubling measures. In particular, let be a nonnegative Radon measure on which only satisfies the polynomial growth condition that there exists a positive constant and such that, for all and , where . Such a measure need not satisfy the doubling condition (2). The analysis with nondoubling measures plays an important role in solving the longstanding open Painlevé problem by Tolsa in [7]. In 2007, Hu et al. introduced the Marcinkiewicz integral with nondoubling measures and got some excellent results in [8]. In 2012, Wang et al. established the boundedness of Marcinkiewicz integrals on in [9].

However, in 2010, Hytönen pointed out that the measures satisfying the polynomial growth condition are different from the doubling measures in [10]. Hytönen introduced a new class of metric measure spaces which satisfy the upper doubling condition and the geometrically doubling condition (resp., see Definitions 1 and 3 below). This new class of metric measure space is called the nonhomogeneous metric measure space, which includes both spaces of homogeneous type and metric spaces with the measures satisfying (3) as special cases. We refer the reader to the monograph [11] for several recent developments on harmonic analysis in this setting.

In this paper, we mainly discuss the boundedness of Marcinkiewicz integrals on with nonhomogeneous metric measure spaces. Now we recall some notations and definitions as follows.

Definition 1. A metric measure space is said to be upper doubling, if is Borel measure on and there exists a dominating function and a positive constant such that, for each , is nondecreasing and, for all and ,

Remark 2. Obviously, a space of homogeneous type is a special case of upper doubling spaces, where one can take the dominating function . Moreover, let be a nonnegative Radon measure on which only satisfies the Polynomial growth condition. By taking , we see that is also an upper doubling measure space.
It was proved that there exists a dominating function related to satisfying the property that there exists a positive constant such that , and, for all with , Based on this, in this paper, we always assume that the dominating function also satisfies (5).

Definition 3. A metric space is said to be geometrically doubling, if there exist some such that, for any ball , there exists a finite ball covering of such that the cardinality of this covering is at most .

Remark 4. Let be a metric space. Hytönen showed that the following statements are mutually equivalent (see [10]):(1) is geometrically doubling.(2)For any and ball , there exists a finite ball covering of such that the cardinality of this covering is at most . Here and in what follows, is as in Definition 3 and .(3)For every , any ball can contain at most centers of disjoint balls with radius .(4)There exists such that any ball can contain at most centers of disjoint balls .

Definition 5. For any two balls , define where above and in that follows, for a ball and , is the center of ball .

Remark 6. The following discrete version, , of defined in Definition 5 was first introduced by Bui and Duong in nonhomogeneous metric measure spaces (see [12]), which is more close to the quantity introduced by Tolsa [13] in the setting of nondoubling measures. For any two balls , let be defined by where and , respectively, denote the radius of the balls and and is the smallest integer satisfying . Obviously, . That was pointed by Bui and Duong in [12]; in general, it is not true that .

Definition 7. Let . A ball is called -doubling if .

It was proved in [10] that if a metric measure space is upper doubling and , then, for every ball , there exists some such that is doubling. Moreover, let be geometrically doubling, with , and is Borel measure on which is finite on bounded sets. In [10] Hytönen also showed that, for almost every , there exist arbitrarily small doubling balls centered at . Furthermore, the radius of these balls may be chosen to be form for and any preassigned number . Throughout this paper, for any and ball , denotes the smallest doubling ball of the form with , where

If , we denote the ball simply by .

Let be a -locally integrable function on . Assume that there exists a positive constant such that, for any with , and, for any ,

The Marcinkiewicz integral associated with the above kernel is defined by

Obviously, by taking , we see that, in the classical Euclidean space , if with homogeneous of degree zero and for some , then satisfies (5) and (8). In 2014, Lin and Yang [14] established the equivalent boundedness of Marcinkiewicz integral with kernel satisfying (9) and (10). In this note, we make some modification for the kernel. Besides satisfying the regular condition (9), also satisfies that, for any and with , there exists such that and, for any and any two positive real numbers with ,

When the kernel satisfies condition (13), it also satisfies (10).

Definition 8. Let . A function is said to be the space , if there exists a positive constant and a number for any ball such that, for all balls , and, for balls , Moreover, the norm of in is defined to be the minimal constant as above and denoted by .

Definition 9. A dominating function is said to satisfy the weak reverse doubling condition if, for all and , there exists a number , depending only on and , such that, for all , and, moreover,

Remark 10. If the dominating function satisfies the weak reverse doubling condition, then for any two balls; see [15].

Throughout this paper, we denote by a positive constant which is independent of the main parameters involved, but it may be different from line to line. For any , we denote by its conjugate index; namely, .

2. Main Result and Its Proof

We give the main result as follows.

Theorem 11. Let be a nonhomogeneous metric measure space with the dominating function satisfying the weak reverse doubling condition, let satisfy conditions (9), (13), and (14), and let be defined as (11). If is bounded on , then, for any , where is a positive constant independent of , only when is finite almost everywhere; otherwise, .

In order to prove the theorem, we need the following two lemmas. The following useful properties of were proved in [10, 16]; see also [17].

Lemma 12. (i) For all balls , .
(ii) For any , there exists a positive constant , depending on , such that, for all balls with , .
(iii) For any , there exists a positive constant , depending on , such that, for all balls , .
(iv) There exists a positive constant such that, for all balls , . In particular, if and are concentric, then .
(v) There exists a positive constant such that, for all balls , ; moreover, if and are concentric, then .

We also need the following characterization of ; see [16, Proposition 2.10] and [18, Lemma 3.2].

Lemma 13. Let , and let be as in (8). For , the following statements are equivalent:(i);(ii)there exists a nonnegative constant such that, for all doubling balls , and that, for all doubling balls , where and in what follows .

Moreover, the minimal constant as above is equivalent to .

Proof of Theorem 11. For , we have that is finite almost everywhere and decompose for any two balls .
By the vanishing (14), we have Furthermore, By applying the Minkowski inequality and vanishing condition (14), we have Then it is easy to get that, for any , which follows that So, we deduce Applying Hölder’s inequality, [10, Corollary 6.3], and boundedness of , we deduce that There exists a nonnegative integer such that Set ; then we get For any , by applying the Minkowski inequality and a similar argument to (29) and (9), we have that By using Definition 8 and Lemma 12, we can get Denote . Let us estimate : There are three cases about the relation of and .
Case I. Consider  ; then by using the vanishing condition (14), we have Case II. Consider  ; then by using the vanishing condition (14), we have Case III. Others. By applying the vanishing condition (14), we also get And, in this case, it is easy to get that .
From this and the Minkowski inequality, it follows that From (31), (32), and (38), we deduce that On the other hand, By applying a similar argument of (29), (32), and (38), we have From the above inequalities, it follows that and by (39), we deduce that Now let us estimate To prove (44), suppose Now we estimate , For , applying the Minkowski inequality and (9), we can deduce that For , by using a similar method of , we have Now let us estimate ; by applying (13), we obtain Combining the , , and , we know that (44) holds. And then we have Finally, we obtain that, for any two balls , Suppose that is a doubling ball; by (51) we have And when are two doubling balls, it can be deduced that Applying Lemma 13 and the above two inequalities, we have If such that is infinite on a set of positive measures, it is easy to prove that we take a ball such that namely, . So, .
We complete the proof of Theorem 11.

Conflict of Interests

The authors declare that they do not have any commercial or associative interest that represents a conflict of interests in connection with the work submitted.

Acknowledgments

This work is supported by National Science Foundation of China (Grant no. 11261055) and College Students’ Innovative Project of Xinjiang University (Grant no. 201410755008).