Estimates for Parameter Littlewood-Paley Functions on Nonhomogeneous Metric Measure Spaces

Abstract

Let be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measure conditions. In this paper, the authors prove that, under the assumption that the kernel of satisfies a certain Hörmander-type condition, is bounded from Lebesgue spaces to Lebesgue spaces for and is bounded from into . As a corollary, is bounded on for . In addition, the authors also obtain that is bounded from the atomic Hardy space into the Lebesgue space .

1. Introduction

In 1958, Stein in [1] firstly introduced the Littlewood-Paley operators of the higher-dimensional case; meanwhile, the author also obtained the boundedness of the Marcinkiewicz integrals and area integrals. In 1970, Fefferman in [2] proved that the Littlewood-Paley function is weak type for and . With further research about Littlewood-Paley operators, some authors turn their attentions to study the parameter Littlewood-Paley operators. For example, in 1999, Sakamoto and Yabuta in [3] considered the parameter function. Since then, many papers focus on the behaviours of the operators; among them we refer readers to see [4–6].

In the past ten years or so, most authors mainly study the classical theory of harmonic analysis on under nondoubling measures which only satisfy the polynomial growth condition; see [7–12]. Exactly, we assume that which is a positive Radon measure on satisfies the following growth conditions; namely, for all and , there exist constant and such that where . The analysis associated with nondoubling measures as in (1) has important applications in solving long-standing open Painlevé’s problem and Vitushkin’s conjecture (see [13, 14]). Besides, Coifman and Weiss have showed that the measure is a key assumption in harmonic analysis on homogeneous-type spaces (see [15, 16]).

However, Hytönen in [17] pointed that the measure as in (1) may not contain the doubling measure as special cases. To solve the problem, in 2010, Hytönen in [17] introduced a new class of metric measure spaces satisfying the so-called upper doubling conditions and the geometrically doubling (resp., see Definitions 1 and 2 below), which are now claimed nonhomogeneous metric measure spaces. Therefore, if we replace the underlying spaces with nonhomogeneous metric measure spaces, many known-consequences have been proved still true; for example, see [18–22].

In this paper, we always assume that is a nonhomogeneous metric measure space. In this setting, we will establish the boundedness of the parameter Littlewood-Paley functions on .

In order to state our main results, we firstly recall some necessary notions and notation. Hytönen in [17] gave out the definition of upper doubling metric spaces as follows.

Definition 1 (see [17]). A metric measure space is said to be upper doubling, if is Borel measure on and there exist a dominating function and a positive constant such that for each , is nondecreasing and, for all and ,

Htyönen et al. in [18] proved that there exists another dominating function such that , and where and . Based on this, from now on, let the dominating function in (2) also satisfy (3).

Now we recall the notion of geometrically doubling conditions given in [17].

Definition 2 (see [17]). A metric space is said to be geometrically doubling, if there exists some such that, for any ball , there exists a finite ball covering of such that the cardinality of this covering is at most .

Remark 3 (see [17]). Let be a metric space. Hytönen in [17] showed that the following statements are mutually equivalent:(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 Definition 2 and (3)For every , any ball can contain at most centers of disjoint balls .(4)There exists such that any ball can contain at most centers of disjoint balls .

Hytönen in [17] introduced the following coefficients analogous to Tolsa’s number in [7].

Given any two balls , set where represents the center of the ball .

Remark 4. Bui and Duong in [21] firstly introduced the following discrete version of as in (4) on , which is very similar to the number introduced in [7] by Tolsa. For any two balls , is defined by where the radii of the balls and are denoted by and , respectively, and is the smallest integer satisfying . It is easy to obtain . Bui and Duong in [21] also pointed out that it is incorrect that .

Now we recall the following notion of -doubling property (see [17]).

Definition 5 (see [17]). Let . A ball is claimed to be -doubling if .

It was stated in [17] that, there exist many balls which have the above -doubling property. In the latter part of the paper, if and are not specified, -doubling ball always stands for -doubling ball with a fixed number , where is considered as a geometric dimension of the space. Moreover, the smallest -doubling ball of the form with is denoted by , and sometimes can be simply denoted by .

Now we give the definition of the parameter Littlewood-Paley functions on .

Definition 6 (see [22]). Let be a locally integrable function on . Assume that there exists a positive constant such that, for all with , and, for all ,

The parameter Marcinkiewicz integral associated with the above which satisfies (6) and (7) is defined by where . The parameter function is defined by where , , and .

Remark 7. (1) When , the operator as in (8) is just the Marcinkiewicz integral on (see [22]).
(2) If we take and , then the parameter function as in (9) is just a parameter Littlewood-Paley operator with nondoubling measures in [8].

The following definition of the atomic Hardy space was introduced by Htyönen et al. (see [18]).

Definition 8 (see [18]). Let and . A function is called a -atomic block if(a)there exists a ball such that ,(b),(c)for any there exist a function supported on ball and a number such that Moreover, let .

We say a function belongs to the atomic Hardy space if there are atomic blocks such that with . The norm of is denoted by , where the infimum is taken over all the possible decompositions of as above.

It was proved by Htyönen et al. in [18] that the definition of is not related to the choice of and the spaces and have the same norms for . Thus, for convenience, we always denote by .

Now we give the Hörmander-type condition on ; that is, there exists a positive , such that Notice this condition is slightly stronger than (7).

Now let us state the main theorems which generalize and improve the corresponding results in [8].

Theorem 9. Let satisfy (6) and (7), and let be as in (9) with and . Then is bounded on for any .

Theorem 10. Let satisfy (6) and (11), and let be as in (9) with and . Then is bounded from into weak ; namely, there exists a positive constant such that, for any and ,

Theorem 11. Let satisfy (6) and (11), and let be as in (9) with and . Suppose that is bounded on . Then, is bounded from into .

Applying the Marcinkiewicz interpolation theorem and Theorems 9 and 10, it is easy to get the following result.

Corollary 12. Under the assumption of Theorem 10, is bounded on for .

The organization of this paper is as follows. In Section 2, we will give some preliminary lemmas. The proofs of the main theorems will be given in Section 3. Throughout this paper, stands for a positive constant which is independent of the main parameters, but it may be different from line to line. For any , we use to denote its characteristic function.

2. Preliminary Lemmas

In this section, we make some preliminary lemmas which are used in the proof of the main results. Firstly, we recall some properties of as in (4) (see [17]).

Lemma 13 (see [17]). (1) For all balls , it holds true that .
(2) For any , there exists a positive constant , such that, for all balls with .
(3) For any , there exists a positive constant , depending on , such that, for all balls .
(4) There exists a positive constant such that, for all balls , In particular, if and are concentric, then .
(5) There exists a positive constant such that, for all balls , ; moreover, if and are concentric, then .

To state the following lemmas, let us give a known-result (see [19]). For , the maximal operator is defined, by setting that, for all and , is bounded on provided that and also bounded from into .

The following lemma is slightly changed from [8].

Lemma 14. Let satisfy (6) and (7), and . Assume that is as in (8) and is as in (9) with and . Then for any nonnegative function , there exists a positive constant such that, for all with ,

Proof. By the definition of , we have Thus, to prove Lemma 14, we only need to estimate that For any and , write For , it is not difficult to obtain that Now we turn to estimate , by (2) and (13); we have Combining the estimates for and , we obtain (16) and hence complete the proof of Lemma 14.

Finally, we recall the Calderón-Zygmund decomposition theorem (see [21]). Suppose that is a fixed positive constant satisfying that , where is as in (2) and as in Remark 3.

Lemma 15 (see [21]). Let , , and . Then(1)there exists a family of finite overlapping balls such that is pairwise disjoint: (2)for each , let be a -doubling ball of the family , and . Then there exists a family of functions that, for each , , has a constant sign on and where is some positive constant depending only on , and there exists a positive constant , independent of , , and , such that if , then and if ,

3. Proofs of Theorems

Proof of Theorem 9. For the case of , assume in Lemma 14; then it is easy to get that which, along with -boundedness of , easily yields that Theorem 9 holds.
For the case of , let be the index conjugate to . By applying Hölder inequality and Lemma 14, we can conclude which is desired. Thus, we complete the proof of Theorem 9.

Proof of Theorem 10. Without loss of generality, we may assume that . It is easy to see that the conclusion of Theorem 10 naturally holds if when . Thus, we only need to discuss the case that . Applying Lemma 15 to at the level and letting , , , and be the same as in Lemma 15, we see that , where and . It is easy to obtain that and . By -boundedness of , we have On the other hand, by (20) with and the fact that the sequence of balls, , is pairwise disjoint, we see that and thus the proof of the Theorem 10 can be reduced to prove that For each fixed , denote the center of by , and let be the positive integer satisfying . We have Firstly, let us estimate and write it as where . By Hölder inequality, (24), and -boundedness of , we have For , by Minkowski inequality and (6), write To this end, let be as in Lemma 15 with and being, respectively, its center and radius. For any and , by (2) and (3), we have where we use the fact that Next we estimate . For any , , and satisfying , , and , we have Finally, for any , , and satisfying , , and , by applying (2), we have Combining the estimates for , , and , we obtain that , where, together with the fact that , we haveNow we turn to estimate for . Let , and write For each fixed , decompose as For any , with , and , and , together with Minkowski inequality and (6), we can conclude For , write For , by Minkowski inequality and (6), we deduce Now we estimate . Applying Minkowski inequality and the vanishing moment, we have With a way similar to that used in the proof of , we have . Thus, we only need to estimate ; by Minkowski inequality and (11), it follows that Combining the estimates for , , , and , we obtain that Next we estimate . For any , , and , we have , , and , and together with this fact, Minkowski inequality, and (6), we get It remains to estimate . Applying Minkowski inequality and (6), we have Now we estimate . For any , , and , it is easy to see . So we haveOn the other hand, by a method similar to that used in the proof of , we have Combining the estimates , , , , and the fact that , we conclude that which, together with , implies (30) and the proof of Theorem 10 is finished.

Proof of Theorem 11. Without loss of generality, we assume . By a standard argument, it suffices to show that, for any -atomic block , Assume that and , where, for , is a function supported in such that and . Write For , we see that Applying the Hölder inequality, -boundedness of , and the fact that for , we haveNow we estimate , with a method similar to that used in the proof of and , and we see that Therefore, .
On the other hand, based on the proof of and Definition 8, it is easy to obtain thatCombining the estimates for and , (53) holds. Thus, Theorem 11 is completed.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final paper.

Acknowledgments

This paper is supported by National Natural Foundation of China (Grant no. 11561062).