# Sharp Weighted Bounds for Multilinear Fractional Type Operators Associated with Bergman Projection

Authors: Juan Zhang; Senhua Lan; Qingying Xue
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We first introduce the multiple weights which are suitable for the study of Bergman type operators. Then, we give the sharp weighted estimates for multilinear fractional Bergman operators and fractional maximal function.


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## Body

## 1. Introduction

LetdV be the area measure in the upper half plane H={z∈C:Imz>0} and dVα(z)=(Imz)αdV(z) for α>-1. Then, the Bergman space Vαp(H) (1<p<∞) is defined to be the space of analytic functions in Lp(dVα). The well-known Bergman projection Pα and its maximal operator Pα+ have the following integral representations: (1)Pαfz=cα∫Hfξz-ξ¯2+αdVαξ,Pα+fz=∫Hfξz-ξ¯2+αdVαξ,where cα is an appropriate real constant. It is well known that [1] both Pα and Pα+ map Lp(dVα) boundedly onto Vαp(H) for 1<p<∞. The boundedness of Pα and Pα+ in weighted Lp spaces was studied by Békollé and Bonami in [2, 3]. They showed that the following assertions are equivalent: (i)
Pα: Lp(ωdVα)↦Lp(ωdVα),(ii)
Pα+: Lp(ωdVα)↦Lp(ωdVα),(iii)
ω∈Bp,α,whereBp,α is the family of positive locally integrable functions for which the following inequality holds: (2)ωBp,α≔supI⊂R1QIα∫QIωzdVαz1QIα∫QIωz1-p′dVαzp-1<∞.Here QI is denoted to be the Carleson cube associated with the interval I (see (18)) and |QI|α≔∫QIdVα≈|QI|1+α/2. Later on, for 1<p<∞, Pott and Reguera [4] demonstrated the sharp Békollé estimates for the maximal Bergman projection as follows:(3)Pα+fLpωdVα≤CωBp,αmax1,1/p-1fLpωdVα,where the constant C depends only on p, α and the estimate is sharp in the sense that the power of [ω]Bp,α is optimal. Since then, several nice works have been done in the direction of the sharp estimate of Bergman type operators, such as [5–7].It is easy to see that the maximal Bergman operatorPα+ coincides with the following operators in the case of γ=0: (4)Tα,γfz≔∫Hfξz-ξ¯2+α-γdVαξ,α>-1,0≤γ<2+α.The operator Tα,γ and the following operator Mα,γ appear naturally in the problem of off-diagonal weighted inequalities for the Bergman operator [8].(5)Mα,γfz≔supI⊂RχQIzQIα1-γ/2+α∫QIfξdVαξ.Obviously, Mα,γ is pointwise dominated by Tα,γ.In order to state some known results, we first introduce the weights classBp,q,α. Let ω be a weight function and let 1<p≤q<∞; we say ω∈Bp,q,α if the following quantity is finite: (6)ωBp,q,α≔supI⊂R1QIα∫QIωzqdVαz1QIα∫QIωz-p′dVαzq/p′.The weights class Bp,q,α is an extension and an analogue of the Ap,q weight class introduced by Muckenhoupt and Wheeden in [9]. Recently, Sehba [8] showed that Pα+ is bounded from Lp(H,ωpdVα) into Lq(H,ωqdVα+(2+α)(q/p-1)) if and only if ω∈Bp,q,α(H) for α>-1, 1<p≤q<∞, qp≥p′2 or p′q′≥q2. Moreover, it holds that(7)ωPα+fLpH,yα+2+αq/p-1dxdy≲ωBp,q,α1/p′+1/qmax⁡1,p′/qωfLpH,yαdxdy,and the power of [ω]Bp,q,α in the above inequality is sharp. In the case p=q, one may note that formula (7) coincides with formula (3). We summarize the results in [8, 10] for the estimates of Tα,γ and Mα,γ as follows.Theorem A (see [8]).
Letα>-1,  0≤γ<2+α,  1<p≤q<∞,  1/q=1/p-γ/2+α, and min{p′/q,q/p′}≤1-γ/2+α. Then Tα,γ is bounded from Lp(ωpdVα) into Lq(ωqdVα) if and only if ω∈Bp,q,α. Moreover, (8)Tα,γLpωpdVα→LqωqdVα≤cωBp,q,α1-γ/2+αmax⁡1,p′/q and the power of [ω]Bp,q,α is sharp.Theorem B (see [10]).
Letα>-1,  0≤γ<2+α,  1<p<2+α/γ, and 1/q=1/p-γ/2+α. If ω∈Bp,q,α then (9)ωMα,γfq,α≲ωBp,q,αp′/q1-γ/2+αωfp,α, where the exponent of [ω]Bp,q,α is sharp.In order to state our results, we first introduce some definitions.Definition 1 (multilinear weightsBp→,q,α).
Let1<p1,…,pm<∞ and α>-1. Suppose that each ωi  (i=1,…,m) is a positive locally integrable function defined on H. Given ω→=(ω1,…,ωm),  μω→=(∏i=1mωi)q, and σi=ωi-pi′. A multiple weight ω→ associated with QI is said to satisfy the Bp→,q,α condition if the following inequality holds: (10)ω→Bp→,q,α=supI⊂R1QIα∫QIμω→zdVαz∏i=1m1QIα∫QIσizdVαzq/pi′<∞.Definition 2 (multilinear fractional Bergman operator).
Forα>-1,  0≤γ<(2+α)m, and f→=(f1,…,fm), the multilinear positive fractional Bergman operator Tα,γ(f→)(z) is defined by (11)Tα,γf→z≔∫Hmf1ξ1f2ξ2⋯fmξmz-ξ¯1+z-ξ¯2+⋯+z-ξ¯m2+αm-γdVαξ→.Definition 3 (multilinear fractional maximal function).
Forα>-1,  0≤γ<(2+α)m,  f→=(f1,…,fm), and fi∈L1(H,dVα), the multilinear fractional maximal function Mα,γ(f→)(z) is defined by (12)Mα,γf→z≔supI⊂R∏i=1mχQIzQIα1-γ/2+αm∫QIfiξidVαξi.
Note that ifγ=0, then denote Tα,0=Tα; it is just the multilinear version of Bergman operators. This paper will be devoted to investigate the sharp weighted bounds for Tα,γ and Mα,γ. We summarize our results as follows.Theorem 4.
Letα>-1,  0≤γ<2+α,  q>0, and 1<p1,…,pm<∞. Suppose that 1/q=∑i=1m1/pi-γ/2+α, ω→=(ω1,…,ωm)∈Bp→,q,α, and (13)minmaxi⁡pi′q,minjmaxi≠j⁡pi′,qpj′≤1-γ2+α.Then, the following inequality holds: (14)Tα,γf→Lqμω→dVα≲ω→Bp→,q,α1-γ/2+αmaxi⁡1,pi′/q∏i=1mfiLpiωipidVα, and the exponent of [ω→]Bp→,q,α is sharp.Theorem 5.
Letα>-1,  0≤γ<2+α,  q>0, and 1<p1,…,pm<∞. Suppose that 1/q=∑i=1m1/pi-γ/2+α,  pi0′=max⁡{pi′:1≤i≤m} for some i0 and pi0′(1-γ/2+α)≥maxi≠i0{pi′}. If  ω→=(ω1,…,ωm)∈Bp→,q,α, then it holds that (15)Mα,γf→Lqμω→dVα≲ω→Bp→,q,α1-γ/2+αmaxi⁡pi′/q∏i=1mfiLpiωipidVα, and the exponent of [ω→]Bp→,q,α is sharp.Although the estimate in Theorem5 is sharp, it can be improved whenever mixed estimates are invoked.Theorem 6.
Letα>-1,  0≤γ<(2+α)m,  1<p1,…,pm<∞,  1/p=∑i=1m1/pi, and p≤q. If (μ,ω→) is an m+1 tuple of weights satisfying σi=ωi-pi′∈B∞,α for all i=1,…,m and [μ,ω→]Bp→,q,α<∞, then (16)Mα,γf→LqμdVα≲μ,ω→Bp→,q,α∏i=1mσiB∞,αp/q1/pi∏i=1mfiLpiωipidVα.As a consequence of Theorem6, if 1/q=1/p-γ/2+α and μ=μω→=(∏i=1mωi)q, then [μω→,ω→]Bp→,q,α=[ω→]Bp→,q,α1/q. The following Corollary 7 follows immediately from Theorem 6.Corollary 7.
Letα>-1,  0≤γ<(2+α)m,  1<p1,…,pm<∞,  1/p=∑i=1m1/pi, and p≤q. Suppose that 1/q=∑i=1m1/pi-γ/2+α and ω→=(ω1,…,ωm)∈Bp→,q,α. Then we have the following: (17)Mα,γf→Lqμω→dVα≲ω→Bp→,q,α1/q∏i=1mσiB∞,α1/pi1-γp/2+α∏i=1mfiLpiωipidVα, and the exponents are sharp.Throughout the paper, the symbolA≲B means that A≤CB, where C>0 is used to denote a constant which is independent of the main parameters.


## 2. Notions and Preliminaries

We begin this section with some basic concepts and notions that will be used in our paper. The space we will focus on is the upper half of complex plane,H. For any subset E of H, we denote |E|ω,α≔∫Eω(z)dVα(z). For an interval I⊂R, the Carleson cube associated with I will be denote by QI, and the upper half associated with I will be denoted by TI; namely,(18)QI=I×0,I,TI=I×I2,I.In order to establish the sparse domination for the Bergman type operators, let us first introduce the following system of dyadic grids [11]Dβ≔{2j([0,1)+m+(-1)jβ):m∈Z,j∈Z}, for β∈{0,1/3}. We need the following well-known lemma given by Pott and Reguera [4].Lemma 8 (see [4]).
Given an intervalI⊂R, there is a dyadic interval J∈Dβ, for some β∈{0,1/3}, such that I⊂J and |J|≤8|I|.For any locally integrable functionf and z∈H, it follows that (19)Mα,γfz≲∑β∈0,1/3Mα,γd,βfz,where Mα,γd,β is defined in the same way as Mα,γ but the supremum is taken only over dyadic intervals of the dyadic grid Dβ.We say thatS={QIjk} is an η-sparse family, if for any I∈S, there exists E(QI)⊂QI, such that |E(Q)I|α≥η|QI|α and the sets {E(QI)} are pointwise disjoint. We need one more definition.Definition 9.
LetDβ be one of the dyadic grids described above, and a sparse family S in Dβ, we define the dyadic fractional Bergman operators by(20)Tα,γSf→z≔∑QI∈SQIαγ/2+α-m∏i=1m∫QIfiξidVαξiχQIz,(21)Tα,γq,Sf→z≔∑QI∈SQIαγ/2+α-m∏i=1m∫QIfiξidVαξiqχQIz1/q.
Then we have the following sparse domination forTα,γ, which will be used in proving Theorems 4 and 5.Proposition 10.
Letα>-1, 0≤γ<(2+α)m; then, it holds that(22)Tα,γf→z≲∑β∈0,1/3Tα,γSf→z,(23)Tα,γf→z≲∑β∈0,1/3Tα,γq,Sf→z,for0<q≤1.Proof.
Let us fixz,ξ∈H, then there exists l∈Z such that (24)22l-2≤Rez-Reξ2+Imz+Imξ2≤22l. It is easy to see that there exists an interval I such that 2l-1<|I|<2l; by Lemma 8, we have I⊂J and |J|≤8|I|. Now it follows that (25)Tα,γf→z≤∫Hmf1ξ1⋯fmξmz-ξ¯1+⋯+z-ξ¯m2+αm-γdVαξ→≤∑l∈z∫2l-1<∑i=1mz-ξi¯<2lf1ξ1⋯fmξmz-ξ¯1+⋯+z-ξ¯m2+αm-γdVαξ→=∑J∈Dβ,J=2l+3,I⊂JχQIz∫J/16<∑i=1mz-ξi¯<J/8f1ξ1⋯fmξm∑i=1mz-ξ¯i2+αm-γdVαξ→≤c∑J∈Dβ,I⊂JJγ-2+αmχQIz∫supi⁡z-ξi¯<J/8f1ξ1⋯fmξmdVαξ→≤c∑J∈Dβ,I⊂JQIγ/2+α-mχQIz∫QImf1ξ1⋯fmξmdVαξ→≲∑β∈0,1/3Tα,γSf→z. Applying the same decomposition to the case 0<q≤1 yields that (26)Tα,γf→z≤∫Hm∏i=1mfiξiz-ξ¯1+⋯+z-ξ¯m2+αm-γdVαξ→≲∑J∈Dβ,I⊂JQIγ/2+α-m∫QIm∏i=1mfiξidVαξ→χQIzq·1/q≤∑J∈Dβ,I⊂JQIγ/2+α-m∫QIm∏i=1mfiξidVαξ→qχQIz1/q≲∑β∈0,1/3Tα,γq,Sf→z.


## 3. Lemmas and the Proof of Theorems

We begin by preparing some lemmas.Lemma 11 (see [10]).
Letα>-1,  0≤γ<2+α,  1<p≤q<∞, and σ be a weight. Assume that q is such that 1/p-1/q=γ/2+α. Then there exists a constant C=C(p,α,γ) such that (27)∫HMσ,α,γd,βfzqσzdVα1/q≤C∫HfzpσzdVα1/p.Lemma 12.
Letα>-1,  0≤γ<2+α,  q>0, and 1<p1,…,pm<∞. Suppose that 1/q=∑i=1m1/pi-γ/2+α,  q(1-γ/2+α)≥max⁡pi′, and ω→=(ω1,…,ωm)∈Bp→,q,α; then we have(28)Tα,γSf1σ1,…,fmσmLqμω→dVα≲ω→Bp→,q,α1-γ/2+α∏i=1mfiLpiσidVα.Proof.
Let1/q=1/p-γ/2+α; we have (29)Tα,γSf1σ1,…,fmσmLqμω→dVα=supgLq′μω→dVα=1∫HTα,γSf1σ1,…,fmσmgμω→dVα=supgLq′μω→dVα=1∑I∈SQIαγ/2+α-m∏i=1m∫QIfiσidVα∫QIgμω→dVα≤ω→Bp→,q,α1-γ/2+αsupgLq′μω→dVα=1∑I∈SQIαm-γ/2+αq/p′∏i=1mQIσi,α1-q/pi′1-γ/2+α×QIμω→,αγ/2+α-1∫QIgμω→dVα∏i=1m1QIσi,α∫fiσidVα. Given QI, we recall that it is upper-half is the set TI. It is clear that if D is a dyadic grid in R, then the family {TI}I∈D forms a tiling of H. As TI⊂QI, and q(1-γ/2+α)≥max⁡pi′, i.e., 1-q/p′(1-γ/2+α)≤0,  |QI|σi,α1-q/p′1-γ/2+α≲|TI|σi,α1-q/p′1-γ/2+α.
We remark that(30)μω→1/m-γ/2+αq∏i=1mσi1/m-γ/2+αpi′=1,1m-γ/2+αq+∑i=1m1m-γ/2+αpi′=1. Then, it follows that(31)TIα=∫TIμω→1/m-γ/2+αq∏i=1mσi1/m-γ/2+αpi′dVα≤TIμω→,α1/m-γ/2+αq∏i=1mTIσi,α1/m-γ/2+αpi′.Since |QI|α⋍|TI|α, we deduce that (32)QIαm-γ/2+αq/p′∏i=1mQIσi,α1-q/pi′1-γ/2+α≲TIαm-γ/2+αq/p′∏i=1mQIσi,α1-q/pi′1-γ/2+α≲TIμω→,α1/p′∏i=1mTIσi,αq/p′pi′∏i=1mQIσi,α1-q/pi′1-γ/2+α≤TIμω→,α1/p′∏i=1mTIσi,α1/pi. Hence, we have (33)∑I∈SQIαm-γ/2+αq/p′∏i=1mQIσi,α1-q/pi′1-γ/2+αQIμω→,αγ/2+α-1∫QIgμωdVα×∏i=1m1QIσi,α∫QIfiσidVα≲∑I∈SQIμω→,αγ/2+α-1∫QIgμωdVαp′TIμω→,α1/p′×∏i=1m∑I∈S1QIσi,α∫QIfiσidVαpiTIσi,α1/pi≤Mα,γ,μω→d,βgLp′μω→dVα∏i=1mMα,0,σid,βfiLpiσdVα≲∏i=1mfiLpiσdVα. Combining the former estimate, we get the desired result, (34)Tα,γSf1σ1,…,fmσmLqμω→dVα≲ω→Bp→,q,α1-γ/2+α∏i=1mfiLpiσidVα.Lemma 13.
Letα>-1,  0≤γ<2+α,  q>0, and 1<p1,…,pm<∞. Suppose that 1/q=∑i=1m1/pi-γ/2+α,  pi0′=max⁡{pi′:1≤i≤m} for some i0,  pi0′(1-γ/2+α)≥maxi≠i0{pi′} and ω→=(ω1,…,ωm)∈Bp→,q,α. Furthermore, assume that S is a sparse family. Then we have (35)Tα,γq,Sf1σ1,…,fmσmLqμω→dVα≲ω→Bp→,q,α1-γ/2+αmax⁡pi′/q∏i=1mfiLpiσidVα.Proof.
Letp1′=max⁡{pi′:1≤i≤m}; one may obtain that (36)Tα,γq,Sf1σ1,…,fmσmLqμω→dVαq=∑I∈SQIαγ/2+α-m∏i=1m∫QIfiσidVαqQIμω→,α=∑I∈SQIμω→,αQIα1-γ/2+αp1′∏i=1mQIσi,αQIα1-γ/2+αqp1′/pi′QIαqm-γ/2+α1-γ/2+αp1′-1×∏i=2mQIσi,αq-1-γ/2+αqp1′/pi′QIμω→,α1-1-γ/2+αp1′QIσ1,αγ/2+αQIσ1,α∫QIf1σ1dVαq×∏i=2m1QIσi,α∫QIfiσidVαq. By (31) and the fact p1′1-γ/2+α≥maxi≠1{pi′}, we have (37)Tα,γq,Sf1σ1,…,fmσmLqμω→dVα≤ω→Bp→,q,α1-γ/2+αp1′∑I∈S1QIσ1,α1-γ/2+α∫QIf1σ1dVαq∏i=2m1QIσi,α∫QIfiσidVαq×TIσ1,αq1-γ/2+α-1/p1′∏i=2mTIσi,αq/pi. Let 1/q1=1/P1-γ/2+α, by the Hölder inequality, we obtain (38)Tα,γq,Sf1σ1,…,fmσmLqμω→dVαq≤ω→Bp→,q,α1-γ/2+αp1′∑I∈S1QIσ1,α1-γ/2+α∫QIf1σ1dVαq1TIσ1,αq/q1×∏i=2m∑QI∈S1QIσi,α∫QIfiσidVαpiTIσi,αq/pi≤ω→Bp→,q,α1-γ/2+αp1′Mα,γ,σ1d,βf1Lq1σ1dVαq∏i=2mMα,0,σid,βfiLpiσidVαq≲ω→Bp→,q,α1-γ/2+αp1′∏i=1mfiLpiσidVαq.We are now ready to prove Theorem4.Proof of Theorem4.
We know that whenq>1, we can use the duality. However, when 0<q<1, we cannot use the method of the duality. Thus, we need to consider these two cases separately, and here we will borrow some ideas from [12, 13].
(i) The Case q > 1. By (22), it is sufficient to show that (39)Tα,γSf→Lqμω→dVα≲ω→Bp→,q,α1-γ/2+αmaxi⁡1,pi′/q∏i=1mfiLpiωipidVα. If q(1-γ/2+α)≥max⁡{pi′}, then using Lemma 12, we may get the desired result. If pj′1-γ/2+α≥maxi≠j{pi′,q}, we assume that p1′(1-γ/2+α)≥maxi≠j{pi′,q}. By duality, we have (40)Tα,γSf→Lp1ω1p1dVα×⋯×LpmωmpmdVα→Lqμω→dVα=Tα,γSf→Lq′∏i=1mωi-q′dVα×Lp2ω2p2dVα×⋯×LpmωmpmdVα→Lp1′ω1-p1′dVα≲ω→Bp→,q,α1-γ/2+αp1′/q.
(ii) The Case q ≤ 1. In this case, we first notice that (41)minjmaxi≠j⁡pi′pj′=min⁡maxi⁡pi′q,minjmaxi≠j⁡pi′,qpj′≤1-γ2+α. By (23), it is enough to show that (42)Tα,γq,Sf→Lqμω→dVα≲ω→Bp→,q,α1-γ/2+αmaxi⁡1,pi′/q∏i=1mfiLpiωipidVα. By Lemma 13, it yields that (43)Tα,γq,Sf→Lqμω→dVα≲ω→Bp→,q,α1-γ/2+αmaxi⁡pi′/q∏i=1mfiLpiωipidVα=ω→Bp→,q,α1-γ/2+αmaxi⁡1,pi′/q∏i=1mfiLpiωipidVα.Proof of Theorem5.
Leta=2(m-γ/2+α)(3+α) and Ωk≔z∈H:ak<Mα,γd,βf1,…,fmz<ak+1. Then we have Ωk⊂∪j=1mQIjk, where Ij∈Dβ and {QIjk}j≥1 is a family of dyadic cube which is maximal with respect to the inclusion and enjoys the properties that (44)22+αm-γak>∏i=1m1QIjkα1-γ/2+αm∫QIjkfi>ak. Define E(QIjk)≔QIjk∖Ωk+1; then it is easy to see that the sets {Ek,j} are disjoint. By the Hölder inequality, it follows that (45)QIjk∩Ωk+1=∑QIlk+1⊂QIjkQIlk+1≤1ak+1∑QIlk+1⊂QIjk∏i=1m∫QIlk+1fi1/m-γ/2+α≤1ak+1∏i=1m∑QIlk+1⊂QIjk∫QIlk+1fi1/m-γ/2+α≤1ak+1∏i=1m∫QIjkfi1/m-γ/2+α≤1ak+122+αm-γakQIjkαm-γ/2+α1/m-γ/2+α=22+αQIjkαa-1/m-γ/2+α=12QIjkα. Hence, we obtain (46)QIjkα≤EQIjkα+QIjk∩Ωk+1α≤EQIjkα+12QIjkα. Thus, |QIjk|α≤2|E(QIjk)|α, and then S is a sparse family.
Therefore, it holds that(47)∫HMα,γd,βf1,…,fmqμω→dVα=∑k∫Ωk∖Ωk+1Mα,γd,βf1,…,fmqμω→dVα≲∑k,j∏i=1m1QIjkα1-γ/2+αm∫QIjkfiξidVαqEQIjkμω→,α≲∫HTα,γq,Sf1,…,fmqμω→dVα. By Lemma 13, we complete the proof of Theorem 5.To prove Theorem6, the following Carlson embedding Theorem (see, for examples, [10, 14, 15]) will be needed. And we give the definition of the Carlson sequence first.Definition 14 (see [10]).
Letα>-1 and ω be a positive weight. For any δ≥1, a sequence of positive numbers {λQI}I∈Dβ is called a (ω,α,δ)-Carleson sequence, if there is a constant C>0 such that for any J∈Dβ, (48)∑I⊂J,I∈DβλQI≤CQJω,αδ. The smallest constant C in the above definition is called the Carleson constant of the sequence.Lemma 15 (Carlson embedding theorem [10]).
Letα>-1, and 0≤γ<2+α. Let ω be a weight on H and s≥1. Assume {λQI}I∈Dβ is a sequence of positive numbers. If there exists some constant A>0 such that for any interval J∈Dβ, it holds that (49)∑I⊂J,I∈DβλQI≤AQJω,αs. Then, for any p∈(1,∞), we have (50)∑I,I∈DβλQI1QIω,α1-γ/2+α∫QIfzωzdVαps≤Aγ∫HMω,α,γd,βfzpωzdVαs.Proof of Theorem6.
Letqi=qpi/p≥pi with 1/q=∑i=1m1/qi and denote βi by βi/2+α=1/pi-1/qi. Using the Hölder inequality, one deduces that (51)∫HMα,γd,βf1,…,fmqμdVα1/q≤∑k,j∏i=1m1QIjkα1-γ/2+αm∫QIjkfiξidVαqEQIjkμ,α1/q≤μ,ωBp→,q,α∑k,j∏i=1m∫QIjkfiξidVαQIjkσi,α-1/pi′q1/q≤μ,ωBp→,q,α∏i=1m∑k,j∫QIjkfiξidVαQIjkσi,α-1/pi′qi1/qi=μ,ωBp→,q,α∏i=1m∑k,j1QIjkσi,α1-βi/2+α∫QIjkfiξidVαqiQIjkσi,α1/qi. We claim that the sequence (52)λQI=QIjkσi,α,QI=QIjkfor somek,j;∅,otherwise is a (σi,α,1)-Carlson sequence. Indeed, when σi∈B∞,α, we have (53)∑Ij⊂J,Ij∈DβQIjkσi,α=∑Ij⊂J,Ij∈DβQIjkσi,αQIjkαQIjkα⋍∑Ij⊂J,Ij∈DβQIjkσi,αQIjkα⋍TIjkα≤∑Ij⊂J,Ij∈Dβ∫TIQIjkσi,αQIjkαdVα≤∑Ij⊂J,Ij∈Dβ∫TIMαχQJσidVα=∫QJMαχQJσidVα≤σiB∞,αQJσi,α. So by Carlson embedding theorem, we have (54)∑k,jλQI1QIjkσi,α1-βi/2+α∫QIjkfiξidVαqi=∑k,jλQI1QIjkσi,α1-βi/2+α∫QIjkfiξiσiξiσi-1ξidVαqi≲∫HMσi,βi,α,γd,βfiσi-1ξiσiξidVα. By Lemma 11, we have (55)∫HMσi,βi,α,γd,βfiσi-1qiσiξidVα1/qi≲∫Hfiσi-1piσiξidVα1/pi=fiLpiωipidVα. Then we have (56)∫HMα,γd,βf1,…,fmqμdVα1/q≲μ,ω→Bp→,q,α∏i=1mσiB∞,α1/qifiLpiωipidVα=μ,ω→Bp→,q,α∏i=1mσiB∞,αp/q1/pi∏i=1mfiLpiωipidVα.


## 4. Examples

In this part, we give some examples to show that our estimates are sharp. First, we show that Theorem5 is sharp, we consider the simple case, m=2, recall that 0≤γ<2+α, let 0<δ<1, we suppose that p1′≥p2′, then consider ω1(z)=|z|2+α-δ/p1′ and ω2(z)=1, and one can easily check that ω→∈Bp→,q,α and [ω→]Bp→,q,α⋍δ-q/p1′. We also consider the function f1(z)=|z|δ-(2+α)χ{z∈H:|z|≤1}(z) and f2(z)=|z|δ-(2+α)/p2χ{z∈H:|z|≤1}(z). We obtained that f1Lp1(ω1p1)⋍δ-1/p1 and f2Lp2(ω2p2)⋍δ-1/p2. Let z∈{z∈H:|z|≤1}; then (57)Mα,γf1,f2z≳z-22+α+γ∫ξ1∈H:ξ1≤zξ1δ-2+αdVαξ1×∫ξ2∈H:ξ2≤zξ2δ-2+α/p2dVαξ2⋍z-22+α+γ∫0π/2∫0zrδ-2+αrαsin⁡ϕαrdrdϕ×∫0π/2∫0zrδ-2+α/p2rαsin⁡ϕαrdrdϕ≳1δz-22+α+γ·zδ·zδ-2+α/p2+α+2=1δzδ-2+α/p2+δ-2+α+γ.Then we have (58)∫HMα,γf1,f2zqω1qω2qdVαz≳δ-q∫z∈H:z≤1zδ-2+αq+q/p2-q/p1′+qγdVαz≳δ-q-1.That is, (59)Mα,γf1,f2Lqω1qω2qdVα≳δ-1-1/q=δ-1/p1′+1/q-1/p2δ-1/p1-1/p2=δ-1-γ/2+αδ-1/p1-1/p2⋍ωBp→,q,α1-γ/2+αp1′/qf1Lp1ω1p1dVαf2Lp2ω2p2dVα.Let us also check that Theorem 4 is sharp. We notice that Mα,γ≲Tα,γ. If maxi{pi′}≥q, taking the same fi and ωi as the above, then we may obtain the sharpness immediately. If p1′=maxi{pi′}<q, we merely prove the case q=maxi{q,p2′⋯pm′}>p1′ is sharp. Indeed, 1/p1+1/p2+⋯+1/pm=1/q+γ/2+α; that is, 1/p1′+1/p2+⋯+1/pm=1/p1′+γ/2+α. But in this case, we have already showed that the result is sharp. So the sharpness just follows from the standard duality argument in the proof of Theorem 4.Finally, we give the example to show Corollary7 is sharp. We proceed to use the similar calculation as before. For α>-1,0≤γ<2+α and 0<δ<1, we take ωi(z)=|z|2+α-δ/pi′ and fi(z)=|z|δ-(2+α)χ{z∈H:|z|≤1},  i=1,…,m; then μω→=|z|(2+α-δ)(m-1/p)q. One can easily check that [ω→]Bp→,q,α⋍δ-q(m-1/p),  [σi]B∞,α≲δ-1, and ∏i=1mfiLpi(ωipidVα)⋍δ-1/p. Since the process is similar as the above, we omit the proof. Let z∈z∈H:z≤1; then (60)Mα,γf→z≳∏i=1m1z2+α-γ/m∫ξi∈H:ξi≤zfiξdVαξi≳1δmzmδ-2+α+γ.Therefore, we have (61)∫HMα,γf→zqμω→dVα1/q≳1δm∫zmqδ-2+α+qγ+2+α-δm-1/pq1/q⋍δm-1/q≳ω→Bp→,q,α1/q∏i=1mσiB∞,α1/pi1-pγ/2+α∏i=1mfiLpiωipidVα.



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*Source: 3485962-2018-06-03.xml*