On the Boundedness of Biparameter Littlewood-Paley g ∗ λ-Function

Copyright © 2016 M. Cao and Q. Xue.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let m, n ≥ 1 and let g∗ λ1 ,λ2 be the biparameter Littlewood-Paley g∗ λ -function defined by g∗ λ1 ,λ2 (f)(x) = (∫∫Rm+1 + (t2/(t2 + |x2 − y2|))mλ2∫∫Rn+1 + (t1/(t1 + |x1 − y1|))nλ1 × |θt1 ,t2f(y1, y2)|2(dy1dt1/tn+1 1 )(dy2dt2/tm+1 2 ))1/2, λ1 > 1, λ2 > 1, where θt1 ,t2f is a nonconvolution kernel defined on R. In this paper we show that the biparameter Littlewood-Paley function g∗ λ1 ,λ2 is bounded from L2(Rn+m) to L2(Rn+m).This is done bymeans of probabilistic methods and by using a new averaging identity over good double Whitney regions.

Recently, Cao et al. [5] gave a characterization of twoweight 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  ∈ (1,2], which means that the averaging identity over good Whitney regions used in [5] is not suitable for the new setting  ∈ (1,2).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 1 theorem was established.In 2014, Hytönen and Martikainen [13] proved a nonhomogeneous version of 1 theorem for certain biparameter singular integral operators.Moreover, they discussed the related nonhomogeneous Journés lemma and product BMO 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  2 (R + ) 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  2 (R + ).
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. where Under certain structural assumptions, we will prove the following  2 (R + ) boundedness of  *  1 , 2 , 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 twoweight 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 R  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].

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 ,  > 0. We use, for minor convenience, ℓ ∞ metrics on R  and R  .
Assumption 1 (standard estimates).The kernel   1 , 2 : R + × R + → C is assumed to satisfy the following estimates: (1) Size condition: (2) Hölder condition: whenever (3) Mixed Hölder and size conditions: whenever whenever Assumption 2 (Carleson condition × standard estimates).If  ⊂ R  is a cube with side length ℓ(), we define the associated Carleson box by Î =  × (0, ℓ()).We assume the following conditions: for every cube  ⊂ R  and  ⊂ R  , it holds the following: (1) Combinations of Carleson and size conditions: (2) Combinations of Carleson and Hölder conditions: We assume the following biparameter Carleson condition: for every D = D  × D  it holds that for all sets Ω ⊂ R + such that |Ω| < ∞ and such that for every  ∈ Ω there exists  ×  ∈ D so that  ∈  ×  ⊂ Ω.
where the implied constant depends only on the assumptions.Remark 3. In Section 6, we shall show that the biparameter Carleson condition is necessary for  *  1 , 2 -function bound on  2 (R + ).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  1 or  2 were used in our assumptions.

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 R  which can be found in [12,16,17].
Journal of Function Spaces 2.1.Random Dyadic Grids.Let   = {   } ∈Z , where    ∈ {0, 1}  .Let D 0  be the standard dyadic grids on R  .We define the new dyadic grid in R  by Similarly, we can define the dyadic grids D  in R  .There is a natural product probability structure on ({0, 1}  (15) depends only on    for 2 − < ℓ().On the other hand, the relative position of  +   with respect to a bigger cube 2 −    (16) depends only on    for ℓ() ≤ 2 − < ℓ().Thus, the position and goodness of  +   are independent.

Haar Functions.
In order to decompose a function  ∈  2 , we next recall the definition of the Haar function on R  .Let ℎ  be an  2 normalized Haar function related to  ∈ D  , where D  is a dyadic grid on R  .With this we mean that where ) for every  = 1, . . ., .Here  , and  , are the left and right halves of the interval   , respectively.If  ̸ = 0, the Haar function is cancellative: ∫ R  ℎ  = 0.All the cancellative Haar functions form an orthonormal basis of  2 (R  ).If  ∈  2 (R  ), we may thus write However, we suppress the finite  summation and just write  = ∑  ⟨, ℎ  ⟩ℎ  .We may expand a function  defined in R + using the corresponding product basis: Indeed, 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: G <,< , G <,≥ , G ≥,< , and G ≥,≥ in the following sections.

The Case
For the sake of convenience, we first present two key lemmas, which will be used later.
Now we turn our attention to the estimate of G <,< .An easy consequence of the Hölder estimates of the kernel Moreover, by Lemma 5, we obtain that Since ℓ( 1 ) < ℓ( 2 ) and ℓ( 1 ) < ℓ( 2 ), then we get Therefore, from Minkowski's integral inequality and Lemma 4, it now follows that ( 4. The Case: ℓ( 1 ) ≥ ℓ( 2 ) and ℓ( 1 ) < ℓ( 2 ) 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  2 is good actually imply that  1 is the ancestor of  2 .Thus, it holds where Now we are in position to estimate the above three terms, respectively.

Lemma 7.
Let  1 , 2 ∈ D  be cubes, and Proof.The first step is to split where From combinations of Carleson and Hölder conditions and Lemma 5, it follows that The mixed Hölder and size estimate gives that Thus, collecting the estimates of Lemma 5 and (55), one can deduce that Therefore, we obtain (74) So far, we have completed the estimate of G ≥,< .
As for the term G <,≥ , it is completely symmetric with the term G ≥,< .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 G <,≥ is also true and we here omit its proof.

The Case
Similar to what we have done before, the summation ℓ( 1 ) ≥ ℓ( 2 ) was decomposed into the separated, Nested, and adjacent terms.A similar splitting in the summation ℓ( 1 ) ≥ ℓ( 2 ) is also performed.This splits the whole summation into nine parts as follows: (75) 5.1.Nested/Nested: G nes,nes .We begin with the term G nes,n , 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 G ≥,≥ 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 BMO type estimates.
The decomposition of ℎ  () in (50) gives that where (77) where in the last step we have used the   (1 <  < ∞) boundedness of the strong maximal function associated with rectangles.(82)

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 P(, ), Q(, ), and R( 2 ,  2 ).So do the rest of terms.Moreover, the initial estimates of P, Q, and R are retained in the inequality (33) and Lemmas 6 and 7, respectively.They do not involve the relationship of side length of cubes  1 ,  2 ,  1 , and  2 .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 G sep,sep , G sep,adj , G adj,adj , and G adj,sep directly.Finally, for the terms G nes,sep and G nes,adj , nes is split into mod and Car.Applying the size condition and the combinations of Carleson and size estimate, we will bound them.The terms G sep,nes and G adj,nes are symmetric with respect to them, respectively.

The Necessity of Biparameter Carleson Condition
We here show that the biparameter Carleson condition is necessary for  * × D  , where D  is a dyadic grid in R  and D  is a dyadic grid in R  .For  ∈ D  , let   =  × (ℓ()/2, ℓ()) be the associated Whitney region.Denote  1 = ,  2 = , and Now we state the main result of this paper.
5.1.3.Estimate of G Car,mod and G mod,Car Lemma 8. Let  ∈ D ,good , ( 2 ,  2 ) ∈   2 and  ∈ N 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, G Car,mod is bounded as below.