# The Commutators of Multilinear Maximal and Fractional-Type Operators on Central Morrey Spaces with Variable Exponent

Authors: Liwei Wang
Journal: Journal of Function Spaces
 (2022)

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## Abstract

We show that the maximal operator associated with multilinear Calderón-Zygmund singular integrals and its commutators are bounded on products of central Morrey spaces with variable exponent. Moreover, some bounded properties are obtained for the commutators of multilinear Calderón-Zygmund operators as well as for the corresponding fractional integrals.


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## Body

## 1. Introduction

LetT be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values into the space of tempered distributions,
(1)T:Sℝn×⋯×Sℝn⟶S′ℝn.Following Grafakos and Torres [1], we say that T is an m-linear Calderón-Zygmund operator if, for some 1≤p1,⋯,pm<∞, it extends to a bounded multilinear operator from Lp1ℝn×⋯×Lpmℝn to Lpℝn, where 1/p=1/p1+⋯+1/pm, and if there exists a function K, defined away from the diagonal x=y1=⋯=ym in ℝnm+1, satisfying
(2)Tf1,⋯,fmx=∫ℝnmKx,y1,⋯,ymf1y1⋯fmymdy1⋯dym,for all x∉⋂j=1msuppfj and f1,⋯,fm are C∞ functions with compact support,
(3)Kx,y1,⋯,ym≤A∑k=1mx−ykmn,for some A>0 and all x,y1,⋯,ym∈ℝnm+1 with x≠yk for some k, and
(4)Kx,y1,⋯,ym−Kx′,y1,⋯,ym≤Ax−x′ε∑k=1mx−ykmn+ε,for some ε>0, whenever x−x′≤1/2maxx−y1,⋯,x−ym, and also that
(5)Kx,y1,⋯,yi,⋯,ym−Kx,y1,⋯,yi′,⋯,ym≤Ayi−yi′ε∑k=1mx−ykmn+ε,whenever yi−yi′≤1/2maxx−y1,⋯,x−ym for all 1≤i≤m.Multilinear Calderón-Zygmund operators were introduced by Coifman and Meyer [2, 3] in the 70s and were systematically studied by Grafakos and Torres [1]. They showed that multilinear Calderón-Zygmund operators map Lp1×⋯×Lpm into Lp for some 1<p1,⋯,pm<∞ and some 0<p<∞ with 1/p=1/p1+⋯+1/pm. Moreover, multilinear Calderón-Zygmund operators satisfy weak endpoint bounds when at least one pi is equal to one. Lerner et al. [4] established the multiple weights theory for multilinear Calderón-Zygmund operators and obtained some weighted estimates for these operators and their commutators. Grafakos and Kalton [5] studied the boundedness of these operators on products of Hardy spaces. Subsequently, under some additional conditions, Hu and Meng [6] proved the boundedness of these operators from product Hardy spaces into Hardy spaces.For convenience, we will writey1,⋯,ym=y⟶ and dy1,⋯,dym=dy⟶. Grafakos and Torres [7] defined the maximal multilinear Calderón-Zygmund operator
(6)T∗f⟶x=supδ>0Tδf1,⋯,fmx,where f⟶=f1,⋯,fm and Tδ are the smooth truncations of T given by
(7)Tδf1,⋯,fmx=∫x−y12+⋯+x−ym2>δ2Kx,y1,⋯,ymf1y1⋯fmymdy⟶.Moreover, they established the boundedness ofT∗ on product of Lebesgue spaces and also obtained some weighted norm inequalities for this maximal operator. Later on, the topic was studied by several authors; see [8–16].A functionb∈Llocℝn is said to belong to space BMOℝn, if
(8)∥b∥BMOℝn≔supx∈B1B∫Bby−bBdy<∞,where B is a ball of ℝn with center at the origin and radius R, B is the Lebesgue measure of B, and bB≔1/B∫Bfxdx.Definition 1.
Given a collection of locally integrable functionsb⟶=b1,⋯,bm, the commutators of b⟶ and the multilinear Calderón-Zygmund operator T are defined by
(9)Tb⟶f1,⋯,fm=∑j=1mTb⟶jf⟶,where each term is the commutator of bj and T in the jth entry of T, that is,
(10)Tb⟶jf⟶=bjTf1,⋯,fj,⋯,fm−Tf1,⋯,bjfj,⋯,fm.
Also, the commutators ofb⟶ and the maximal multilinear Calderón-Zygmund operator T∗ are defined by
(11)Tb⟶∗f1,⋯,fm=∑j=1mTb⟶∗jf⟶,where each term is the commutator of bj and T∗ in the jth entry of T∗, that is,
(12)Tb⟶∗jf⟶=supδ>0bjTδf1,⋯,fj,⋯,fm−Tδf1,⋯,bjfj,⋯,fm.This definition coincides with the commutator of Coifman et al. [17] when m=1. The multilinear commutators Tb⟶ were considered by Lerner et al. in [4]; they proved that if b⟶∈BMOm, 1<p1,⋯,pm<∞, and p defined by 1/p=1/p1+⋯+1/pm, then
(13)Tb⟶:Lp1ω1×⋯×Lpmωm⟶Lpνω⟶,where νω⟶=∏j=1mωj1/m and ω⟶=ω1,⋯,ωm satisfies the Ap1,⋯,pm condition, that is,
(14)supQ1Q∫Qνω⟶1/p∏j=1m1Q∫Qωj1−pj′1/p′j<∞.For further contributions in the study of the multilinear commutatorsTb⟶ and Tb⟶∗, we refer to [18, 19] and the references therein.On the other hand, as one of the most important operators, the multilinear fractional operator has also been intensively studied in the recent years. Kenig and Stein [20] considered the following multilinear fractional operator Iα,0<α<mn,
(15)Iαf⟶x=∫ℝnmf1y1f2y2⋯fmymx−y1,x−y2,⋯,x−ymmn−αdy1dy2⋯dym,where m,n denote the nonnegative integers with m≥1,n≥2. They proved that Iα is of strong type Lp1×⋯×Lpm,Lq and weak type Lp1×⋯×Lpm,Lq,∞. Moen [21] developed a weighted theory that adapts to the multilinear fractional integral operators. He established the multiple weighted norm inequalities for the multilinear fractional integral operators and the corresponding multilinear fractional maximal operators. Following the work of [4], we define the commutators of b⟶ and the multilinear fractional operator Iα as follows:
(16)Iα,b⟶f1,⋯,fm=∑j=1mIα,b⟶jf⟶,where each term is the commutator of bj and Iα in the jth entry of Iα, that is,
(17)Iα,b⟶jf⟶=bjIαf1,⋯,fj,⋯,fm−Iαf1,⋯,bjfj,⋯,fm.The commutatorsIα,b⟶ were first studied by Chen and Xue [22], in which some weighted strong bounds and LlogL type endpoint estimates for Iα,b⟶ are obtained.Lu and Yang [23, 24] introduced the central bounded mean oscillation space CṀOqℝn, 1<q<∞, which satisfies the following condition:
(18)fCṀOqℝn≔supR>01B0,R∫B0,Rfy−fB0,Rqdy1/q<∞.This space can be regarded as a local version ofBMOℝn at the origin, but they have quite different properties. For example, the well-known John-Nirenberg inequality shows that the functions in BMOℝn can be described by means of the condition
(19)supB⊂ℝn1B0,R∫B0,Rfy−fB0,Rqdy1/q<∞.However, the spaceCṀOqℝn depends on q. More precisely, if q1<q2, then CṀOq2ℝn⊂CṀOq1ℝn. Therefore, there is no analogy of the classical John-Nirenberg inequality of BMOℝn for the space CṀOqℝn. One can imagine that the behavior of CṀOqℝn may be quite different from that of BMOℝn..Definition 2.
Letλ<1/n and 1<q<∞. The λ-central bounded mean oscillation space CṀOq,λℝn is defined by
(20)fCṀOq,λℝn≔supR>01B0,R1+λq∫B0,Rfy−fB0,Rqdy1/q<∞.Remark 3.
Whenλ=0, the space CṀOq,λℝn is just the space CṀOqℝn defined above. If two functions which differ by a constant are regarded as a function in the space CṀOq,λℝn, then it becomes a Banach space. Obviously, (20) is equivalent to the following condition:
(21)supR>0infc∈ℂ1B0,R1+λq∫B0,Rfy−cqdy1/q<∞.Definition 4.
Letλ∈ℝ and 1<q<∞. The central Morrey space Bq,λℝn consists of all f∈Llocqℝn such that
(22)fBq,λℝn≔supR>01B0,R1+λq∫B0,Rfyqdy1/q<∞.The aboveλ-central BMO spaces and central Morrey spaces were introduced in Alvarez et al. [25]; since then, various operators were studied in these spaces. For some recent development, we mention that Fu et al. [26] established λ-central BMO estimates for a class of multisublinear operators on the product of central Morrey spaces. As its special cases, the corresponding results of multilinear Calderón-Zygmund operators can be deduced. Si and Xue [18] established λ-central BMO estimates for commutators of maximal multilinear Calderón-Zygmund operators and multilinear fractional operators on central Morrey spaces.In the last two decades, following the fundamental work of Kovác˘ik and Rákosnk [27] on variable Lebesgue spaces Lp·ℝn and variable Sobolev spaces Wk,p·ℝn (here and below, the exponent p· is a function and not a constant), function spaces with variable exponent, such as variable Morrey spaces, variable Herz spaces, and variable Hardy spaces, have been widely studied by a significant number of authors; see [28–37] and the references therein. These spaces are of interest in their own right and also have applications to image restoration [38], fluid dynamics [39], and PDEs with nonstandard growth conditions [40].In 2019, Fu et al. [41] introduced the λ-central BMO spaces and the central Morrey spaces with variable exponent and proved the boundedness of the fractional singular integrals and its commutator on those spaces. Subsequently, Wang and Xu [42] further obtained the boundedness of multilinear fractional integral operators and their commutators on central Morrey spaces with variable exponent. The λ-central BMO estimates for m-linear Calderón-Zygmund operators and their commutators on the product of central Morrey spaces with variable exponent are independently obtained by Wang [43] and Wang et al. [44].Motivated by [18, 41], the aim of this paper is to establish λ-central BMO estimates for the maximal multilinear Calderón-Zygmund operators T∗ and its commutators Tb⟶∗ on central Morrey spaces with variable exponent. Moreover, the similar boundedness properties for the multilinear commutators Tb⟶ and Iα,b⟶ are obtained.Throughout this paper, the symbolℕ stands for the set of all natural numbers. We denote B0,R≔y∈ℝn:y<r simply by B, and B0,lR by lB for l∈ℕ. fB denotes the integral average of f on B, i.e., fB=1/B∫Bfxdx. If E⊂ℝn is a measurable set, then E means the Lebesgue measure of E and χE denotes its characteristic function. p′· is the conjugate exponent function defined by 1/p·+1/p′·=1. The letter C stands for a positive constant whose value may change from appearance to appearance.


## 2. Preliminaries and Main Results

Let us first recall some basic properties of Lebesgue spaces with variable exponent; we refer to the surveys [45, 46] and the monographs [47, 48] for further details.The variable Lebesgue spaces are a generalization of the classicalLp spaces with the exponent p replaced by a measurable function p·:ℝn⟶0,∞. It consists of all measurable functions f on ℝn such that
(23)Ip·f≔∫Rnfxpxdx<∞.Ifx≥1 a.e., then this becomes a Banach space when equipped with the norm
(24)fLp·ℝn=infλ>0:Ip·fλ≤1.Given an open setΩ⊂ℝn, the space Llocp·Ω is defined by
(25)Llocp·Ω=f:f∈Lp·FforallcompactsubsetsF⊂Ω.In what follows, we definePℝn to be the set of measurable function p·:ℝn⟶1,∞ such that
(26)p−≔essinfx∈ℝnpx>1,p+≔esssupx∈ℝnpx<∞.Bℝn≔p·∈Pℝn:MisboundedonLp·Rn,where M is the Hardy-Littlewood maximal operator defined by
(27)Mfx=supx∈ℝn,r>0r−n∫Bx,rfydy.A measurable functionp·∈Pℝn is called globally log-Ho¨lder continuous if it satisfies
(28)px−py≤C−logx−y,x−y≤1/2,(29)px−py≤Cloge+x,y≥x.The set ofp· satisfying (28) and (29) is denoted by LHℝn. In Cruz-Uribe et al. [49], Theorem 1.1 shows that if p·∈Pℝn⋂LHℝn, then p·∈Bℝn.Supposep·∈Pℝn, then for all f∈Lp·ℝn and all g∈Lp′·ℝn, the generalized Ho¨lder inequality holds in the form
(30)∫ℝnfxgxdx≤rpfLp·ℝngLp′·ℝn,with rp=1+1/p−−1/p+; see [27], Theorem 14.The next Lemmas5 and 6 are due to Izuki [33], Page 203.Lemma 5.
Letp·∈BRn, then we have
(31)1BχBLp·ℝnχBLp′·ℝn≤C.Lemma 6.
Letp·∈Bℝn, then we have for all measurable subsets E⊂B,
(32)χBLp·ℝnχELp·ℝn≤CBE,χELp·ℝnχBLp·ℝn≤CEBδ1,χELp′·ℝnχBLp′·ℝn≤CEBδ2,where δ1,δ2 are constants with 0<δ1,δ2<1.Lemma 7.
Letp·,q·,r·∈Pℝn be such that
(33)1rx=1px+1qx,for almost every x∈ℝn. Then, we have
(34)fgLr·ℝn≤2fLp·ℝngLq·ℝnfor all f∈Lp·ℝn and all g∈Lq·ℝn.Lemma 8.
Suppose thatp·∈LHℝn and 0<p−≤px≤p+<∞.
(i) For all cubes (or balls)Q≤2n and any x∈Q, we have
(35)χQLp·ℝn≈Q1/px.
(ii) For all cubes (or balls)Q≥1, we have
(36)χQLp·ℝn≈Q1/p∞,where p∞=limx⟶∞px.The proofs of Lemmas7 and 8 can be found in [48]. Tan et al. [50] obtain the following result.Lemma 9.
Letm∈ℕ,0<α<mn. If q·,p1·,⋯,pm·∈Pℝn⋂LHℝn such that 1/q·=1/p1·+⋯+1/pm·−α/n, then we have
(37)Iαf⟶Lq·ℝn≤C∏i=1mfiLpi·ℝn,with the constant C>0 independent of f1,⋯,fm.Now, we recall that theλ-central bounded mean oscillation space and the central Morrey space in [41] are defined as follows.Definition 10.
Letλ<1/n and q·∈Pℝn. The λ-central bounded mean oscillation space with variable exponent CṀOq·,λℝn is defined by
(38)CṀOq·,λℝn=f∈Llocq·ℝn:fCṀOq·,λℝn<∞,
where(39)fCṀOq·,λℝn≔supR>0f−fB0,RχB0,RLq·ℝnB0,RλχB0,RLq·ℝn.Definition 11.
Letλ∈ℝ and q·∈Pℝn. The central Morrey space with variable exponent Ḃq·,λℝn is defined by
(40)Ḃq·,λℝn=f∈Llocq·ℝn:fḂq·,λℝn<∞,
where(41)fḂq·,λℝn≔supR>0fχB0,RLq·ℝnB0,RλχB0,RLq·ℝn.Remark 12.
Whenλ=0, the space CṀOq·,λℝn is just the space CBMOq·ℝn defined in [51]. If q·=q is constant, then we can immediately get Definitions 2 and 4, respectively.Remark 13.
Denote byCMOq·,λℝn and Bq·,λℝn the inhomogeneous versions of the λ-central bounded mean oscillation space and the central Morrey space with variable exponent, which are defined, respectively, by taking the supremum over R≥1 in Definitions 10 and 11 instead of R>0 there. Obviously, CṀOq·,λℝn⊂CMOq·,λℝn for q·∈Pℝn and λ<1/n, and Ḃq·,λℝn⊂Bq·,λℝn for q·∈Pℝn and λ∈ℝ.Our main results can be stated as follows.Theorem 14.
Suppose thatp·,pi·∈Pℝn⋂LHℝn, λi<0, i=1,⋯,m, λ=λ1+⋯+λm and 1/p·=1/p1·+⋯+1/pm·, then the maximal multilinear Calderón-Zygmund operator T∗ is bounded from Ḃp1·,λ1ℝn×⋯×Ḃpm·,λmℝn to Ḃp·,λℝn and satisfies
(42)T∗f⟶Ḃp·,λℝn≤C∏i=1mfiḂpi·,λiℝn.Theorem 15.
Suppose thatp·,pj·,sj·∈Pℝn⋂LHℝn, 1<pj′·<sj·<∞, 0<uj<1/n, bj∈CṀOsj·,ujℝn, j=1,⋯,m, 1/p·=1/sj·+∑i=1m1/pi· and λ=uj+∑i=1mvi with uj+vj<0, then the commutators of the multilinear Calderón-Zygmund operator Tb⟶∗ defined as in (11) are bounded from Ḃp1·,v1ℝn×⋯×Ḃpm·,vmℝn to Ḃp·,λℝn and satisfy
(43)Tb⟶∗f⟶Ḃp·,λℝn≤CsupjbjCṀOsj·,ujℝn∏i=1mfiḂpi·,viℝn.Theorem 16.
Let0<α<mn. Suppose that q·,pj·,sj·∈Pℝn⋂LHℝn, 1<pj′·<sj·<∞, 0<βj<1/n, bj∈CṀOsj·,βjℝn, j=1,⋯,m, γi<−α/mn, λ=βj+∑i=1mγi+α/n with βj+γj<−α/n, 1/p·=∑i=1m1/pi·−α/n>0 and 1/q·=1/sj·+1/p·, then the commutators of the multilinear fractional operator Iα,b⟶ defined as in (16) are bounded from Ḃp1·,γ1ℝn×⋯×Ḃpm·,γmℝn to Ḃq·,λℝn and satisfy
(44)Iα,b⟶f⟶Ḃq·,λℝn≤CsupjbjCṀOsj·,βjℝn∏i=1mfiḂpi·,γiℝn.Remark 17.
In fact, under the same conditions of Theorem15, the commutators of the multilinear Calderón-Zygmund operator Tb⟶ are bounded from Ḃp1·,v1ℝn×⋯×Ḃpm·,vmℝn to Ḃp·,λℝn. We omit its proof since it is essentially similar to the proof of Theorem 2.2Remark 18.
We mention that, even whenq· and pi·i=1,⋯,m are constant, Theorem 14 is completely new. Theorems 15 and 16 can be seen as generalizations of the corresponding main results in [18] with different approaches based on the theory of variable exponent and on generalization of the λ-central BMO norms. Moreover, our results remain true for the inhomogeneous versions of λ-central BMO spaces and central Morrey spaces with variable exponent.


## 3. Proof of Theorem 2.1

Without loss of generality, we may assume thatm=2. Let f1,f2 be functions in Ḃp1·,λ1ℝn and Ḃp2·,λ2ℝn, respectively. For simplicity, we denote by Lp·=Lp·ℝn and Ḃp·,λ=Ḃp·,λℝn. We need to prove
(45)χBT∗f1,f2Lp·≤CBλχBLp·∏i=12fiḂpi·,λi.It follows from the Minkowski inequality that(46)χBT∗f1,f2Lp·≤χBT∗f1χ2B,f2χ2BLp·+χBT∗f1χ2B,f2χ2BcLp·+χBT∗f1χ2Bc,f2χ2BLp·+χBT∗f1χ2Bc,f2χ2BcLp·≕E1+E2+E3+E4.ForE1, noting that T∗ is of type Lp1·×⋯×Lpm·,Lp· (see [44], Theorem 1.3]), by Lemmas 6 and 8, we deduce that
(47)E1≤Cf1χ2BLp1·f2χ2BLp2·≤C∏i=122Bλiχ2BLpi·fiḂpi·,λi≤C∏i=12BλiχBLpi·fiḂpi·,λi≤CBλχBLp·∏i=12fiḂpi·,λi.ForE2, since x−y1,x−y22n≥x−y22n~2lB2 for x∈B and y2∈2lBc, using Ho¨lder’s inequality and Lemma 5, we have
(48)T∗f1χ2B,f2χ2Bcx=supδ>0∫x−y12+x−y22>δ2Kx,y1,y2f1y1χ2By1f2y2χ2Bcy2dy1dy2≤C∫ℝn2f1y1χ2By1f2y2χ2Bcy2x−y1,x−y22ndy1dy2≤C∫2Bf1y1dy1∫2Bcf2y2x−y22ndy2≤C∫2Bf1y1dy1∑l=1∞∫2l+1B\2lBf2y2x−y22ndy2≤Cf1χ2BLp1·χ2BLp1′·∑l=1∞2lB−2f2χ2l+1BLp2·χ2l+1BLp2′·≤C2Bλ1χ2BLp1·f1Ḃp1·,λ1χ2BLp1′·×∑l=1∞2lB−22l+1Bλ2χ2l+1BLp2·f2Ḃp2·,λ2χ2l+1BLp2′·≤CBλ1+λ2∏i=12fiḂpi·,λi∑l=1∞2lnλ2−1≤CBλ∏i=12fiḂpi·,λi.Therefore, we have(49)E2≤CBλχBLp·∏i=12fiḂpi·,λi.ForE3, as in the estimation of E2, we get
(50)E3≤CBλχBLp·∏i=12fiḂpi·,λi.ForE4, we note that x−y1,x−y22n≥x−y1n·x−y2n~2l1B·2l2B for x∈B, y1∈2lBcand y2∈2lBc, an application of Ho¨lder’s inequality and Lemma 5 gives
(51)T∗f1χ2Bc,f2χ2Bcx≤C∫ℝn2f1y1χ2Bcy1f2y2χ2Bcy2x−y1,x−y22ndy1dy2≤C∫2Bcf1y1x−y1ndy1∫2Bcf2y2x−y2ndy2≤C∑l1=1∞∑l2=1∞∏i=12∫2li+1B\2liBfiyix−yindyi≤C∏i=12∑li=1∞2liB−1∫2li+1Bfiyidyi≤C∏i=12∑li=1∞2liB−1fiχ2li+1BLpi·χ2li+1BLpi′·≤CBλ1+λ2∏i=12fiḂpi·,λi∑li=1∞2linλi≤CBλ∏i=12fiḂpi·,λi.Hence, we derive the estimate(52)E4≤CBλχBLp·∏i=12fiḂpi·,λi.The proof of Theorem 2.1 is complete.


## 4. Proof of Theorem 2.2

By linearity, it is enough to consider the operator with only one symbol. Fix thenb1∈CṀOs1·,u1ℝn and consider the operator
(53)Tb⟶∗1f⟶x=supδ>0b1xTδf1,⋯,fm−Tδb1f1,⋯,fm.Without loss of generality, we may assume thatm=2. Let f1,f2 be functions in Ḃp1·,v1ℝn and Ḃp2·,v2ℝn, respectively. We need to prove
(54)χBTb⟶∗1f⟶Lp·ℝn≤Cb1CṀOs1·,u1ℝnBλχBLp·ℝn∏i=12fiḂpi·,viℝn.Applying the Minkowski inequality, we obtain(55)∥χBTb⟶∗1f⟶∥Lp·≤∥χBb1−b1BT∗f1,f2∥Lp·+∥χBT∗b1−b1Bf1,f2∥Lp·≕I+II.ForI, we have
(56)I≤χBb1−b1BT∗f1χ2B,f2χ2BLp·+χBb1−b1BT∗f1χ2Bc,f2χ2BLp·+χBb1−b1BT∗f1χ2B,f2χ2BcLp·+χBb1−b1BT∗f1χ2Bc,f2χ2BcLp·≕I1+I2+I3+I4.ForI1, set 1/r·=1/p1·+1/p2·, then 1/p·=1/s1·+1/r·. By Lemmas 7, 6, and 8, we get
(57)I1≤χBb1−b1BLs1·χBT∗f1χ2B,f2χ2BLr·≤Cb1CṀOs1·,u1Bu1χBLs1·∏i=12fiχ2BLpi·≤Cb1CṀOs1·,u1Bu1χBLs1·∏i=12fiḂpi·,vi2Bviχ2BLpi·≤Cb1CṀOs1·,u1Bu1+v1+v2χBLs1·∏i=12fiḂpi·,viχBLpi·≤Cb1CṀOs1·,u1ℝnBλχBLp·ℝn∏i=12fiḂpi·,viℝn.ForI2, noting that x−y1,x−y22n≥x−y12n~2kB2 for x∈B and y1∈2kBc, by Ho¨lder’s inequality and Lemma 5, we get
(58)T∗f1χ2Bc,f2χ2Bx≤C∫2Bf2y2dy2∫2Bcf1y1x−y12ndy1≤C∫2Bf2y2dy2∑l=1∞∫2l+1B\2lBf1y1x−y12ndy1≤Cf2χ2BLp2·χ2BLp2′·∑l=1∞2lB−2f1χ2l+1BLp1·χ2l+1BLp1′·≤C2Bv2χ2BLp2·f2Ḃp2·,v2χ2BLp2′·×∑l=1∞2lB−22l+1Bv1∥χ2l+1B∥Lp1·f1Ḃp1·,v1χ2l+1BLp1′·≤CBv1+v2∏i=12fiḂpi·,vi∑l=1∞2lnv1−1≤CBv1+v2∏i=12fiḂpi·,vi.Thus, from Lemmas7 and 8, it follows that
(59)I2≤CBv1+v2∏i=12fiḂpi·,viχBb1−b1BLp·≤CBv1+v2∏i=12fiḂpi·,viχBLr·χBb1−b1BLs1·≤CBv1+v2∏i=12fiḂpi·,viχBLr·Bu1χBLs1·b1CṀOs1·,u1≤Cb1CṀOs1·,u1BλχBLp·∏i=12∥fi∥Ḃpi·,vi.As argued in the estimation ofE2 and E4, we also have
(60)I3≤Cb1CṀOs1·,u1BλχBLq·∏i=12fiḂpi·,vi,I4≤Cb1CṀOs1·,u1BλχBLq·∏i=12fiḂpi·,vi.ForII, we have the following decomposition
(61)II≤χBT∗b1−b1Bf1χ2B,f2χ2BLp·+χBT∗b1−b1Bf1χ2B,f2χ2BcLp·+χBT∗b1−b1Bf1χ2Bc,f2χ2BLp·+χBT∗b1−b1Bf1χ2Bc,f2χ2BcLp·≕II1+II2+II3+II4.ForII1, set 1/ϕ·=1/s1·+1/p1·, then 1/p·=1/ϕ·+1/p2·. By Ho¨lder’s inequality, Lemmas 7, 5, and 8, we get
(62)II1≤Cb1−b1Bf1χ2BLϕ·f2χ2BLp2·≤Cb1−b1Bχ2BLs1·f1χ2BLp1·f2χ2BLp2·≤Cb1−b12Bχ2BLs1·+b12B−b1Bχ2BLs1·f1χ2BLp1·f2χ2BLp2·≤Cb1−b12Bχ2BLs1·+1Bb1−b12Bχ2BLs1·χ2BLs1′·χ2BLs1·×f1χ2BLp1·f2χ2BLp2·≤Cb1−b12Bχ2BLs1·f1χ2BLp1·f2χ2BLp2·≤Cb1CṀOs1·,u1BλχBLp·∏i=12fiḂpi·,vi,where in the penultimate inequality we use the estimates(63)b12B−b1B≤1B∫Bb1−b12Bdx≤1B∫2Bb1−b12Bdx≤1Bb1−b12Bχ2BLs1·χ2BLs1′·.ForII2, set 1/ψ·=1/p1′·−1/s1·. By Ho¨lder’s inequality, Lemmas 5–8, we deduce that
(64)T∗b1−b1Bf1χ2B,f2χ2Bcx≤C∫2Bf1y1b1y1−b1Bdy1∫2Bcf2y2x−y22ndy2≤C∫2Bf1y1b1y1−b1Bdy1×∑l=1∞2lB−2∫2l+1B\2lBf2y2dy2≤Cb1−b1Bχ2BLs1·f1χ2BLp1·χ2BLψ·×∑l=1∞2lB−2f2χ2l+1BLp2·χ2l+1BLp2′·≤Cb1−b12Bχ2BLs1·+b12B−b1Bχ2BLs1·f1χ2BLp1·χ2BLψ·×∑l=1∞2lB−22l+1Bv2χ2l+1BLp2·f2Ḃp2·,v2χ2l+1BLp2′·≤C∥b1∥CṀOs1·,u12Bu1+v1χ2BLs1·χ2BLp1·χ2BLψ·∥f1∥Ḃp1·,v1×∑l=1∞2lB−22l+1Bv2χ2l+1BLp2·f2Ḃp2·,v2χ2l+1BLp2′·≤Cb1CṀOs1·,u1Bu1+v1+1f1Ḃp1·,v1∑l=1∞Bv2−1f2Ḃp2·,v22lnv2−1≤Cb1CṀOs1·,u1Bλf1Ḃp1·,v1f2Ḃp2·,v2∑l=1∞2lnv2−1≤Cb1CṀOs1·,u1Bλ∏i=12fiḂpi·,vi.This implies that(65)II2≤Cb1CṀOs1·,u1Bλb1Lp·∏i=12fiḂpi·,vi.ForII3, noting the fact that (see [17], Page 516])
(66)b12l+1B−b1B≤Cb1CṀOs1·,u1l+12l+1Bu1.Using Ho¨lder’s inequality, Lemmas 5–8, we obtain
(67)T∗b1−b1Bf1χ2Bc,f2χ2Bx≤C∫2Bf2y2dy2∫2Bcf1y1b1y1−b1Bx−y12ndy1≤C∫2Bf2y2dy2∑l=1∞2lB−2∫2l+1B\2lBf1y1b1y1−b1Bdy1≤Cf2χ2BLp2·χ2BLp2′·×∑l=1∞2lB−2b1−b1Bχ2l+1BLs1·b1−b1Bχ2l+1BLp1·χ2l+1BLψ·≤Cf2χ2BLp2·χ2BLp2′·∑l=1∞2lB−2b1−b12l+1Bχ2l+1BLs1·+b12l+1B−b1Bχ2l+1BLs1·f1χ2l+1BLp1·χ2l+1BLψ·≤CBu1+v1+v2∏i=12fiḂpi·,vib1CṀOs1·,u1∑l=1∞l2lnu1+v1−1≤Cb1CṀOs1·,u1Bλ∏i=12fiḂpi·,vi.Thus, we get(68)II3≤Cb1CṀOs1·,u1BλχBLp·∏i=12fiḂpi·,vi.ForII4, in view of uj+vj<0, as argued before, we arrive at the inequality
(69)T∗b1−b1Bf1χ2Bc,f2χ2Bcx≤C∫2Bcf1y1b1y1−b1Bx−y1ndy1∫2Bcf2y2x−y2ndy2≤C∑l1=1∞2l1B−1∫2l1+1B\2l1Bf1y1b1y1−b1Bdy1×∑l2=1∞2l2B−1∫2l2+1B\2l2Bf2y2dy2≤C∑l1=1∞2l1B−1f1χ2l1+1BLp1·b1−b1Bχ2l1+1BLs1·χ2l1+1BLψ·×∑l2=1∞2l2B−1f2χ2l2+1BLp2·f2χ2l2+1BLp2′·≤C∑l1=1∞2l1B−12l1+1Bv1χ2l1+1BLp1·f1Ḃp1·,v1χ2l1+1BLψ·×b1CṀOs1·,u12l1+1Bu1χ2l1+1BLs1·+l1+12l1+1Bu1χ2l1+1BLs1·×∑l2=1∞2l2B−12l2+1Bv2χ2l2+1BLp2·f2Ḃp2·,v2χ2l2+1BLp2′·≤Cb1CṀOs1·,u1Bλ∏i=12fiḂpi·,vi∑l1=1∞l12l1nu1+v1∑l2=1∞2l2nv2≤Cb1CṀOs1·,u1Bλ∏i=12fiḂpi·,vi.Thus, we get(70)II4≤Cb1CṀOs1·,u1BλχBLp·∏i=12fiḂpi·,vi.The proof of Theorem 2.2 is complete.


## 5. Proof of Theorem 2.3

Fixb1∈CṀOs1·,β1ℝn; as in the proof of Theorem 2.2, we consider the operator
(71)Iα,b⟶1f⟶x=b1x−b1BIαf1,f2x−Iαb1−b1Bf1,f2x.Without loss of generality, we may assume thatm=2. Let f1,f2 be functions in Ḃp1·,γ1ℝn and Ḃp2·,γ2ℝn, respectively. We need to prove
(72)χBIα,b⟶1f⟶Lq·ℝn≤Cb1CṀOs1·,β1ℝnBλχBLq·ℝn∏i=12fiḂpi·,γiℝn.The Minkowski inequality implies that(73)χBIα,b⟶1f⟶Lq·≤χBb1−b1BIαf1,f2Lq·+χBIαb1−b1Bf1,f2Lq·≕J+JJ.ForJ, we decompose
(74)J≤χBb1−b1BIαf1χ2B,f2χ2BLq·+χBb1−b1BIαf1χ2B,f2χ2BcLq·+χBb1−b1BIαf1χ2B,f2χ2BcLq·+χBb1−b1BIαf1χ2Bc,f2χ2BcLq·≕J1+J2+J3+J4.ForJ1, since λ=β1+γ1+γ2+α/n, 1/p·=1/p1·+1/p2·−α/n, and 1/q·=1/s1·+1/p·, we use Lemmas 6–9 and obtain
(75)J1≤χBb1−b1BLs1·χBIαf1χ2B,f2χ2BLp·≤Cb1CṀOs1·,β1Bβ1χBLs1·∏i=12fiχ2BLpi·≤Cb1CṀOs1·,β1Bβ1χBLs1·∏i=12fiḂpi·,γi2Bγiχ2BLpi·≤Cb1CṀOs1·,β1Bβ1+γ1+γ2+α/n+1/q·∏i=12fiḂpi·,γi≤Cb1CṀOs1·,β1BλχBLq·∏i=12fiḂpi·,γi.ForJ2, noting that x−y1,x−y22n−α≥x−y22n−α~2kB2−α/n for x∈B and y2∈2kBc, by Ho¨lder’s inequality and Lemma 5, we get
(76)Iαf1χ2B,f2χ2Bcx≤C∫2Bf1y1dy1∫2Bcf2y2x−y22n−αdy2≤C∫2Bf1y1dy1∑l=1∞∫2l+1B\2lBf2y2x−y22n−αdy2≤Cf1χ2BLp1·χ2BLp1′·∑k=1∞2kB−2+α/nf2χ2l+1BLp2·χ2l+1BLp2′·≤C2Bγ1χ2BLp1·f1Ḃp1·,γ1χ2BLp1′·×∑l=1∞2lB−2+α/n2l+1Bγ2χ2l+1BLp2·f2Ḃp2·,γ2χ2l+1BLp2′·≤CBγ1+γ2+α/n∏i=12fiḂpi·,γi∑l=1∞2lnγ2+α/n−1≤CBγ1+γ2+α/n∏i=12fiḂpi·,γi.Then, from Lemmas7 and 8, it follows that
(77)J2≤CBγ1+γ2+α/n∏i=12fiḂpi·,γiχBb1−b1BLq·≤CBγ1+γ2+α/n∏i=12fiḂpi·,γiχBLp·χBb1−b1BLs1·≤CBγ1+γ2+α/n∏i=12fiḂpi·,γiχBLp·Bβ1χBLs1·b1CṀOs1·,β1≤Cb1CṀOs1·,β1BλχBLq·∏i=12fiḂpi·,γi.Similarly, we have(78)J3≤Cb1CṀOs1·,β1BλχBLq·∏i=12fiḂpi·,γi.ForJ4, noting that x−y1,x−y22n−α~x−y1n−α/2·x−y2n−α/2~2k1B1−α/2n·2k2B1−α/2n for x∈B, y1∈2kBc, and y2∈2kBc, by Ho¨lder’s inequality and Lemma 5, we get
(79)Iαf1χ2Bc,f2χ2Bcx≤C∫2Bcf1y1x−y1n−α/2dy1∫2Bcf2y2x−y2n−α/2dy2≤C∑l1=1∞∑l2=1∞∏i=12∫2li+1B\2liB∣fiyi∣x−yin−α/2dyi≤C∏i=12∑li=1∞2liB−1+α2n∫2li+1Bfiyidyi≤C∏i=12∑li=1∞2liB−1+α2nfiχ2li+1BLpi·χ2li+1BLpi′·≤CBγ1+γ2+αn∏i=12fiḂpi·,γi∑li=1∞2linγi+α2n≤CBγ1+γ2+αn∏i=12fiḂpi·,γi.Therefore, as in the estimation ofJ2, we have
(80)J4≤Cb1CṀOs1·,β1BλχBLq·∏i=12fiḂpi·,γi.ForJJ, we use the following decomposition
(81)JJ≤χBIαb1−b1Bf1χ2B,f2χ2BLq·\+χBIαb1−b1Bf1χ2B,f2χ2BcLq·+χBIαb1−b1Bf1χ2Bc,f2χ2BLq·+χBIαb1−b1Bf1χ2Bc,f2χ2BcLq·≕JJ1+JJ2+JJ3+JJ4.ForJJ1, set 1/φ·=1/s1·+1/p1·, then 1/q·=1/φ·+1/p2·−α/n. By Lemmas 9, 7, and 5, we get
(82)JJ1≤Cb1−b1Bf1χ2BLφ·f2χ2BLp2·≤Cb1−b1Bχ2BLs1·∏i=12fiχ2BLpi·≤Cb1−b12Bχ2BLs1·+b12B−b1Bχ2BLs1·∏i=12fiχ2BLpi·≤C∏i=12fiχ2BLpi·b1−b12Bχ2BLs1·+1Bb1−b12Bχ2BLs1·χ2BLs1′·χ2BLs1·≤Cb1−b12Bχ2BLs1·∏i=12fiχ2BLpi·≤Cb1CṀOs1·,β1BλχBLq·∏i=12fiḂpi·,γi.ForJJ2, set 1/ϕ·=1−1/p1·−1/s1·. By Ho¨lder’s inequality, Lemmas 5–8, we derive
(83)Iαb1−b1Bf1χ2B,f2χ2Bcx≤C∫2Bf1y1b1y1−b1Bdy1∫2Bcf2y2x−y22n−αdy2≤C∫2Bf1y1b1y1−b1Bdy1×∑l=1∞2lB−2+α/n∫2l+1B\2lBf2y2dy2≤Cb1−b1Bχ2BLs1·f1χ2BLp1·χ2BLϕ·×∑l=1∞2lB−2+α/nf2χ2l+1BLp2·χ2l+1BLp2′·≤Cb1−b12Bχ2BLs1·+b12B−b1Bχ2BLs1·f1χ2BLp1·χ2BLϕ·×∑l=1∞2lB−2+α/n2l+1Bγ2χ2l+1BLp2·f2Ḃp2·,γ2χ2l+1BLp2′·≤Cb1CṀOs1·,β12Bβ1+γ1χ2BLs1·χ2BLp1·χ2BLϕ·f1Ḃp1·,γ1×∑l=1∞2lB−2+α/n2l+1Bγ2χ2l+1BLp2·f2Ḃp2·,γ2χ2l+1BLp2′·≤Cb1CṀOs1·,β1Bβ1+γ1+1f1Ḃp1·,γ1∑l=1∞Bγ2+α/n−1f2Ḃp2·,γ22lnγ2+α/n−1≤Cb1CṀOs1·,β1Bλf1Ḃp1·,γ1f2Ḃp2·,γ2∑l=1∞2lnγ2+α/n−1≤Cb1CṀOs1·,β1Bλ∏i=12fiḂpi·,γi.Therefore, we have(84)JJ2≤Cb1CṀOs1·,β1BλχBLq·∏i=12fiḂpi·,γi.ForJJ3, we note that β1+γ1<−α/n and derive
(85)Iαb1−b1Bf1χ2Bc,f2χ2Bx≤C∫2Bf2y2dy2∫2Bcf1y1b1y1−b1Bx−y12n−αdy1≤C∫2Bf2y2dy2∑l=1∞2lB−2+α/n∫2l+1B\2lBf1y1b1y1−b1Bdy1≤Cf2χ2BLp2·χ2BLp2′·×∑l=1∞2lB−2+α/nb1−b1Bχ2l+1BLs1·f1χ2l+1BLp1·χ2l+1BLϕ·≤Cf2χ2BLp2·χ2BLp2′·∑l=1∞2lB−2+α/nb1−b12l+1Bχ2l+1BLs1·+b12l+1B−b1Bχ2l+1BLs1·f1χ2l+1BLp1·χ2l+1BLϕ·≤CBβ1+γ1+γ2+α/n∏i=12fiḂpi·,γib1CṀOs1·,β1∑l=1∞l2lnβ1+γ1+α/n−1≤Cb1CṀOs1·,β1Bλ∏i=12fiḂpi·,γi.Then, we conclude that(86)JJ3≤Cb1CṀOs1·,β1BλχBLq·∏i=12fiḂpi·,γi.ForJJ4, since β1+γ1<−α/n, then β1+γ1+α/2n<0, using Ho¨lder’s inequality, Lemma 5, and the fact that γ2<−α/2n, we have
(87)Iαb1−b1Bf1χ2Bc,f2χ2Bcx≤C∫2Bcf1y1∣b1y1−b1B∣x−y1n−α/2dy1∫2Bcf2y2x−y2n−α/2dy2≤C∑l1=1∞2l1B−1+α/2n∫2l1+1B\2l1Bf1y1b1y1−b1Bdy1×∑l2=1∞2l2B−1+α/2n∫2l2+1B\2l2Bf2y2dy2≤C∑l1=1∞2l1B−1+α/2nf1χ2l1+1BLp1·∥b1−b1Bχ2l1+1B∥Ls1·χ2l1+1BLϕ·×∑l2=1∞2l2B−1+α/2nf2χ2l2+1BLp2·χ2l2+1BLp2′·≤C∑l1=1∞2l1B−1+α/2n2l1+1Bγ1χ2l1+1BLp1·f1Ḃp1·,γ1∥χ2l1+1B∥Lϕ·×b1CṀOs1·,β12l1+1Bβ1χ2l1+1BLs1·+l1+12l1+1Bβ1χ2l1+1BLs1·×∑l2=1∞2l2B−1+α/2n2l2+1Bγ2χ2l2+1BLp2·f2Ḃp2·,γ2χ2l2+1BLp2′·≤Cb1CṀOs1·,β1Bλ∏i=12fiḂpi·,γi∑l1=1∞l12l1nβ1+γ1+α/2n∑l2=1∞2l2nγ2+α/2n≤Cb1CṀOs1·,β1Bλ∏i=12fiḂpi·,γi.Thus, we get(88)JJ4≤Cb1CṀOs1·,β1BλχBLq·∏i=12fiḂpi·,γi.The proof of Theorem 2.3 is complete.



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