Research Article | Open Access
Mingming Cao, Qingying Xue, "On the Boundedness of Biparameter Littlewood-Paley -Function", Journal of Function Spaces, vol. 2016, Article ID 3605639, 15 pages, 2016. https://doi.org/10.1155/2016/3605639
On the Boundedness of Biparameter Littlewood-Paley -Function
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.
1.1. Background and Motivation
It is well known that -function originated in the work of Littlewood and Paley  in the 1930s. In 1961, Stein  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 . Moreover, in , 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  considered the endpoint weak estimates of -function when and .
Recently, Cao et al.  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  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  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  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 . As a consequence, a new version of the product space theorem was established. In 2014, Hytönen and Martikainen  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 . 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  directly, since addition terms appear in Definition 1. In fact, we will use much more weaker conditions than the conditions used in  (see assumptions in the following subsection). Unlike the one-parameter case and two-weight case , 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  in the study of the biparameter Calderón-Zygmund integrals and later appeared in . 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.
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 , 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
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 . 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 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