# On the Boundedness of Biparameter Littlewood-Paleygλ⁎-Function

Authors: Mingming Cao; Qingying Xue
Journal: Journal of Function Spaces
 (2016)

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## Abstract

Letm,n≥1 and let gλ1,λ2⁎ be the biparameter Littlewood-Paley gλ⁎-function defined by gλ1,λ2⁎fx = ∬R+m+1t2/t2+x2-y2mλ2∬R+n+1t1/t1+x1-y1nλ1×θt1,t2fy1,y22dy1dt1/t1n+1dy2dt2/t2m+11/2,  λ1>1,  λ2>1, where θt1,t2f is a nonconvolution kernel defined on Rm+n. In this paper we show that the biparameter Littlewood-Paley function gλ1,λ2⁎ is bounded from L2Rn+m to L2Rn+m. This is done by means of probabilistic methods and by using a new averaging identity over good double Whitney regions.


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## Body

## 1. Introduction

### 1.1. Background and Motivation

It is well known thatgλ∗-function originated in the work of Littlewood and Paley [1] in the 1930s. In 1961, Stein [2] introduced and studied the following higher dimensional (n≥2) Littlewood-Paley gλ∗-function:(1)gλ∗fx=∬R+n+1tt+x-ynλ∇Ptfy,t2dydttn-11/2,λ>1,where Ptf(y,t)=pt∗f(y), pt(y)=t-np(y/t) denotes the Poisson kernel, and ∇=(∂/∂y1,…,∂/∂yn,∂/∂t). It plays important roles in harmonic analysis and other fields. It is easy to show that gλ∗ is an isometry on L2(Rn). With much greater difficulty, it can be proved that, for any 1<p<∞, gλ∗(f)Lp(Rn) and fLp(Rn) are equivalent norms [3]. Moreover, in [3], Stein also proved that if λ>2 then gλ∗ is of weak type (1,1) and is of strong type (p,p) for 1<p<∞. In 1970, as a replacement of weak (1,1) bounds for 1<λ<2, Fefferman [4] considered the endpoint weak (p,p) estimates of gλ∗-function when p>1 and λ=2/p.Recently, Cao et al. [5] gave a characterization of two-weight norm inequalities for the classical gλ∗-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 Tb theorem for the nonhomogeneous Littlewood-Paley gλ∗-function with nonconvolution type kernels and upper power bound measure μ. It was the first time to investigate gλ∗-function in the simultaneous presence of three attributes: local, nonhomogeneous, and Lp-testing condition. It is important to note that the testing condition here is Lp type with p∈(1,2], which means that the averaging identity over good Whitney regions used in [5] is not suitable for the new setting p∈(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 T1 theorem was established. In 2014, Hytönen and Martikainen [13] proved a nonhomogeneous version of T1 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 L2(Rn+m) 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 L2(Rn+m).Motivated by the above works, in this paper, we keep on studying the Littlewood-Paleygλ∗-function but in biparameter setting. To state more clearly, we first introduce the definition of the biparameter Littlewood-Paley gλ∗-function.Definition 1.
Letλ1,λ2>1, for any x=(x1,x2)∈Rn+m, and the biparameter Littlewood-Paley gλ∗-function is defined by (2)gλ1,λ2∗fx≔∬R+m+1t2t2+x2-y2mλ2∬R+n+1t1t1+x1-y1nλ1×θt1,t2fy1,y22dy1dt1t1n+1dy2dt2t2m+11/2,where θt1,t2fy1,y2 =∬Rn+mKt1,t2y1,y2,z1,z2f(z1,z2)dz1dz2.Under certain structural assumptions, we will prove the followingL2(Rn+m) boundedness of gλ1,λ2∗, in other words, the following inequality:(3)gλ1,λ2∗fL2Rn+m≲fL2Rn+m.Compared to the biparameter vertical square function, the biparameter Littlewood-Paleygλ∗-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 gλ∗-function does not involve the stopping cubes and martingale differences decomposition. In fact, the decomposition associated with Haar function in Rn 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 thatα,β>0. We use, for minor convenience, l∞ metrics on Rn and Rm.Assumption 1 (standard estimates).
The kernelKt1,t2:Rn+m×Rn+m→C is assumed to satisfy the following estimates:(1)
Size condition:(4)Kt1,t2x,y≲t1αt1+x1-y1n+αt2βt2+x2-y2m+β.(2)
Hölder condition:(5)Kt1,t2x,y-Kt1,t2x,y1,y2′-Kt1,t2x,y1′,y2+Kt1,t2x,y′≲y1-y1′αt1+x1-y1n+αy2-y2′βt2+x2-y2m+β,whenevery1-y1′<t1/2 and y2-y2′<t2/2.(3)
Mixed Hölder and size conditions:(6)Kt1,t2x,y-Kt1,t2x,y1,y2′≲t1αt1+x1-y1n+αy2-y2′βt2+x2-y2m+β,whenevery2-y2′<t2/2 and(7)Kt1,t2x,y-Kt1,t2x,y1′,y2≲y1-y1′αt1+x1-y1n+αt2βt2+x2-y2m+β,whenevery1-y1′<t1/2.Assumption 2 (Carleson condition× standard estimates).
IfI⊂Rn is a cube with side length l(I), we define the associated Carleson box by I^=I×(0,l(I)). We assume the following conditions: for every cube I⊂Rn and J⊂Rm, it holds the following:(1)
Combinations of Carleson and size conditions:(8)∬I^∫Rn∫IKt1,t2x-y,z1,z2dz12t1t1+y1nλ1dy1t1ndx1dt1t11/2≲I1/2t2βt2+x2-y2-z2m+β,∬J^∫Rm∫JKt1,t2x-y,z1,z2dz22t2t2+y2nλ1dy2t2ndx2dt2t21/2≲J1/2t1αt1+x1-y1-z1n+α.(2)
Combinations of Carleson and Hölder conditions:(9)∬I^∫Rn∫IKt1,t2x-y,z1,z2-Kt1,t2x-y,z1,z2′dz12t1t1+y1nλ1dy1t1ndx1dt1t11/2≲I1/2z2-z2′βt2+x2-y2-z2m+β,wheneverz2-z2′<t2/2. And(10)∬J^∫Rm∫JKt1,t2x-y,z1,z2-Kt1,t2x-y,z1′,z2dz22t2t2+y2mλ2dy2t2ndx2dt2t21/2≲J1/2z1-z1′αt1+x1-y1-z1n+α,wheneverz1-z1′<t1/2.Assumption 3 (biparameter Carleson condition).
LetD=Dn×Dm, where Dn is a dyadic grid in Rn and Dm is a dyadic grid in Rm. For I∈Dn, let WI=I×lI/2,lI be the associated Whitney region. Denote n1=n,  n2=m, and(11)CIJD=∬WJ∬WI∬Rn+mθt1,t21y1,y22∏i=12titi+xi-yiniλidy1dy2t1nt2mdx1dt1t1dx2dt2t2.We assume the followingbiparameter Carleson condition: for every D=Dn×Dm it holds that(12)∑I×J∈DI×J⊂ΩCIJD≲Ωfor all sets Ω⊂Rn+m such that Ω<∞ and such that for every x∈Ω there exists I×J∈D so that x∈I×J⊂Ω.Now we state the main result of this paper.Theorem 2.
Letλ1,λ2>2, 0<α≤nλ1-2/2, and 0<β≤mλ2-2/2. Assume that the kernel Kt1,t2 satisfies the Assumptions 1–3. Then it holds that(13)gλ1,λ2∗fL2Rn+m≲fL2Rn+m,where the implied constant depends only on the assumptions.Remark 3.
In Section6, we shall show that the biparameter Carleson condition is necessary for gλ1,λ2∗-function bound on L2(Rn+m). 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 y1 or y2 were used in our assumptions.


## 1.1. Background and Motivation

It is well known thatgλ∗-function originated in the work of Littlewood and Paley [1] in the 1930s. In 1961, Stein [2] introduced and studied the following higher dimensional (n≥2) Littlewood-Paley gλ∗-function:(1)gλ∗fx=∬R+n+1tt+x-ynλ∇Ptfy,t2dydttn-11/2,λ>1,where Ptf(y,t)=pt∗f(y), pt(y)=t-np(y/t) denotes the Poisson kernel, and ∇=(∂/∂y1,…,∂/∂yn,∂/∂t). It plays important roles in harmonic analysis and other fields. It is easy to show that gλ∗ is an isometry on L2(Rn). With much greater difficulty, it can be proved that, for any 1<p<∞, gλ∗(f)Lp(Rn) and fLp(Rn) are equivalent norms [3]. Moreover, in [3], Stein also proved that if λ>2 then gλ∗ is of weak type (1,1) and is of strong type (p,p) for 1<p<∞. In 1970, as a replacement of weak (1,1) bounds for 1<λ<2, Fefferman [4] considered the endpoint weak (p,p) estimates of gλ∗-function when p>1 and λ=2/p.Recently, Cao et al. [5] gave a characterization of two-weight norm inequalities for the classical gλ∗-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 Tb theorem for the nonhomogeneous Littlewood-Paley gλ∗-function with nonconvolution type kernels and upper power bound measure μ. It was the first time to investigate gλ∗-function in the simultaneous presence of three attributes: local, nonhomogeneous, and Lp-testing condition. It is important to note that the testing condition here is Lp type with p∈(1,2], which means that the averaging identity over good Whitney regions used in [5] is not suitable for the new setting p∈(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 T1 theorem was established. In 2014, Hytönen and Martikainen [13] proved a nonhomogeneous version of T1 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 L2(Rn+m) 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 L2(Rn+m).Motivated by the above works, in this paper, we keep on studying the Littlewood-Paleygλ∗-function but in biparameter setting. To state more clearly, we first introduce the definition of the biparameter Littlewood-Paley gλ∗-function.Definition 1.
Letλ1,λ2>1, for any x=(x1,x2)∈Rn+m, and the biparameter Littlewood-Paley gλ∗-function is defined by (2)gλ1,λ2∗fx≔∬R+m+1t2t2+x2-y2mλ2∬R+n+1t1t1+x1-y1nλ1×θt1,t2fy1,y22dy1dt1t1n+1dy2dt2t2m+11/2,where θt1,t2fy1,y2 =∬Rn+mKt1,t2y1,y2,z1,z2f(z1,z2)dz1dz2.Under certain structural assumptions, we will prove the followingL2(Rn+m) boundedness of gλ1,λ2∗, in other words, the following inequality:(3)gλ1,λ2∗fL2Rn+m≲fL2Rn+m.Compared to the biparameter vertical square function, the biparameter Littlewood-Paleygλ∗-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 gλ∗-function does not involve the stopping cubes and martingale differences decomposition. In fact, the decomposition associated with Haar function in Rn 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 thatα,β>0. We use, for minor convenience, l∞ metrics on Rn and Rm.Assumption 1 (standard estimates).
The kernelKt1,t2:Rn+m×Rn+m→C is assumed to satisfy the following estimates:(1)
Size condition:(4)Kt1,t2x,y≲t1αt1+x1-y1n+αt2βt2+x2-y2m+β.(2)
Hölder condition:(5)Kt1,t2x,y-Kt1,t2x,y1,y2′-Kt1,t2x,y1′,y2+Kt1,t2x,y′≲y1-y1′αt1+x1-y1n+αy2-y2′βt2+x2-y2m+β,whenevery1-y1′<t1/2 and y2-y2′<t2/2.(3)
Mixed Hölder and size conditions:(6)Kt1,t2x,y-Kt1,t2x,y1,y2′≲t1αt1+x1-y1n+αy2-y2′βt2+x2-y2m+β,whenevery2-y2′<t2/2 and(7)Kt1,t2x,y-Kt1,t2x,y1′,y2≲y1-y1′αt1+x1-y1n+αt2βt2+x2-y2m+β,whenevery1-y1′<t1/2.Assumption 2 (Carleson condition× standard estimates).
IfI⊂Rn is a cube with side length l(I), we define the associated Carleson box by I^=I×(0,l(I)). We assume the following conditions: for every cube I⊂Rn and J⊂Rm, it holds the following:(1)
Combinations of Carleson and size conditions:(8)∬I^∫Rn∫IKt1,t2x-y,z1,z2dz12t1t1+y1nλ1dy1t1ndx1dt1t11/2≲I1/2t2βt2+x2-y2-z2m+β,∬J^∫Rm∫JKt1,t2x-y,z1,z2dz22t2t2+y2nλ1dy2t2ndx2dt2t21/2≲J1/2t1αt1+x1-y1-z1n+α.(2)
Combinations of Carleson and Hölder conditions:(9)∬I^∫Rn∫IKt1,t2x-y,z1,z2-Kt1,t2x-y,z1,z2′dz12t1t1+y1nλ1dy1t1ndx1dt1t11/2≲I1/2z2-z2′βt2+x2-y2-z2m+β,wheneverz2-z2′<t2/2. And(10)∬J^∫Rm∫JKt1,t2x-y,z1,z2-Kt1,t2x-y,z1′,z2dz22t2t2+y2mλ2dy2t2ndx2dt2t21/2≲J1/2z1-z1′αt1+x1-y1-z1n+α,wheneverz1-z1′<t1/2.Assumption 3 (biparameter Carleson condition).
LetD=Dn×Dm, where Dn is a dyadic grid in Rn and Dm is a dyadic grid in Rm. For I∈Dn, let WI=I×lI/2,lI be the associated Whitney region. Denote n1=n,  n2=m, and(11)CIJD=∬WJ∬WI∬Rn+mθt1,t21y1,y22∏i=12titi+xi-yiniλidy1dy2t1nt2mdx1dt1t1dx2dt2t2.We assume the followingbiparameter Carleson condition: for every D=Dn×Dm it holds that(12)∑I×J∈DI×J⊂ΩCIJD≲Ωfor all sets Ω⊂Rn+m such that Ω<∞ and such that for every x∈Ω there exists I×J∈D so that x∈I×J⊂Ω.Now we state the main result of this paper.Theorem 2.
Letλ1,λ2>2, 0<α≤nλ1-2/2, and 0<β≤mλ2-2/2. Assume that the kernel Kt1,t2 satisfies the Assumptions 1–3. Then it holds that(13)gλ1,λ2∗fL2Rn+m≲fL2Rn+m,where the implied constant depends only on the assumptions.Remark 3.
In Section6, we shall show that the biparameter Carleson condition is necessary for gλ1,λ2∗-function bound on L2(Rn+m). 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 y1 or y2 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 onRn which can be found in [12, 16, 17].


### 2.1. Random Dyadic Grids

Letβn=βnjj∈Z, where βnj∈0,1n. Let Dn0 be the standard dyadic grids on Rn. We define the new dyadic grid in Rn by(14)Dn=I+βn;I∈Dn0≔I+∑j:2-j<lI2-jβnj;I∈Dn0.Similarly, we can define the dyadic grids Dm in Rm. There is a natural product probability structure on 0,1nZ and 0,1mZ. So we have independent random dyadic grids Dn and Dm in Rn and Rm, respectively. Even if n=m we need two independent grids.


### 2.2. Good and Bad Cubes

A cubeI∈Dn is said to bebad if there exists a J∈Dn with l(J)≥2rl(I) such that dist⁡(I,∂J)≤l(I)γnl(J)1-γn. Otherwise, I is calledgood. Here r∈Z+ and γn∈(0,1/2) are given parameters. Denote πgoodn=Pβn(I+βnisgood)=Eβn(1good(I+βn)). Then πgoodn is independent of I∈Dn0, and the parameter r is a fixed constant so that πgoodn, πgoodm>0.Throughout this article, we takeγn=α/2(n+α), where α>0 appears in the kernel estimates. Moreover, roughly speaking, a dyadic cube I 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 I∈Dn0 are independent. Indeed, according to the definition, the spatial position of(15)I+βn=I+∑j:2-j<lI2-jβnjdepends only on βnj for 2-j<l(I). On the other hand, the relative position of I+βn with respect to a bigger cube(16)J+βn=I+∑j:2-j<lI2-jβnj+∑j:lI≤2-j<lJ2-jβnjdepends only on βnj for l(I)≤2-j<l(J). Thus, the position and goodness of I+βn are independent.


### 2.3. Haar Functions

In order to decompose a functionf∈L2, we next recall the definition of the Haar function on Rn. Let hI be an L2 normalized Haar function related to I∈Dn, where Dn is a dyadic grid on Rn. With this we mean that hI,  I=I1×⋯×In, is one of the 2n functions hIη,  η=η1,…,ηn∈{0,1}n, defined by(17)hIη=hI1η1⊗⋯⊗hInηn,where hIi0=Ii-1/21Ii and hIi1=Ii-1/2(1Ii,l-1Ii,r) for every i=1,…,n. Here Ii,l and Ii,r are the left and right halves of the interval Ii, respectively. If η≠0, the Haar function is cancellative: ∫Rn‍hI=0. All the cancellative Haar functions form an orthonormal basis of L2(Rn). If a∈L2(Rn), we may thus write(18)a=∑I∈Dn∑η∈0,1n∖0a,hIηhIη.However, we suppress the finite η summation and just write a=∑Ia,hIhI. We may expand a function f defined in Rn+m using the corresponding product basis:(19)f=∑I,JfIJhI×J≔∑I,Jf,hI⊗hJhI⊗hJ.


### 2.4. Averaging over Good Whitney Regions

Letf∈L2(Rn+m). Let always I1,I2∈Dn and J1,J2∈Dm. Note that the position and goodness of I+βn are independent. Therefore, one can write(20)gλ1,λ2∗fL2Rn+m2=∬R+m+1∬R+n+1∬Rn+mθt1,t2fx-y2t1t1+y1nλ1×t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2=1πgoodn1πgoodmEβn,βm∑I2,J2:good∬WJ2∬WI2∬Rn+mθt1,t2fx-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2=1πgoodn1πgoodmEβn,βm∑I2,J2:good∬WJ2∬WI2∬Rn+m∑I1,J1fI1J1θt1,t2hI1×J1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2.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 (21)G≔∑I2,J2:good∬WJ2∬WI2∬Rn+m∑I1,J1fI1J1θt1,t2hI1×J1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2.Furthermore, we can carry out the decomposition (22)G≲G<,<+G<,≥+G≥,<+G≥,≥,where (23)G<,<≔∑I2,J2:good∬WJ2∬WI2∬Rn+m∑I1,J1lI1<lI2lJ1<lJ2fI1J1θt1,t2hI1×J1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,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.


## 2.1. Random Dyadic Grids

Letβn=βnjj∈Z, where βnj∈0,1n. Let Dn0 be the standard dyadic grids on Rn. We define the new dyadic grid in Rn by(14)Dn=I+βn;I∈Dn0≔I+∑j:2-j<lI2-jβnj;I∈Dn0.Similarly, we can define the dyadic grids Dm in Rm. There is a natural product probability structure on 0,1nZ and 0,1mZ. So we have independent random dyadic grids Dn and Dm in Rn and Rm, respectively. Even if n=m we need two independent grids.


## 2.2. Good and Bad Cubes

A cubeI∈Dn is said to bebad if there exists a J∈Dn with l(J)≥2rl(I) such that dist⁡(I,∂J)≤l(I)γnl(J)1-γn. Otherwise, I is calledgood. Here r∈Z+ and γn∈(0,1/2) are given parameters. Denote πgoodn=Pβn(I+βnisgood)=Eβn(1good(I+βn)). Then πgoodn is independent of I∈Dn0, and the parameter r is a fixed constant so that πgoodn, πgoodm>0.Throughout this article, we takeγn=α/2(n+α), where α>0 appears in the kernel estimates. Moreover, roughly speaking, a dyadic cube I 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 I∈Dn0 are independent. Indeed, according to the definition, the spatial position of(15)I+βn=I+∑j:2-j<lI2-jβnjdepends only on βnj for 2-j<l(I). On the other hand, the relative position of I+βn with respect to a bigger cube(16)J+βn=I+∑j:2-j<lI2-jβnj+∑j:lI≤2-j<lJ2-jβnjdepends only on βnj for l(I)≤2-j<l(J). Thus, the position and goodness of I+βn are independent.


## 2.3. Haar Functions

In order to decompose a functionf∈L2, we next recall the definition of the Haar function on Rn. Let hI be an L2 normalized Haar function related to I∈Dn, where Dn is a dyadic grid on Rn. With this we mean that hI,  I=I1×⋯×In, is one of the 2n functions hIη,  η=η1,…,ηn∈{0,1}n, defined by(17)hIη=hI1η1⊗⋯⊗hInηn,where hIi0=Ii-1/21Ii and hIi1=Ii-1/2(1Ii,l-1Ii,r) for every i=1,…,n. Here Ii,l and Ii,r are the left and right halves of the interval Ii, respectively. If η≠0, the Haar function is cancellative: ∫Rn‍hI=0. All the cancellative Haar functions form an orthonormal basis of L2(Rn). If a∈L2(Rn), we may thus write(18)a=∑I∈Dn∑η∈0,1n∖0a,hIηhIη.However, we suppress the finite η summation and just write a=∑Ia,hIhI. We may expand a function f defined in Rn+m using the corresponding product basis:(19)f=∑I,JfIJhI×J≔∑I,Jf,hI⊗hJhI⊗hJ.


## 2.4. Averaging over Good Whitney Regions

Letf∈L2(Rn+m). Let always I1,I2∈Dn and J1,J2∈Dm. Note that the position and goodness of I+βn are independent. Therefore, one can write(20)gλ1,λ2∗fL2Rn+m2=∬R+m+1∬R+n+1∬Rn+mθt1,t2fx-y2t1t1+y1nλ1×t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2=1πgoodn1πgoodmEβn,βm∑I2,J2:good∬WJ2∬WI2∬Rn+mθt1,t2fx-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2=1πgoodn1πgoodmEβn,βm∑I2,J2:good∬WJ2∬WI2∬Rn+m∑I1,J1fI1J1θt1,t2hI1×J1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2.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 (21)G≔∑I2,J2:good∬WJ2∬WI2∬Rn+m∑I1,J1fI1J1θt1,t2hI1×J1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2.Furthermore, we can carry out the decomposition (22)G≲G<,<+G<,≥+G≥,<+G≥,≥,where (23)G<,<≔∑I2,J2:good∬WJ2∬WI2∬Rn+m∑I1,J1lI1<lI2lJ1<lJ2fI1J1θt1,t2hI1×J1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,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.


## 3. The Case:lI1<lI2 and lJ1<lJ2

For the sake of convenience, we first present two key lemmas, which will be used later.Lemma 4 (see [8, 14]).
Let(24)AI1I2=lI1α/2lI2α/2DI1,I2n+αI11/2I21/2,where the long distance D(I1,I2)=l(I1)+l(I2)+d(I1,I2), I1,I2∈Dn and α>0. Then for any xI1,yI2≥0, we have the following estimate: (25)∑I1,I2AI1I2xI1yI22≲∑I1xI12×∑I2yI22.In particular, it holds that (26)∑I2∑I1AI1I2xI12≲∑I1xI12.Lemma 5.
Let0<α≤n(λ1-2)/2. For a given cube I2∈Dn and (x1,t1)∈WI2, it holds (27)∫Rn∫I1dz1t1+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≲I1lI2+dI1,I2n+α.Proof.
Fixed(x1,t1)∈WI2. If y1≤1/2d(I1,I2), then(28)t1+x1-y1-z1≳lI2+x1-z1-y1≳lI2+dI1,I2.Thus (29)∫y1≤1/2dI1,I2∫I1dz1t1+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≲I1lI2+dI1,I2n+α.
Ify1>1/2d(I1,I2), then(30)t1t1+y1nλ11t1n≲lI2nλ1-nlI2+dI1,I2nλ1.Hence, (31)∫y1>1/2dI1,I2∫I1dz1t1+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≲lI2nλ1/2-n/2lI2+dI1,I2nλ1/2ψt1∗1I1L2Rn≲lI2nλ1/2-n/2lI2+dI1,I2nλ1/2ψt1L2RnI1≲lI2nλ1/2-n-αlI2+dI1,I2nλ1/2I1≲I1lI2+dI1,I2n+α,where ψt1(z1)=t1+z1-n-α and we have used the condition 0<α≤n(λ1-2)/2 in the last step.Now we turn our attention to the estimate ofG<,<. An easy consequence of the Hölder estimates of the kernel Kt1,t2 is that (32)θt1,t2hI1×J1x-y≲I1-1/2∫I1lI1αt1+x1-y1-z1n+αdz1×J1-1/2∫J1lJ1βt2+x2-y2-z2m+βdz2.Moreover, by Lemma 5, we obtain that(33)Px,t≔∬Rn+mθt1,t2hI1×J1x-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2m1/2≲I1-1/2∫Rn∫I1lI1αdz1t1+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2×J1-1/2∫Rm∫J1lJ1βdz2t2+x2-y2-z2m+β2t2t2+y2mλ2dy2t2n1/2≲lI1αlI2+dI1,I2n+αI11/2lJ1βlJ2+dJ1,J2m+βJ11/2.Since l(I1)<l(I2) and l(J1)<l(J2), then we get (34)Px,t≲AI1I2I2-1/2·AJ1J2J2-1/2.Therefore, from Minkowski’s integral inequality and Lemma 4, it now follows that (35)G<,<≲∑I2,J2:good∬WJ2∬WI2∑lI1<lI2lJ1<lJ2fI1J1∬Rn+mθt1,t2hI1×J1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2m1/22dx1dt1t1dx2dt2t2≲∑J2∑I2∑I1AI1I2∑J1AJ1J2fI1J12≲∑J2∑I1∑J1AJ1J2fI1J12≲∑I1∑J1fI1J12=fL2Rn+m2.


## 4. The Case:l(I1)≥l(I2) and lJ1 < lJ2

In any case, we perform the splitting(36)∑lI1≥lI2=∑lI1≥lI2dI1,I2>lI2γnlI11-γn+∑lI1>2rlI2dI1,I2≤lI2γnlI11-γn+∑lI2≤lI1≤2rlI2dI1,I2≤lI2γnlI11-γn.These three parts are called separated, Nested, and adjacent, respectively. The term Nested makes sense, since the summing conditions that I2 is good actually imply that I1 is the ancestor of I2. Thus, it holds (37)G≥,<≲Gsep,<+Gnes,<+Gadj,<,where (38)Gsep,<=∑I2,J2:good∬WJ2∬WI2∬Rn+m∑I1:lI1≥lI2dI1,I2>lI2γnlI11-γn∑J1:lJ1<lJ2fI1J1×θt1,t2hI1×J1x-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,Gnes,<=∑I2,J2:good∬WJ2∬WI2∬Rn+m∑I1:lI1>2rlI2dI1,I2≤lI2γnlI11-γn∑J1:lJ1<lJ2fI1J1×θt1,t2hI1×J1x-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,Gadj,<=∑I2,J2:good∬WJ2∬WI2∬Rn+m∑I1:lI2≤lI1≤2rlI2dI1,I2≤lI2γnlI11-γn∑J1:lJ1<lJ2fI1J1×θt1,t2hI1×J1x-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2.Now we are in position to estimate the above three terms, respectively.


### 4.1. Separated PartGsep,<

In this case, we note that the following inequality holds:(39)t1αlI2+dI1,I2n+αI11/2≲AI1I2I2-1/2.Indeed, if d(I1,I2)≥l(I1), then D(I1,I2)~d(I1,I2). Therefore, we get (40)t1αlI2+dI1,I2n+αI11/2≲AI1I2I2-1/2.If d(I1,I2)<l(I1), then D(I1,I2)~l(I1). Moreover, noticing that γn(n+α)=α/2 and d(I1,I2)>lI2γnlI11-γn, one may conclude that (41)t1αlI2+dI1,I2n+αI11/2≲lI1α/2lI2α/2lI1n+αI11/2≲AI1I2I2-1/2.It is obvious that the mixed Hölder and size condition implies that(42)θt1,t2hI1×J1x-y≲I1-1/2∫I1t1αt1+x1-y1-z1n+αdz1×J1-1/2∫J1lJ1βt2+x2-y2-z2m+βdz2.Thus, combining Lemma5 with (39), one can obtain(43)
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.Consequently, by the similar argument as G<,<, we have (44)Gsep,<≲fL2Rn+m2.


### 4.2. Adjacent PartGadj,<

The summation conditionsl(I2)≤l(I1)≤2rl(I2) and d(I1,I2)≤lI2γnlI11-γn indicate that l(I1)~l(I2)~D(I1,I2). Thus, (45)lI2αlI2+dI1,I2n+αI11/2≲lI2-n~lI1α/2lI2α/2DI1,I2n+α.It follows from (43) that (46)Px,t≲lI2αlI2+dI1,I2n+αI11/2·AJ1J2J2-1/2≲AI1I2I2-1/2·AJ1J2J2-1/2.Therefore, exactly as we have seen before, (47)Gadj,<≲fL2Rn+m2.


### 4.3. Nested PartGnes,<

We useI(k)∈Dn to denote the unique cube for which l(I(k))=2kl(I) and I⊂I(k). We call I(k) as the k generation older dyadic ancestor of I. In this case, by the goodness of I2, it must actually have I2⊊I1. That is, I1 is the ancestor of I2. This enables us to write (48)Gnes,<=∑I,J2:good∬WJ2∬WI∬Rn+m∑k=1∞∑J1:lJ1<lJ2fIkJ1θt1,t2hIk×J1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2.Introduce the notation(49)sIk=-1Ik-1chIkIk-1+∑I′∈chIkI′≠Ik-11I′hIk.Then, it is easy to check that(50)hIk=sIk+hIkIk-1,supp⁡sIk⊂Ik-1c, and sIk≲Ik-1/2.DenotefJ1=f,hJ1 so that fJ1=∫Rm‍f(y1,y2)dy2, y1∈Rn. Then, we split (51)Gnes,<≲Gmod,<+GCar,<,where(52)Gmod,<=∑I,J2:good∬WJ2∬WI∬Rn+m∑k=1∞∑J1:lJ1<lJ2fIkJ1θt1,t2sIk⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,GCar,<=∑I,J2:good∬WJ2∬WI∬Rn+m∑J1:lJ1<lJ2fIkJ1θt1,t21⊗hJ1x-y×∑k=1∞ΔIkfJ1Ik-12t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2=∑I,J2:good∬WJ2∬WI∬Rn+m∑J1:lJ1<lJ2fJ1Iθt1,t21⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2.


#### 4.3.1. Estimate ofGmod,<

We need the following lemma.Lemma 6.
Let0<β≤m(λ2-2)/2 and k∈N+. Given cubes I∈Dn, J1,J2∈Dm, (x1,t1)∈WI, and (x2,t2)∈WJ1, the following estimate holds: (53)Qx,t≔∬Rn+mθt1,t2sIk⊗hJ1x-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2m1/2≲2-αk/2Ik-1/2J11/2lJ1βlJ2+dJ1,J2m+β.Proof.
By the mixed Hölder and size condition, it yields that(54)θt1,t2sIk⊗hJ1x-y≲Ik-1/2∫Ik-1ct1αt1+x1-y1-z1n+αdz1×J1-1/2∫J1lJ1βt2+x2-y2-z2m+βdz2.Proceeding as we did in (33), we only need to show(55)K≔∫Rn∫Ik-1ct1αdz1t1+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≲2-αk/2.Indeed, if k≤r, (56)K≲∫Rn∫RnlIαdz1lI+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≲∫RnlIαlI+x1-z1n+αdz1≲lI-nI+lIα∫Icdz1z1-x1n+α≲1~2-αk/2.If k>r, we have by the goodness of I that (57)dI,Ik-1c>lIγnlIk-11-γn=2k-11-γnlI≳2k/2lI.Thus, we obtain (58)∫Ik-1clIαz1-x1n+αdz1≤∫Bx1,dI,Ik-1clIαz1-x1n+αdz1≲lIαdI,Ik-1c-α≲2-αk/2.Given y1∈Rn, we introduce the notation (59)E1=z1∈Ik-1c;z1-x1≥2y1,E2=z1∈Ik-1c;z1-x1<2y1.Then, it follows that (60)K≲∫Rn∫E1lIαdz1lI+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2+∫Rn∫E2lIαdz1lI+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≔K1+K2.Note that (61)lI+x1-y1-z1>z1-x1-y1≥12z1-x1,wheneverz1∈E1.This yields that (62)K1≲∫Ik-1clIαz1-x1n+αdz1·∫Rnt1t1+y1nλ1dy1t1n1/2≲2-αk/2.As for K2, we by Young’s inequality have the following estimate: (63)K2≲lIα∫Rn∫E2lInλ1/2-n/2lI+x1-z1nλ1/21t1+y1-z1n+αdz12dy11/2≤lIα∫Rn∫E2lIn/2+αlI+x1-z1n+α1t1+y1-z1n+αdz12dy11/2=lIn/2+2αξ∗ηL2Rn≤lIn/2+2αξL2RnηL1Rn≲lIα∫Ik-1clIαz1-x1n+αdz1≲2-αk/2,where(64)ξz1=1t1+z1n+α,ηz1=1E2z1t1+z1-x1n+α.Accordingly, Minkowski’s integral inequality and Lemma6 give that(65)Gmod,<≲∑I,J2:good∬WJ2∬WI∑k=1∞∑J1:lJ1<lJ2fIkJ1∬Rn+mθt1,t2sIk⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2m1/22dx1dt1t1dx2dt2t2≤∑I∑J2∬WJ2∬WI∑k=1∞2-αk/2∑J1:lJ1<lJ2AJ1J2fIkJ1Ik-1/2J11/22dx1dt1t1dx2dt2t2≲∑I∑J2∑k=1∞2-αk/2IIk∑J1:lJ1<lJ2AJ1J2fIkJ11/22≤∑k=1∞2-αk/4·2-αk/4∑IIIk∑J2∑J1:lJ1<lJ2AJ1J2fIkJ121/22≲∑k=1∞2-αk/2∑IIIk∑J2∑J1:lJ1<lJ2AJ1J2fIkJ12≲∑k=1∞2-αk/2∑Q,J1fQJ12Q∑I:Ik=QI≲fL2Rn+m2.


#### 4.3.2. Estimate ofGCar,<

We also need to use the following lemma.Lemma 7.
LetJ1,J2∈Dm be cubes, and (x2,t2)∈WJ2. Then the Carleson condition holds (66)Rx2,t2≔∑I′⊂I∬WI′∬Rn+mθt1,t21⊗hJ1x-y2∏i=12titi+yinλidy1t1ndy2t2mdx1dt1t1≲IlJ1βJ11/2lJ2+dJ1,J2m+β2.Proof.
The first step is to split(67)Rx2,t2≲R1x2,t2+R2x2,t2,where(68)R1x2,t2=∬3I^∬Rn+mθt1,t213I⊗hJ1x-y2∏i=12titi+yinλidy1dy2t1nt2mdx1dt1t1,R2x2,t2=∬I^∬Rn+mθt1,t213Ic⊗hJ1x-y2∏i=12titi+yinλidy1dy2t1nt2mdx1dt1t1≔∬I^Hx,tdx1dt1t1.
From combinations of Carleson and Hölder conditions and Lemma5, it follows that (69)R1x2,t2≲J1-1∫Rm∬3I^∫Rn∫J1∫3IKt1,t2x-y,z1,z2-Kt1,t2x-y,z1,z2+cJ1dz1dz22×t1t1+y1nλ1dy1t1ndx1dt1t1t2t2+y2mλ2dy2t2m≤J1-1∫Rm∫J1∬3I^∫Rn∫3IKt1,t2x-y,z1,z2-Kt1,t2x-y,z1,z2+cJ1dz12×t1t1+y1nλ1dy1t1ndx1dt1t11/2dz22t2t2+y2mλ2dy2t2m≲IJ1-1∫Rm∫J1lJ1βdz2t2+x2-y2-z2m+β2t2t2+y2mλ2dy2t2m≲IlJ1βJ11/2lJ2+dJ1,J2m+β2.
The mixed Hölder and size estimate gives that(70)θt1,t213Ic⊗hJ1x-y≲J1-1/2∫3Ict1αt1+x1-y1-z1n+αdz1×∫J1lJ1βt2+x2-y2-z2m+βdz2.Thus, collecting the estimates of Lemma 5 and (55), one can deduce that (71)Hx,t≲J1-1∫Rn∫3Ict1αdz1t1+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n×∫Rm∫J1lJ1βdz2t2+x2-y2-z2m+β2t2t2+y2mλ2dy2t2n≲t12αlI-2αlJ1βJ11/2lJ2+dJ1,J2m+β2.Therefore, we obtain(72)R2x2,t2=∬I^Hx,tdx1dt1t1≲IlI-2α∫0lIt12α-1dt1·lJ1βJ11/2lJ2+dJ1,J2m+β2≲IlJ1βJ11/2lJ2+dJ1,J2m+β2.Thus, we finish the proof of Lemma 7.Now we boundGCar,<. If l(J1)<l(J2), then we have (73)Rx2,t2≲IAJ1,J2J2-1/22.Therefore, we obtain the following estimate (74)GCar,<=∑J2:good∬WJ2∑I:good∬WI∬Rn+m∑J1:lJ1<lJ2fJ1Iθt1,t21⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2≤∑J2∬WJ2∑I∑J1:lJ1<lJ2∬WI∬Rn+mfJ1Iθt1,t21⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t11/22dx2dt2t2≤∑J2∬WJ2∑J1:lJ1<lJ2∑IfJ1I2∬WI∬Rn+mθt1,t21⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t11/22dx2dt2t2≲∑J2∑J1:lJ1<lJ2AJ1J2fJ1L2Rn2≲∑J1fJ1L2Rn2≲fL2Rn+m2.So far, we have completed the estimate of G≥,<.As for the termG<,≥, 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.


## 4.1. Separated PartGsep,<

In this case, we note that the following inequality holds:(39)t1αlI2+dI1,I2n+αI11/2≲AI1I2I2-1/2.Indeed, if d(I1,I2)≥l(I1), then D(I1,I2)~d(I1,I2). Therefore, we get (40)t1αlI2+dI1,I2n+αI11/2≲AI1I2I2-1/2.If d(I1,I2)<l(I1), then D(I1,I2)~l(I1). Moreover, noticing that γn(n+α)=α/2 and d(I1,I2)>lI2γnlI11-γn, one may conclude that (41)t1αlI2+dI1,I2n+αI11/2≲lI1α/2lI2α/2lI1n+αI11/2≲AI1I2I2-1/2.It is obvious that the mixed Hölder and size condition implies that(42)θt1,t2hI1×J1x-y≲I1-1/2∫I1t1αt1+x1-y1-z1n+αdz1×J1-1/2∫J1lJ1βt2+x2-y2-z2m+βdz2.Thus, combining Lemma5 with (39), one can obtain(43)
P
x
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t
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I
1
-
1
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∫
R
n
∫
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1
t
1
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2
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1
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1
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m
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J
1
l
J
1
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d
z
2
t
2
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x
2
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y
2
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2
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+
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2
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2
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2
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2
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2
J
2
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1
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2
.Consequently, by the similar argument as G<,<, we have (44)Gsep,<≲fL2Rn+m2.


## 4.2. Adjacent PartGadj,<

The summation conditionsl(I2)≤l(I1)≤2rl(I2) and d(I1,I2)≤lI2γnlI11-γn indicate that l(I1)~l(I2)~D(I1,I2). Thus, (45)lI2αlI2+dI1,I2n+αI11/2≲lI2-n~lI1α/2lI2α/2DI1,I2n+α.It follows from (43) that (46)Px,t≲lI2αlI2+dI1,I2n+αI11/2·AJ1J2J2-1/2≲AI1I2I2-1/2·AJ1J2J2-1/2.Therefore, exactly as we have seen before, (47)Gadj,<≲fL2Rn+m2.


## 4.3. Nested PartGnes,<

We useI(k)∈Dn to denote the unique cube for which l(I(k))=2kl(I) and I⊂I(k). We call I(k) as the k generation older dyadic ancestor of I. In this case, by the goodness of I2, it must actually have I2⊊I1. That is, I1 is the ancestor of I2. This enables us to write (48)Gnes,<=∑I,J2:good∬WJ2∬WI∬Rn+m∑k=1∞∑J1:lJ1<lJ2fIkJ1θt1,t2hIk×J1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2.Introduce the notation(49)sIk=-1Ik-1chIkIk-1+∑I′∈chIkI′≠Ik-11I′hIk.Then, it is easy to check that(50)hIk=sIk+hIkIk-1,supp⁡sIk⊂Ik-1c, and sIk≲Ik-1/2.DenotefJ1=f,hJ1 so that fJ1=∫Rm‍f(y1,y2)dy2, y1∈Rn. Then, we split (51)Gnes,<≲Gmod,<+GCar,<,where(52)Gmod,<=∑I,J2:good∬WJ2∬WI∬Rn+m∑k=1∞∑J1:lJ1<lJ2fIkJ1θt1,t2sIk⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,GCar,<=∑I,J2:good∬WJ2∬WI∬Rn+m∑J1:lJ1<lJ2fIkJ1θt1,t21⊗hJ1x-y×∑k=1∞ΔIkfJ1Ik-12t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2=∑I,J2:good∬WJ2∬WI∬Rn+m∑J1:lJ1<lJ2fJ1Iθt1,t21⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2.


### 4.3.1. Estimate ofGmod,<

We need the following lemma.Lemma 6.
Let0<β≤m(λ2-2)/2 and k∈N+. Given cubes I∈Dn, J1,J2∈Dm, (x1,t1)∈WI, and (x2,t2)∈WJ1, the following estimate holds: (53)Qx,t≔∬Rn+mθt1,t2sIk⊗hJ1x-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2m1/2≲2-αk/2Ik-1/2J11/2lJ1βlJ2+dJ1,J2m+β.Proof.
By the mixed Hölder and size condition, it yields that(54)θt1,t2sIk⊗hJ1x-y≲Ik-1/2∫Ik-1ct1αt1+x1-y1-z1n+αdz1×J1-1/2∫J1lJ1βt2+x2-y2-z2m+βdz2.Proceeding as we did in (33), we only need to show(55)K≔∫Rn∫Ik-1ct1αdz1t1+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≲2-αk/2.Indeed, if k≤r, (56)K≲∫Rn∫RnlIαdz1lI+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≲∫RnlIαlI+x1-z1n+αdz1≲lI-nI+lIα∫Icdz1z1-x1n+α≲1~2-αk/2.If k>r, we have by the goodness of I that (57)dI,Ik-1c>lIγnlIk-11-γn=2k-11-γnlI≳2k/2lI.Thus, we obtain (58)∫Ik-1clIαz1-x1n+αdz1≤∫Bx1,dI,Ik-1clIαz1-x1n+αdz1≲lIαdI,Ik-1c-α≲2-αk/2.Given y1∈Rn, we introduce the notation (59)E1=z1∈Ik-1c;z1-x1≥2y1,E2=z1∈Ik-1c;z1-x1<2y1.Then, it follows that (60)K≲∫Rn∫E1lIαdz1lI+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2+∫Rn∫E2lIαdz1lI+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≔K1+K2.Note that (61)lI+x1-y1-z1>z1-x1-y1≥12z1-x1,wheneverz1∈E1.This yields that (62)K1≲∫Ik-1clIαz1-x1n+αdz1·∫Rnt1t1+y1nλ1dy1t1n1/2≲2-αk/2.As for K2, we by Young’s inequality have the following estimate: (63)K2≲lIα∫Rn∫E2lInλ1/2-n/2lI+x1-z1nλ1/21t1+y1-z1n+αdz12dy11/2≤lIα∫Rn∫E2lIn/2+αlI+x1-z1n+α1t1+y1-z1n+αdz12dy11/2=lIn/2+2αξ∗ηL2Rn≤lIn/2+2αξL2RnηL1Rn≲lIα∫Ik-1clIαz1-x1n+αdz1≲2-αk/2,where(64)ξz1=1t1+z1n+α,ηz1=1E2z1t1+z1-x1n+α.Accordingly, Minkowski’s integral inequality and Lemma6 give that(65)Gmod,<≲∑I,J2:good∬WJ2∬WI∑k=1∞∑J1:lJ1<lJ2fIkJ1∬Rn+mθt1,t2sIk⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2m1/22dx1dt1t1dx2dt2t2≤∑I∑J2∬WJ2∬WI∑k=1∞2-αk/2∑J1:lJ1<lJ2AJ1J2fIkJ1Ik-1/2J11/22dx1dt1t1dx2dt2t2≲∑I∑J2∑k=1∞2-αk/2IIk∑J1:lJ1<lJ2AJ1J2fIkJ11/22≤∑k=1∞2-αk/4·2-αk/4∑IIIk∑J2∑J1:lJ1<lJ2AJ1J2fIkJ121/22≲∑k=1∞2-αk/2∑IIIk∑J2∑J1:lJ1<lJ2AJ1J2fIkJ12≲∑k=1∞2-αk/2∑Q,J1fQJ12Q∑I:Ik=QI≲fL2Rn+m2.


### 4.3.2. Estimate ofGCar,<

We also need to use the following lemma.Lemma 7.
LetJ1,J2∈Dm be cubes, and (x2,t2)∈WJ2. Then the Carleson condition holds (66)Rx2,t2≔∑I′⊂I∬WI′∬Rn+mθt1,t21⊗hJ1x-y2∏i=12titi+yinλidy1t1ndy2t2mdx1dt1t1≲IlJ1βJ11/2lJ2+dJ1,J2m+β2.Proof.
The first step is to split(67)Rx2,t2≲R1x2,t2+R2x2,t2,where(68)R1x2,t2=∬3I^∬Rn+mθt1,t213I⊗hJ1x-y2∏i=12titi+yinλidy1dy2t1nt2mdx1dt1t1,R2x2,t2=∬I^∬Rn+mθt1,t213Ic⊗hJ1x-y2∏i=12titi+yinλidy1dy2t1nt2mdx1dt1t1≔∬I^Hx,tdx1dt1t1.
From combinations of Carleson and Hölder conditions and Lemma5, it follows that (69)R1x2,t2≲J1-1∫Rm∬3I^∫Rn∫J1∫3IKt1,t2x-y,z1,z2-Kt1,t2x-y,z1,z2+cJ1dz1dz22×t1t1+y1nλ1dy1t1ndx1dt1t1t2t2+y2mλ2dy2t2m≤J1-1∫Rm∫J1∬3I^∫Rn∫3IKt1,t2x-y,z1,z2-Kt1,t2x-y,z1,z2+cJ1dz12×t1t1+y1nλ1dy1t1ndx1dt1t11/2dz22t2t2+y2mλ2dy2t2m≲IJ1-1∫Rm∫J1lJ1βdz2t2+x2-y2-z2m+β2t2t2+y2mλ2dy2t2m≲IlJ1βJ11/2lJ2+dJ1,J2m+β2.
The mixed Hölder and size estimate gives that(70)θt1,t213Ic⊗hJ1x-y≲J1-1/2∫3Ict1αt1+x1-y1-z1n+αdz1×∫J1lJ1βt2+x2-y2-z2m+βdz2.Thus, collecting the estimates of Lemma 5 and (55), one can deduce that (71)Hx,t≲J1-1∫Rn∫3Ict1αdz1t1+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n×∫Rm∫J1lJ1βdz2t2+x2-y2-z2m+β2t2t2+y2mλ2dy2t2n≲t12αlI-2αlJ1βJ11/2lJ2+dJ1,J2m+β2.Therefore, we obtain(72)R2x2,t2=∬I^Hx,tdx1dt1t1≲IlI-2α∫0lIt12α-1dt1·lJ1βJ11/2lJ2+dJ1,J2m+β2≲IlJ1βJ11/2lJ2+dJ1,J2m+β2.Thus, we finish the proof of Lemma 7.Now we boundGCar,<. If l(J1)<l(J2), then we have (73)Rx2,t2≲IAJ1,J2J2-1/22.Therefore, we obtain the following estimate (74)GCar,<=∑J2:good∬WJ2∑I:good∬WI∬Rn+m∑J1:lJ1<lJ2fJ1Iθt1,t21⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2≤∑J2∬WJ2∑I∑J1:lJ1<lJ2∬WI∬Rn+mfJ1Iθt1,t21⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t11/22dx2dt2t2≤∑J2∬WJ2∑J1:lJ1<lJ2∑IfJ1I2∬WI∬Rn+mθt1,t21⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t11/22dx2dt2t2≲∑J2∑J1:lJ1<lJ2AJ1J2fJ1L2Rn2≲∑J1fJ1L2Rn2≲fL2Rn+m2.So far, we have completed the estimate of G≥,<.As for the termG<,≥, 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.


## 4.3.1. Estimate ofGmod,<

We need the following lemma.Lemma 6.
Let0<β≤m(λ2-2)/2 and k∈N+. Given cubes I∈Dn, J1,J2∈Dm, (x1,t1)∈WI, and (x2,t2)∈WJ1, the following estimate holds: (53)Qx,t≔∬Rn+mθt1,t2sIk⊗hJ1x-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2m1/2≲2-αk/2Ik-1/2J11/2lJ1βlJ2+dJ1,J2m+β.Proof.
By the mixed Hölder and size condition, it yields that(54)θt1,t2sIk⊗hJ1x-y≲Ik-1/2∫Ik-1ct1αt1+x1-y1-z1n+αdz1×J1-1/2∫J1lJ1βt2+x2-y2-z2m+βdz2.Proceeding as we did in (33), we only need to show(55)K≔∫Rn∫Ik-1ct1αdz1t1+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≲2-αk/2.Indeed, if k≤r, (56)K≲∫Rn∫RnlIαdz1lI+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≲∫RnlIαlI+x1-z1n+αdz1≲lI-nI+lIα∫Icdz1z1-x1n+α≲1~2-αk/2.If k>r, we have by the goodness of I that (57)dI,Ik-1c>lIγnlIk-11-γn=2k-11-γnlI≳2k/2lI.Thus, we obtain (58)∫Ik-1clIαz1-x1n+αdz1≤∫Bx1,dI,Ik-1clIαz1-x1n+αdz1≲lIαdI,Ik-1c-α≲2-αk/2.Given y1∈Rn, we introduce the notation (59)E1=z1∈Ik-1c;z1-x1≥2y1,E2=z1∈Ik-1c;z1-x1<2y1.Then, it follows that (60)K≲∫Rn∫E1lIαdz1lI+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2+∫Rn∫E2lIαdz1lI+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n1/2≔K1+K2.Note that (61)lI+x1-y1-z1>z1-x1-y1≥12z1-x1,wheneverz1∈E1.This yields that (62)K1≲∫Ik-1clIαz1-x1n+αdz1·∫Rnt1t1+y1nλ1dy1t1n1/2≲2-αk/2.As for K2, we by Young’s inequality have the following estimate: (63)K2≲lIα∫Rn∫E2lInλ1/2-n/2lI+x1-z1nλ1/21t1+y1-z1n+αdz12dy11/2≤lIα∫Rn∫E2lIn/2+αlI+x1-z1n+α1t1+y1-z1n+αdz12dy11/2=lIn/2+2αξ∗ηL2Rn≤lIn/2+2αξL2RnηL1Rn≲lIα∫Ik-1clIαz1-x1n+αdz1≲2-αk/2,where(64)ξz1=1t1+z1n+α,ηz1=1E2z1t1+z1-x1n+α.Accordingly, Minkowski’s integral inequality and Lemma6 give that(65)Gmod,<≲∑I,J2:good∬WJ2∬WI∑k=1∞∑J1:lJ1<lJ2fIkJ1∬Rn+mθt1,t2sIk⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2m1/22dx1dt1t1dx2dt2t2≤∑I∑J2∬WJ2∬WI∑k=1∞2-αk/2∑J1:lJ1<lJ2AJ1J2fIkJ1Ik-1/2J11/22dx1dt1t1dx2dt2t2≲∑I∑J2∑k=1∞2-αk/2IIk∑J1:lJ1<lJ2AJ1J2fIkJ11/22≤∑k=1∞2-αk/4·2-αk/4∑IIIk∑J2∑J1:lJ1<lJ2AJ1J2fIkJ121/22≲∑k=1∞2-αk/2∑IIIk∑J2∑J1:lJ1<lJ2AJ1J2fIkJ12≲∑k=1∞2-αk/2∑Q,J1fQJ12Q∑I:Ik=QI≲fL2Rn+m2.


## 4.3.2. Estimate ofGCar,<

We also need to use the following lemma.Lemma 7.
LetJ1,J2∈Dm be cubes, and (x2,t2)∈WJ2. Then the Carleson condition holds (66)Rx2,t2≔∑I′⊂I∬WI′∬Rn+mθt1,t21⊗hJ1x-y2∏i=12titi+yinλidy1t1ndy2t2mdx1dt1t1≲IlJ1βJ11/2lJ2+dJ1,J2m+β2.Proof.
The first step is to split(67)Rx2,t2≲R1x2,t2+R2x2,t2,where(68)R1x2,t2=∬3I^∬Rn+mθt1,t213I⊗hJ1x-y2∏i=12titi+yinλidy1dy2t1nt2mdx1dt1t1,R2x2,t2=∬I^∬Rn+mθt1,t213Ic⊗hJ1x-y2∏i=12titi+yinλidy1dy2t1nt2mdx1dt1t1≔∬I^Hx,tdx1dt1t1.
From combinations of Carleson and Hölder conditions and Lemma5, it follows that (69)R1x2,t2≲J1-1∫Rm∬3I^∫Rn∫J1∫3IKt1,t2x-y,z1,z2-Kt1,t2x-y,z1,z2+cJ1dz1dz22×t1t1+y1nλ1dy1t1ndx1dt1t1t2t2+y2mλ2dy2t2m≤J1-1∫Rm∫J1∬3I^∫Rn∫3IKt1,t2x-y,z1,z2-Kt1,t2x-y,z1,z2+cJ1dz12×t1t1+y1nλ1dy1t1ndx1dt1t11/2dz22t2t2+y2mλ2dy2t2m≲IJ1-1∫Rm∫J1lJ1βdz2t2+x2-y2-z2m+β2t2t2+y2mλ2dy2t2m≲IlJ1βJ11/2lJ2+dJ1,J2m+β2.
The mixed Hölder and size estimate gives that(70)θt1,t213Ic⊗hJ1x-y≲J1-1/2∫3Ict1αt1+x1-y1-z1n+αdz1×∫J1lJ1βt2+x2-y2-z2m+βdz2.Thus, collecting the estimates of Lemma 5 and (55), one can deduce that (71)Hx,t≲J1-1∫Rn∫3Ict1αdz1t1+x1-y1-z1n+α2t1t1+y1nλ1dy1t1n×∫Rm∫J1lJ1βdz2t2+x2-y2-z2m+β2t2t2+y2mλ2dy2t2n≲t12αlI-2αlJ1βJ11/2lJ2+dJ1,J2m+β2.Therefore, we obtain(72)R2x2,t2=∬I^Hx,tdx1dt1t1≲IlI-2α∫0lIt12α-1dt1·lJ1βJ11/2lJ2+dJ1,J2m+β2≲IlJ1βJ11/2lJ2+dJ1,J2m+β2.Thus, we finish the proof of Lemma 7.Now we boundGCar,<. If l(J1)<l(J2), then we have (73)Rx2,t2≲IAJ1,J2J2-1/22.Therefore, we obtain the following estimate (74)GCar,<=∑J2:good∬WJ2∑I:good∬WI∬Rn+m∑J1:lJ1<lJ2fJ1Iθt1,t21⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2≤∑J2∬WJ2∑I∑J1:lJ1<lJ2∬WI∬Rn+mfJ1Iθt1,t21⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t11/22dx2dt2t2≤∑J2∬WJ2∑J1:lJ1<lJ2∑IfJ1I2∬WI∬Rn+mθt1,t21⊗hJ1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t11/22dx2dt2t2≲∑J2∑J1:lJ1<lJ2AJ1J2fJ1L2Rn2≲∑J1fJ1L2Rn2≲fL2Rn+m2.So far, we have completed the estimate of G≥,<.As for the termG<,≥, 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.


## 5. The Case:l(I1)≥l(I2) and l(J1)≥l(J2)

Similar to what we have done before, the summationl(I1)≥l(I2) was decomposed into the separated, Nested, and adjacent terms. A similar splitting in the summation l(J1)≥l(J2) is also performed. This splits the whole summation into nine parts as follows:(75)G≥,≥≲Gsep,sep+Gsep,nes+Gsep,adj+Gnes,sep+Gnes,nes+Gnes,adj+Gadj,sep+Gadj,nes+Gadj,adj.


### 5.1. Nested/Nested:Gnes,nes

We begin with the termGnes,nes, 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 ofhI(k) in (50) gives that (76)Gnes,nes≲Gmod,mod+GCar,Car+Gmod,Car+GCar,mod,where(77)Gmod,mod=∑I,J:good∬WJ∬WI∬Rn+m∑k=1∞∑i=1∞fIkJiθt1,t2sIk⊗sJix-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,Gmod,Car=∑I,J:good∬WJ∬WI∬Rn+m∑k=1∞fIkJθt1,t2sIk⊗1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,GCar,mod=∑I,J:good∬WJ∬WI∬Rn+m∑l=1∞fJiIθt1,t21⊗sJix-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,GCar,Car=∑I,J:goodfI×J2∬WJ∬WI∬Rn+mθt1,t21x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2.


#### 5.1.1. Estimate ofGmod,mod

We proceed using the standard argument as in Lemma6. The size condition and (55) lead to the bound (78)∬Rn+mθt1,t2sIk⊗sJix-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2m1/2≲2-αk/2Ik-1/2·2-βiJi-1/2.It is similar to estimate Gmod,< to analyze Gmod,mod.(79)Gmod,mod≲∑k,i2-αk/22-βi∑Q,RfQR21Q∑I:Ik=QI·1R∑J:Ji=RJ≲fL2Rn+m2.


#### 5.1.2. Estimate ofGCar,Car

Applying the biparameter Carleson condition, it immediately yields that(80)GCar,Car=∑I,JfI×J2CIJD=2∫0∞∑I,JfI×J>tCIJDtdt≲∫0∞∑I,JI×J⊂MDf>tCIJDtdt≲∫0∞MDf>ttdt≲MDfL2Rn+m2≲fL2Rn+m2,where in the last step we have used the Lp(1<p<∞) boundedness of the strong maximal function associated with rectangles.


#### 5.1.3. Estimate ofGCar,mod and Gmod,Car

Lemma 8.
LetJ∈Dm,good,  (x2,t2)∈WJ2 and i∈N be fixed. Then the Carleson condition is satisfied as follows: (81)∑I′⊂I∬WI′∬Rn+mθt1,t21⊗sJix-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1≲2-βiI·Ji-1.Proof.
The proof of Lemma8 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,GCar,mod is bounded as below. (82)GCar,mod≤∑J:good∬WJ∑I∑i=1∞fJiI∬WI∬Rn+mθt1,t21⊗sJix-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t11/22dx2dt2t2≲∑i=1∞2-βi/2∑RfRL2Rn21R∑J:Ji=RI≲fL2Rn+m2.


### 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 dominateP(x,t), Q(x,t), and R(x2,t2). 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 I1, I2, J1, and J2. 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 forGsep,sep, Gsep,adj, Gadj,adj, and Gadj,sep directly. Finally, for the terms Gnes,sep and Gnes,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 Gsep,nes and Gadj,nes are symmetric with respect to them, respectively.


## 5.1. Nested/Nested:Gnes,nes

We begin with the termGnes,nes, 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 ofhI(k) in (50) gives that (76)Gnes,nes≲Gmod,mod+GCar,Car+Gmod,Car+GCar,mod,where(77)Gmod,mod=∑I,J:good∬WJ∬WI∬Rn+m∑k=1∞∑i=1∞fIkJiθt1,t2sIk⊗sJix-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,Gmod,Car=∑I,J:good∬WJ∬WI∬Rn+m∑k=1∞fIkJθt1,t2sIk⊗1x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,GCar,mod=∑I,J:good∬WJ∬WI∬Rn+m∑l=1∞fJiIθt1,t21⊗sJix-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2,GCar,Car=∑I,J:goodfI×J2∬WJ∬WI∬Rn+mθt1,t21x-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2.


### 5.1.1. Estimate ofGmod,mod

We proceed using the standard argument as in Lemma6. The size condition and (55) lead to the bound (78)∬Rn+mθt1,t2sIk⊗sJix-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2m1/2≲2-αk/2Ik-1/2·2-βiJi-1/2.It is similar to estimate Gmod,< to analyze Gmod,mod.(79)Gmod,mod≲∑k,i2-αk/22-βi∑Q,RfQR21Q∑I:Ik=QI·1R∑J:Ji=RJ≲fL2Rn+m2.


### 5.1.2. Estimate ofGCar,Car

Applying the biparameter Carleson condition, it immediately yields that(80)GCar,Car=∑I,JfI×J2CIJD=2∫0∞∑I,JfI×J>tCIJDtdt≲∫0∞∑I,JI×J⊂MDf>tCIJDtdt≲∫0∞MDf>ttdt≲MDfL2Rn+m2≲fL2Rn+m2,where in the last step we have used the Lp(1<p<∞) boundedness of the strong maximal function associated with rectangles.


### 5.1.3. Estimate ofGCar,mod and Gmod,Car

Lemma 8.
LetJ∈Dm,good,  (x2,t2)∈WJ2 and i∈N be fixed. Then the Carleson condition is satisfied as follows: (81)∑I′⊂I∬WI′∬Rn+mθt1,t21⊗sJix-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1≲2-βiI·Ji-1.Proof.
The proof of Lemma8 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,GCar,mod is bounded as below. (82)GCar,mod≤∑J:good∬WJ∑I∑i=1∞fJiI∬WI∬Rn+mθt1,t21⊗sJix-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t11/22dx2dt2t2≲∑i=1∞2-βi/2∑RfRL2Rn21R∑J:Ji=RI≲fL2Rn+m2.


## 5.1.1. Estimate ofGmod,mod

We proceed using the standard argument as in Lemma6. The size condition and (55) lead to the bound (78)∬Rn+mθt1,t2sIk⊗sJix-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2m1/2≲2-αk/2Ik-1/2·2-βiJi-1/2.It is similar to estimate Gmod,< to analyze Gmod,mod.(79)Gmod,mod≲∑k,i2-αk/22-βi∑Q,RfQR21Q∑I:Ik=QI·1R∑J:Ji=RJ≲fL2Rn+m2.


## 5.1.2. Estimate ofGCar,Car

Applying the biparameter Carleson condition, it immediately yields that(80)GCar,Car=∑I,JfI×J2CIJD=2∫0∞∑I,JfI×J>tCIJDtdt≲∫0∞∑I,JI×J⊂MDf>tCIJDtdt≲∫0∞MDf>ttdt≲MDfL2Rn+m2≲fL2Rn+m2,where in the last step we have used the Lp(1<p<∞) boundedness of the strong maximal function associated with rectangles.


## 5.1.3. Estimate ofGCar,mod and Gmod,Car

Lemma 8.
LetJ∈Dm,good,  (x2,t2)∈WJ2 and i∈N be fixed. Then the Carleson condition is satisfied as follows: (81)∑I′⊂I∬WI′∬Rn+mθt1,t21⊗sJix-y2t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t1≲2-βiI·Ji-1.Proof.
The proof of Lemma8 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,GCar,mod is bounded as below. (82)GCar,mod≤∑J:good∬WJ∑I∑i=1∞fJiI∬WI∬Rn+mθt1,t21⊗sJix-y2×t1t1+y1nλ1t2t2+y2mλ2dy1t1ndy2t2mdx1dt1t11/22dx2dt2t2≲∑i=1∞2-βi/2∑RfRL2Rn21R∑J:Ji=RI≲fL2Rn+m2.


## 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 dominateP(x,t), Q(x,t), and R(x2,t2). 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 I1, I2, J1, and J2. 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 forGsep,sep, Gsep,adj, Gadj,adj, and Gadj,sep directly. Finally, for the terms Gnes,sep and Gnes,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 Gsep,nes and Gadj,nes are symmetric with respect to them, respectively.


## 6. The Necessity of Biparameter Carleson Condition

We here show that the biparameter Carleson condition is necessary forgλ1,λ2∗-function bound on L2(Rn+m).Suppose thatθt1,t2=θt1n⊗θt2m is bounded on L2(Rn+m), where θt1n has a kernel st1n(x1,y1) and θt2n has a kernel st2m(x2,y2),  x1,y1∈Rn,  x2,y2∈Rm, and  t1,t2>0. We assume that these satisfy the size condition and the corresponding L2 bounds in Rn and Rm. We shall show that the biparameter Carleson condition (12) holds.DefineΩ~=MD1Ω>1/2 and Ω^=M1Ω~>c for a small enough dimensional constant c=c(n,m), where MD denotes the strong maximal function related to the grid D and M denotes the strong maximal function. From the endpoint estimates for M and MD, it follows that Ω^≲Ω~≲Ω. Hence, it is enough to show that (83)∑I×J∈DI×J⊂Ω∬WJ∬WI∬Rn+mθt1,t21Ω^cy1,y22t1t1+x1-y1nλ1×t2t2+x2-y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2≲Ω.For everyJ∈Dm we let FJ consist of the maximal F∈Dn for which F×J⊂Ω~. Then we define FJ≔⋃F∈FJ2F. Moreover, for fixed I∈Dn, let GI be the family of the maximal G∈Dm, for which I×G⊂Ω, and IG∈Dn be the maximal cube for which IG⊃I and IG×G⊂Ω~. So, we only need to show the following inequalities:(84)G1=∑I×J∈DI×J⊂Ω∬WJ∬WI∬Rn+mθt1,t21Ω^c1FJy1,y22t1t1+x1-y1nλ1×t2t2+x2-y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2≔∑J∬WJGJx2,t2dx2dt2t2≲Ω,(85)G2=∑I×J∈DI×J⊂Ω∬WJ∬WI∬Rn+mθt1,t21Ω^c1FJcy1,y22t1t1+x1-y1nλ1×t2t2+x2-y2mλ2dy1t1ndy2t2mdx1dt1t1dx2dt2t2≔∑I∬WIGIx1,t1dx1dt1t1≲Ω.To attain the goal, we need to first estimate GJ(x2,t2) and GI(x1,t1). Actually, Minkowski’s integral inequality and size estimate yield that(86)GJx2,t2≲∫Rm∬R+n+1∬Rn+mKt2my2,z22θt1,t21Ω^c1FJy12t1t1+x1-y1nλ1×t2t2+x2-y2mλ2dy1t1ndy2t2mdx1dt1t11/2dz22≲∫Rm∫Rmt2βt2+y2-z2m+β2t2t2+x2-y2mλ2dy2t2m1/2×1Ω^c·,z21FJL2Rndz22≲∫RmlJβlJ+x2-z2m+β1Ω^c·,z21FJL2Rndz22≲∫RmlJβlJ+x2-z2m+β1Ω^c·,z21FJL2Rn2dz2≲∫RmlJβz2-cJm+β1Ω^c·,z21FJL2Rn2dz2=∫Rn1FJz1∫RmlJβz2-cJm+β1Ω^cz1,z2dz2dz1.Similarly, we may estimate (87)GIx1,t1=∑G∈GI∑J:J⊂G∬WJ∬Rn+mθt1,t21Ω^c1FJcy1,y22t1t1+x1-y1nλ1×t2t2+x2-y2mλ2dy1t1ndy2t2mdx2dt2t2≲∫RnlIαlI+x1-z1n+α∑G∈GI12IGcz1G1/2dz12≲∑G∈GIG∫RnlIαlI+x1-z1n+α12IGcz1dz1≲∑G∈GIG∫IGclIαz1-cIGn+αdz1≲∑G∈GIGlIlIGα.The remaining calculation is a routine application of the idea of [14]. We here omit the details. Finally, we obtain (88)G1=∑J∬WJGJx2,t2dx2dt2t2≲Ω,G2=∑I∬WIGIx1,t1dx1dt1t1≲Ω.Thus, we have proved the necessity.



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*Source: 3605639-2016-12-22.xml*