Abstract
Let and let be the biparameter Littlewood-Paley -function defined by = , , where is a nonconvolution kernel defined on . In this paper we show that the biparameter Littlewood-Paley function is bounded from to . This is done by means of probabilistic methods and by using a new averaging identity over good double Whitney regions.
1. Introduction
1.1. Background and Motivation
It is well known that -function originated in the work of Littlewood and Paley [1] in the 1930s. In 1961, Stein [2] introduced and studied the following higher dimensional () Littlewood-Paley -function:where , denotes the Poisson kernel, and . It plays important roles in harmonic analysis and other fields. It is easy to show that is an isometry on . With much greater difficulty, it can be proved that, for any , and are equivalent norms [3]. Moreover, in [3], Stein also proved that if then is of weak type and is of strong type for . In 1970, as a replacement of weak bounds for , Fefferman [4] considered the endpoint weak estimates of -function when and .
Recently, Cao et al. [5] gave a characterization of two-weight norm inequalities for the classical -function. The first step of the proof is to reduce the case to good Whitney regions. In addition, the random dyadic grids and martingale differences decomposition are used. The core of the proof is the construction of stopping cubes, which is a modern and effective technique to deal with two-weight problems. The stopping cubes were first introduced to handle two-weight boundedness for Hilbert transform [6, 7]. Then the related consequences and applications were given, as demonstrated in [5, 8, 9]. Still, more recently, Cao and Xue [10] established a local theorem for the nonhomogeneous Littlewood-Paley -function with nonconvolution type kernels and upper power bound measure . It was the first time to investigate -function in the simultaneous presence of three attributes: local, nonhomogeneous, and -testing condition. It is important to note that the testing condition here is type with , which means that the averaging identity over good Whitney regions used in [5] is not suitable for the new setting . Thus, some new methods and more complicated techniques are needed.
When it comes to the multiparameter harmonic analysis, there is a very large existing theory. In terms of singular integrals, it was initiated in the work of Fefferman and Stein [11] on biparameter singular integral operators and then continued by many authors. In 2012, a dyadic representation theorem for biparameter singular integrals was presented by Martikainen [12]. As a consequence, a new version of the product space theorem was established. In 2014, Hytönen and Martikainen [13] proved a nonhomogeneous version of theorem for certain biparameter singular integral operators. Moreover, they discussed the related nonhomogeneous Journés lemma and product theory with more general type of measures. Still, in 2014, a class of biparameter kernels and related vertical square functions in the upper half-space were first introduced by Martikainen [14]. Using modern dyadic probabilistic techniques adapted to the biparameter situation, the author gave a criterion for the boundedness of these square functions. It is worth pointing out that the kernels are assumed to satisfy some estimates, including a natural size condition, a Hölder estimate and two symmetric mixed Hölder and size estimates, the mixed Carleson and size conditions, the mixed Carleson and Hölder estimates, and a biparameter Carleson condition. Moreover, it should be noted that the biparameter Carleson condition is necessary for the square function to be bounded in .
Motivated by the above works, in this paper, we keep on studying the Littlewood-Paley -function but in biparameter setting. To state more clearly, we first introduce the definition of the biparameter Littlewood-Paley -function.
Definition 1. Let , for any , and the biparameter Littlewood-Paley -function is defined by where =.
Under certain structural assumptions, we will prove the following boundedness of , in other words, the following inequality:
Compared to the biparameter vertical square function, the biparameter Littlewood-Paley -function is significantly much more difficult to be dealt with. Actually, in biparameter case, additional integrals make most of the corresponding estimates more complicated. We could not use the assumptions in [14] directly, since addition terms appear in Definition 1. In fact, we will use much more weaker conditions than the conditions used in [14] (see assumptions in the following subsection). Unlike the one-parameter case and two-weight case [5], the proof of biparameter -function does not involve the stopping cubes and martingale differences decomposition. In fact, the decomposition associated with Haar function in provides a foundation for our analysis. And modern techniques, including probabilistic methods and dyadic analysis, will be used efficiently again. They were first used by Martikainen [12] in the study of the biparameter Calderón-Zygmund integrals and later appeared in [14]. For more applications, one can refer to [13, 15].
1.2. Assumptions and Main Result
To state our main results, the natural framework is to give some appropriate assumptions. From now on, we always assume that . We use, for minor convenience, metrics on and .
Assumption 1 (standard estimates). The kernel is assumed to satisfy the following estimates:(1)Size condition:(2) Hölder condition: whenever and .(3)Mixed Hölder and size conditions: whenever and whenever .
Assumption 2 (Carleson condition standard estimates). If is a cube with side length , we define the associated Carleson box by . We assume the following conditions: for every cube and , it holds the following:(1)Combinations of Carleson and size conditions:(2)Combinations of Carleson and Hölder conditions: whenever . And whenever .
Assumption 3 (biparameter Carleson condition). Let , where is a dyadic grid in and is a dyadic grid in . For , let be the associated Whitney region. Denote , , andWe assume the following biparameter Carleson condition: for every it holds thatfor all sets such that and such that for every there exists so that .
Now we state the main result of this paper.
Theorem 2. Let , , and . Assume that the kernel satisfies the Assumptions 1–3. Then it holds thatwhere the implied constant depends only on the assumptions.
Remark 3. In Section 6, we shall show that the biparameter Carleson condition is necessary for -function bound on . Moreover, Assumptions 2 and 3 are much weaker than the similar conditions used in [14], since here two terms (both less than one) were added and more integrals related to or were used in our assumptions.
2. The Probabilistic Reduction
In this section, our goal is to simplify the proof of the main result. First, we recall the definitions of random dyadic grids, good/bad cubes, Haar function on which can be found in [12, 16, 17].
2.1. Random Dyadic Grids
Let , where . Let be the standard dyadic grids on . We define the new dyadic grid in bySimilarly, we can define the dyadic grids in . There is a natural product probability structure on and . So we have independent random dyadic grids and in and , respectively. Even if we need two independent grids.
2.2. Good and Bad Cubes
A cube is said to be bad if there exists a with such that . Otherwise, is called good. Here and are given parameters. Denote . Then is independent of , and the parameter is a fixed constant so that , .
Throughout this article, we take , where appears in the kernel estimates. Moreover, roughly speaking, a dyadic cube will be bad if it is relatively close to the boundary of a much bigger dyadic cube. It is important to observe that the position and goodness of a cube are independent. Indeed, according to the definition, the spatial position ofdepends only on for . On the other hand, the relative position of with respect to a bigger cubedepends only on for . Thus, the position and goodness of are independent.
2.3. Haar Functions
In order to decompose a function , we next recall the definition of the Haar function on . Let be an normalized Haar function related to , where is a dyadic grid on . With this we mean that , , is one of the functions , , defined bywhere and for every . Here and are the left and right halves of the interval , respectively. If , the Haar function is cancellative: . All the cancellative Haar functions form an orthonormal basis of . If , we may thus writeHowever, we suppress the finite summation and just write . We may expand a function defined in using the corresponding product basis:
2.4. Averaging over Good Whitney Regions
Let . Let always and . Note that the position and goodness of are independent. Therefore, one can writeIndeed, to get this equality, we only need to apply the similar argument to one-parameter case twice. For more details in one-parameter setting, see [5]. Consequently, we are reduced to bound the sum Furthermore, we can carry out the decomposition where and the others are completely similar.
Sequentially, it is enough to focus on estimating the four pieces: , , , and in the following sections.
3. The Case: and
For the sake of convenience, we first present two key lemmas, which will be used later.
Lemma 4 (see [8, 14]). Letwhere the long distance , and . Then for any , we have the following estimate: In particular, it holds that
Lemma 5. Let . For a given cube and , it holds
Proof. Fixed . If , thenThus If , thenHence, where and we have used the condition in the last step.
Now we turn our attention to the estimate of . An easy consequence of the Hölder estimates of the kernel is that Moreover, by Lemma 5, we obtain thatSince and , then we get Therefore, from Minkowski’s integral inequality and Lemma 4, it now follows that
4. The Case: and <
In any case, we perform the splitting These three parts are called separated, Nested, and adjacent, respectively. The term Nested makes sense, since the summing conditions that is good actually imply that is the ancestor of . Thus, it holds where
Now we are in position to estimate the above three terms, respectively.
4.1. Separated Part
In this case, we note that the following inequality holds:Indeed, if , then . Therefore, we get If , then . Moreover, noticing that and , one may conclude that
It is obvious that the mixed Hölder and size condition implies that
Thus, combining Lemma 5 with (39), one can obtainConsequently, by the similar argument as , we have
4.2. Adjacent Part
The summation conditions and indicate that . Thus, It follows from (43) that Therefore, exactly as we have seen before,
4.3. Nested Part
We use to denote the unique cube for which and . We call as the generation older dyadic ancestor of . In this case, by the goodness of , it must actually have . That is, is the ancestor of . This enables us to write
Introduce the notationThen, it is easy to check that, and .
Denote so that , . Then, we split where
4.3.1. Estimate of
We need the following lemma.
Lemma 6. Let and . Given cubes , , , and , the following estimate holds:
Proof. By the mixed Hölder and size condition, it yields that Proceeding as we did in (33), we only need to showIndeed, if , If , we have by the goodness of that Thus, we obtain Given , we introduce the notation Then, it follows that Note that This yields that As for , we by Young’s inequality have the following estimate: where
Accordingly, Minkowski’s integral inequality and Lemma 6 give that
4.3.2. Estimate of
We also need to use the following lemma.
Lemma 7. Let , be cubes, and . Then the Carleson condition holds
Proof. The first step is to splitwhereFrom combinations of Carleson and Hölder conditions and Lemma 5, it follows that The mixed Hölder and size estimate gives thatThus, collecting the estimates of Lemma 5 and (55), one can deduce that Therefore, we obtainThus, we finish the proof of Lemma 7.
Now we bound . If , then we have Therefore, we obtain the following estimate So far, we have completed the estimate of .
As for the term , it is completely symmetric with the term . It is worth noting that the mixed Hölder and size estimate and the combination of Carleson and Hölder estimate are symmetric, respectively. Thus the estimate for is also true and we here omit its proof.
5. The Case: and
Similar to what we have done before, the summation was decomposed into the separated, Nested, and adjacent terms. A similar splitting in the summation is also performed. This splits the whole summation into nine parts as follows:
5.1. Nested/Nested:
We begin with the term , where the new biparameter phenomena will appear. Noting that although this is only one of the many cases one needs to discuss in order to obtain a full estimate for term all the main difficulties in other cases are in fact already embedded in Nested/Nested. The fact will become more and more clear throughout the proof. Similarly, for the singular integral operators including biparameter and multiparameter cases, the Nested part is also the most difficult one. Because it involves in some paraproduct estimates and all the type estimates.
The decomposition of in (50) gives that where
5.1.1. Estimate of
We proceed using the standard argument as in Lemma 6. The size condition and (55) lead to the bound It is similar to estimate to analyze .
5.1.2. Estimate of
Applying the biparameter Carleson condition, it immediately yields that where in the last step we have used the boundedness of the strong maximal function associated with rectangles.
5.1.3. Estimate of and
Lemma 8. Let , and be fixed. Then the Carleson condition is satisfied as follows:
Proof. The proof of Lemma 8 is similar to Lemma 7. The size condition and mixed Carleson and size estimate are used. In addition, the inequality (55) is used twice.
Therefore, is bounded as below.
5.2. The Rest of Terms
As for the estimates of the remaining terms, they are simply combinations of the techniques we have used above. Thereby, we here only present certain key points.
When reviewing the above proof, one will realize that the central part is to dominate , , and . So do the rest of terms. Moreover, the initial estimates of , , and are retained in the inequality (33) and Lemmas 6 and 7, respectively. They do not involve the relationship of side length of cubes , , , and . Thus, based on the inequality (33) and Lemmas 6 and 7, one only needs to add the corresponding the relationship of side length.
Consequently, using the size condition or the mixed Hölder and size condition, it yields the bounds for , , , and directly. Finally, for the terms and , is split into and . Applying the size condition and the combinations of Carleson and size estimate, we will bound them. The terms and are symmetric with respect to them, respectively.
6. The Necessity of Biparameter Carleson Condition
We here show that the biparameter Carleson condition is necessary for -function bound on .
Suppose that is bounded on , where has a kernel and has a kernel , , , and . We assume that these satisfy the size condition and the corresponding bounds in and . We shall show that the biparameter Carleson condition (12) holds.
Define and for a small enough dimensional constant , where denotes the strong maximal function related to the grid and denotes the strong maximal function. From the endpoint estimates for and , it follows that . Hence, it is enough to show that
For every we let consist of the maximal for which . Then we define . Moreover, for fixed , let be the family of the maximal , for which , and be the maximal cube for which and . So, we only need to show the following inequalities:To attain the goal, we need to first estimate and . Actually, Minkowski’s integral inequality and size estimate yield thatSimilarly, we may estimate The remaining calculation is a routine application of the idea of [14]. We here omit the details. Finally, we obtain Thus, we have proved the necessity.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
The second author was supported partly by NSFC (nos. 11471041 and 11671039), the Fundamental Research Funds for the Central Universities (no. 2014KJJCA10), and NCET-13-0065.