# Rough Multilinear Fractional Integrals on Weighted Morrey Spaces

Authors: Xiao Li; Runqing Cui
Journal: Journal of Function Spaces
 (2015)

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## Abstract

It is showed that a class of multilinear fractional operators with rough kernels, which are similar to the higher-order commutators for the rough fractional integrals, are bounded on the weighted Morrey spaces.


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## Body

## 1. Introduction and Results

Suppose thatSn-1 denotes the unit sphere of Rn(n≥2) equipped with the usual Lebesgue measure, Ω∈Ls(Sn-1)(s>1) is homogeneous of degree zero, and Aj, j=1,…,k, are functions defined on Rn. Consider the following multilinear fractional integral with rough kernel defined by(1)TΩ,αA1,…,Akfx=∫RnΩx-yx-yn-α+N∏j=1kRmjAj;x,yfydy,where 0<α<n,  N=∑j=1k(mj-1), and(2)RmjAj;x,y=Ajx-∑γ<mj1γ!DγAjyx-yγ.Whenk=1, A1=A, and m1=1, then TΩ,αA is just the commutator of the rough fractional integral TΩ,α with the function A:(3)A,TΩ,αfx=∫RnΩx-yx-yn-αAx-Ayfydy.If m1≥2, then TΩ,αA is nontrivial generalization of the above commutator. The weighted (Lp,Lq)-boundedness of the operator TΩ,αA was given by Wu and Yang in [1].Whenk>1, Aj=A for j=1,…,k, and mj=1, then TΩ,αA1,…,Ak is the higher-order commutator of the fractional integral TΩ,α with the function A:(4)TΩ,αA,kfx=∫RnΩx-yx-yn-αAx-Aykfydy.The weighted (Lp,Lq)-boundedness of TΩ,αA,k was given by Ding and Lu in [2].Ding and Lu in [3] proved the following result.Theorem 1.
Let0<α<n,  1<p<n/α, let 1/q=1/p-α/n, and let Ω be homogeneous of degree zero with Ω∈Ls(Sn-1),s>1. Moreover, for 1≤j≤k, |γ|=mj-1, mj≥2, and DγAj∈BMO(Rn). If ωs′∈A(p/s′,q/s′), then there exists a constant C, independent of Aj(1≤j≤k) and f, such that(5)TΩ,αA1,…,AkfLqωq,Rn≤C∏j=1k∑γ=mj-1DγAj∗fLpωp,Rn.Here and in the sequel,p′ always denotes the conjugate index of any p>1; that is, 1/p+1/p′=1, and C stands for a constant which is independent of the main parameters, but it may vary from line to line.The purpose of this paper is to discuss the boundedness properties of the rough fractional multilinear integral operatorsTΩ,αA1,…,Ak on appropriate weighted Morrey spaces.The classical Morrey spaces were introduced by Morrey [4] in 1938 and have been studied intensively by various authors and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations [5–7].Let1≤p<∞ and 0≤λ≤n; Morrey spaces are defined by (6)Lp,λRn=f∈Lloc⁡pRn:fLp,λRn<∞,where(7)fLp,λRn=supx∈Rn,r>0⁡r-λ/pfLpBx,r.Note that Lp,0(Rn)=Lp(Rn) and Lp,n(Rn)=L∞(Rn). If λ<0 or λ>n, then Lp,λ(Rn)=Θ, where Θ is the set of all functions equivalent to 0 on Rn.LetΦ(r),  r>0, be a growth function, that is, a positive increasing function in (0,∞), which satisfies doubling condition (8)Φ2r≤DΦr,∀r>0,where D=D(Φ)≥1 is a doubling constant independent of r. Mizuhara in [8] gave generalization Morrey spaces Lp,Φ(Rn) considering Φ(r) instead of rλ in (7).Komori and Shirai [9] introduced a version of the weighted Morrey space Lp,κ(ω,Rn), which is a natural generalization of the weighted Lebesgue spaces Lp(ω,Rn).Let1≤p<∞,  0<κ<1, and ω a weight function. The spaces Lp,κ(ω,Rn) are defined by(9)Lp,κω,Rn=f∈Lloc⁡pω:fLp,κω,Rn<∞,where(10)fLp,κω,Rn=supx∈Rn,r>0⁡1ωBx,rκ·∫Bx,rfypωydy1/p.In order to deal with the fractional order case, we need to consider the weighted Morrey spaces with two weight functions; they were also introduced by Komori and Shirai in [9].Let1≤p<∞, 0<κ<1. For two weight functions u and v, the spaces Lp,κ(u,v)(Rn) are defined by(11)Lp,κu,vRn=f∈Lloc⁡pu:fLp,κu,v<∞,where(12)fLp,κu,vRn=supx∈Rn,r>0⁡1vBx,rκ∫Bx,rfypuydy1/p.Let1≤p<∞,  φ a positive measurable function on Rn×(0,∞), and ω a nonnegative measurable function on Rn. We denote by Mφp(ω,Rn) other weighted Morrey spaces, the spaces of all functions f∈Lloc⁡p(ω) with finite norm(13)fMφpω,Rn=supx∈Rn,r>0⁡1φx,r1wBx,rfLpω,Bx,rp1/p<∞,where (14)fLpω,Bx,r=∫Bx,rfypωydy.Remark 2.
(1) Ifω=1 and φ(x,r)=r(λ-n)/p with 0≤λ≤n, then Mφp(ω,Rn)=Lp,λ(Rn).
(2) Ifω=1,φ(x,r)p=Φ(r)r-n, then Mφp(ω,Rn)=Lp,Φ(Rn).
(3) Ifφ(x,r)=ω(B(x,r))(κ-1)/p, then Mφp(ω,Rn)=Lp,κ(ω,Rn).
(4) Ifφ(x,r)=ν(B(x,r))κ/pω(B(x,r))-1/p, then Mφp(ω,Rn)=Lp,κ(ν,ω)(Rn).Our main results can be formulated as follows.Theorem 3.
Let0<α<n, 1<p<n/α, let 1/q=1/p-α/n, and let Ω be homogeneous of degree zero with Ω∈Ls(Sn-1), s>1. Suppose ωs′∈A(p/s′,q/s′) and (φ1,φ2) satisfies the condition(15)∫s∞1+ln⁡rsk·essinfr<t<∞⁡φ1x,tωpBx,t1/pωqBx,r1/q1rdr≤C0φ2x,s,where C0 does not depend on x and r. If Aj have derivatives of order mj-1 in BMORn, j=1,…,k, then there is a constant C>0, independent of Aj(1≤j≤k) and f, such that(16)TΩ,αA1,…,AkfMφ2qωq,Rn≤C∏j=1k∑γ=mj-1DγAj∗fMφ1pωp,Rn.Remark 4.
Letφ1(x,t)=(Φ(t)t-n)1/p, let φ2(x,t)=(Φ(t))1/ptn/q, and let ω=1. If 1≤D(Φ)≤2n, it is easy to prove that (φ1,φ2) satisfies condition (15).Remark 5.
Letφ1(x,t)=(ωq(B(x,t)))κ/p(ωp(B(x,t)))-1/p, φ2(x,t)=(ωq(B(x,t)))κ/p-1/q, 0<κ<p/q, and wq∈A∞(Rn). By [10], we know that (φ1,φ2) satisfies condition (15).Corollary 6.
Let0<α<n, 1<p<n/α, let 1/q=1/p-α/n, and let Ω be homogeneous of degree zero with Ω∈Ls(Sn-1),s>1. Suppose 1≤D(Φ)≤2n, and Aj have derivatives of order mj-1 in BMO(Rn), j=1,…,k; then TΩ,αA1,…,Ak is bounded from Lp,Φ(Rn) to Lq,Φq/p(Rn) and(17)TΩ,αA1,…,AkfLq,Φq/pRn≤C∏j=1k∑γ=mj-1DγAj∗fLp,ΦRn.Remark 7.
In Corollary6, let k=1 and m1=1; then we obtain the main result in [11].Corollary 8.
Let0<α<n, 1<p<n/α, let 1/q=1/p-α/n, and let Ω be homogeneous of degree zero with Ω∈Ls(Sn-1), s>1. Suppose 0<κ<p/q, ωs′∈A(p/s′,q/s′), and Aj have derivatives of order mj-1 in BMO(Rn), j=1,…,k; then TΩ,αA1,…,Ak is bounded from Lp,κ(ωp,ωq)(Rn) to Lq,κq/p(ωq,Rn) and(18)TΩ,αA1,…,AkfLq,κq/pωq,Rn≤C∏j=1k∑γ=mj-1DγAj∗fLp,κωp,ωqRn.Remark 9.
In Corollary8, if k=1 and m1=1, then we obtain the main result in [12]; if Ω=1 and ω=1, then we obtain the main result in [13].


## 2. Some Preliminaries

We begin with some properties ofAp(Rn) weights which play a great role in the proofs of our main results.Weightω is a nonnegative, locally integrable function on Rn. Let B=B(x0,rB) denote the ball with the center x0 and radius rB and let λB=B(x0,λrB) for any λ>0. For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by |E| and set weighted measure ω(E)=∫Eω(x)dx. For any given weight function ω on Rn, X⫅Rn, and 0<p<∞, denote by Lp(ω,X) the space of all functions f satisfying (19)fLpω,X=∫Xfxpωxdx1/p<∞.Weightω is said to belong to Ap(Rn) for 1<p<∞, if there exists a constant C such that(20)1B∫Bωxdx1B∫Bωx1-p′dxp-1≤C.A1(Rn) denotes the set consisting of all weight functions ω:(21)1B∫Bωydy≤C·essinfx∈B⁡ωxfor  every  ballB⊂Rn.Weight ω is said to belong to A∞(Rn) if there are positive numbers C and δ such that (22)ωEωB≤CEBδfor all balls B and all measurable E⊂B. It is well known that (23)A∞Rn=⋃1≤p<∞ApRn.The classicalAp weight theory was first introduced by Muckenhoupt in the study of weighted Lp-boundedness of Hardy-Littlewood maximal function in [14].Lemma 10 (see [14, 15]).
Supposeω∈Ap(Rn). The following statements hold: (i)
For any1≤p<∞, there is a positive number C such that(24)ωBkωBj≤C2npk-jfork>j.(ii)
For any1≤p<∞, there is a positive number C and δ such that(25)ωBkωBj≥C2δk-jfork>j.(iii)
For any1<p<∞, one has ω1-p′∈Ap′.We also need another weight classA(p,q) introduced by Muckenhoupt and Wheeden in [16] to study weighted boundedness of fractional integral operators.Given1≤p≤q<∞, we say that ω∈A(p,q) if there exists a constant C such that, for every ball B⊂Rn, the inequality(26)1B∫Bωy-p′dy1/p′1B∫Bωyqdy1/q≤Cholds when 1<p<∞, and for every ball B⊂Rn the inequality(27)1B∫Bωyqdy1/q≤C·essinfx∈B⁡wxholds when p=1.By (26), we have(28)∫Bωy-p′dy1/p′∫Bωyqdy1/q≤CB1/p′+1/q.We summarize some properties about weightsA(p,q); see [15, 16].Lemma 11.
Given1≤p≤q<∞(i)
ω∈A(p,q) if and only if ωq∈A1+q/p′;(ii)
ω∈A(p,q) if and only if ω-p′∈A1+p′/q;(iii)
ifp1<p2 and q2>q1, then A(p1,q1)⊂A(p2,q2).John and Nirenberg introduced the function space of BMO in [17]. A locally integrable function b is said to be in BMO(Rn) if (29)supB⊂Rn⁡1B∫Bbx-bBdx=b∗<∞,where(30)bB=1B∫Bbydy.Lemma 12 (see [10]).
Supposeω∈A∞ and b∈BMO(Rn). Then for any 1≤p<∞ and r1,r2>0, we have(31)1ωBx0,r1·∫Bx0,r1bx-bBx0,r2pωxdx1/p≤Cb∗·1+ln⁡r1r2.Below we recall some conclusions aboutRm(A;x,y).Lemma 13 (see [9]).
Letb be a function on Rn with the mth derivatives in Lq(Rn), q>n. Then (32)Rmb;x,y≤Cx-ym∑γ=m1Bx,5nx-y·∫Bx,5nx-yDγbzqdz1/q.Lemma 14 (see [18]).
For fixedx∈Rn, let (33)A¯x=Ax-∑γ=m-11γ!DγABx,5nx-yxγ.Then Rm(A;x,y)=Rm(A¯;x,y).Lemma 15.
Letx∈B(x0,l), and let y∈B(x0,2j+1l)∖B(x0,2jl). Then(34)RmA;x,y≤Cx-ym-1j∑γ=m-1DγA∗+∑γ=m-1DγAy-DγABx0,l.Proof.
From Lemma14, we have(35)RmA;x,y=RmA¯;x,y≤Rm-1A¯;x,y+∑γ=m-11γ!DγA¯yx-ym-1.By Lemma 13,(36)Rm-1A¯;x,y≤Cx-ym-1∑γ=m-1DγA∗.
Ifx∈B(x0,l), y∈B(x0,2j+1l)∖B(x0,2jl), we can easily see that 2j-1l≤|x-y|<2j+2l. We get (37)Bx0,2j-1l⊂Bx,5nx-y⊂100nBx0,2jl.Then (38)100nBx0,2jlBx,5nx-y≤100nBx0,2jlBx0,2j-1l≤C.Thus (39)DγABx,5nx-y-DγABx0,2jl≤1Bx,5nx-y·∫Bx,5nx-yDγAy-DγABx0,2jldy≤C100nBx0,2jl·∫100nBx0,2jlDγAy-DγABx0,2jldy≤CDγA∗.Note that (40)DγABx0,2jl-DγABx0,l≤∑k=1jDγABx0,2kl-DγABx0,2k-1l≤2njDγA∗.Then (41)DγABx,5nx-y-DγABx0,l≤DγABx,5nx-y-DγABx0,2jl+DγABx0,2jl-DγABx0,l≤CjDγA∗.Thus(42)DγA¯y=DγAy-DγABx,5nx-y≤DγAy-DγABx0,l+DγABx,5nx-y-DγABx0,l≤DγAy-DγABx0,l+CjDγA∗.
Combining with (35), (36), and (42), then (34) is proved.Finally, we recall a relationship between essential supremum and essential infimum.Lemma 16 (see [19]).
Letf be a real-valued nonnegative function and measurable on E. Then (43)essinfx∈E⁡fx-1=esssupx∈E⁡1fx.


## 3. A Local Estimate

To prove Theorem3, we first investigate the following local estimate.Theorem 17.
Let0<α<n, 1<p<n/α, let 1/q=1/p-α/n, and let Ω be homogeneous of degree zero with Ω∈Ls(Sn-1), s>1. Suppose ωs′∈A(p/s′,q/s′) and Aj have derivatives of order mj-1 in BMO(Rn), j=1,…,k; then, for any l>0, there is a constant C independent of f such that(44)TΩ,αA1,…,AkfLqωq,Bx0,l≤C∏j=1k∑γ=mj-1DγAj∗ωqBx0,l1/q·∫2l∞fLpωp,Bx0,rωqBx0,r-1/q1rdr.Proof.
To simplify the proof process of Theorem17 in the following discussion we consider only the case k=2. The method can be used to deal with the case k>2 without any essential difficulty.
We writef as f=f1+f2, where f1(y)=f(y)χB(x0,2l)(y), and χB(x0,2l) denotes the characteristic function of B(x0,2l). Then(45)TΩ,αA1,A2fLpωp,Bx0,l≤TΩ,αA1,A2f1Lpωp,Bx0,l+TΩ,αA1,A2f2Lpωp,Bx0,l.
Sincef1∈Lp(ωp,Rn), by the boundedness of TΩ,αA1,A2 from Lp(ωp,Rn) to Lq(ωq,Rn) (Theorem 1) we get(46)TΩ,αA1,A2f1Lqωq,Bx0,l≤TΩ,αA1,A2f1Lqωq,Rn≤C∏j=12∑γ=mj-1DγAj∗f1Lpωp,Rn=C∏j=12∑γ=mj-1DγAj∗fLpωp,Bx0,2l.Noting that q>p>1 and s′p/p′(p-s′)≥1, then by Hölder’s inequality (47)1≤1B∫Bωypdy1/p1B∫Bωy-p′dy1/p′≤1B∫Bωyqdy1/q·1B∫Bωy-s′p/p-s′dyp-s′/s′p.This means (48)ln/s′-α≤ωqBx0,l1/qω-1Ls′p/p-s′Bx0,l.Thus (49)fLpωp,Bx0,2l≤Cln/s′-αfLpω,Bx0,2l·∫2l∞1rn/s′-α+1dr≤CωqBx0,l1/q·ω-1Ls′p/p-s′Bx0,l∫2l∞fLpω,Bx0,r·1rn/s′-α+1dr≤CωqBx0,l1/q·∫2l∞fLpω,Bx0,rω-1Ls′p/p-s′Bx0,r1rn/s′-α+1dr.Since ωs′∈A(p/s′,q/s′), by (28), we get(50)ωqBx0,r1/qω-1Ls′p/p-s′Bx0,r≤Crn/s′-αholds for all r>0. Then(51)TΩ,αA1,A2f1Lqωq,Bx0,l≤C∏j=12∑γ=mj-1DγAj∗·ωqBx0,l1/q·∫2l∞fLpωp,Bx0,rωqBx0,r-1/q1rdr.
LetN=m1+m2-2, and let Δi=(B(x0,2j+1l))∖(B(x0,2jl)). Suppose x∈B(x0,l). By Lemma 15,(52)TΩ,αA1,A2f2x≤∫Bx0,2lcΩx-yx-yn-α+NRm1A1;x,yRm2A2;x,yfydy≤C∑j=1∞∫ΔiΩx-yfyx-yn-α∏k=12j∑γ=mk-1DγAk∗+∑γ=mk-1DγAky-DγAkBx0,ldy≤C∏j=12∑γ=mj-1DγAj∗∑j=1∞j2∫ΔiΩx-yfyx-yn-αdy+C∑γ=m1-1DγA1∗∑γ=m2-1∑j=1∞j∫ΔiΩx-yfyx-yn-αDγA2y-DγA2Bx0,ldy+C∑γ=m2-1DγA2∗∑γ=m1-1∑j=1∞j∫ΔiΩx-yfyx-yn-αDγA1y-Dγ1A1Bx0,ldy+C∑γ=m1-1∑γ=m2-1∑j=1∞∫ΔiΩx-yfyx-yn-α∏k=12DγAky-DγAkBx0,ldy=I1+I2+I3+I4.
By Hölder’s inequalities,(53)∫ΔiΩx-yfyx-yn-αdy≤∫ΔiΩx-ysdy1/s·∫Δifys′x-yn-αs′dy1/s′.When x∈B(x0,s) and y∈Δi, then by a direct calculation, we can see that 2j-1l≤|y-x|<2j+1l. Then(54)∫ΔiΩx-ysdy1/s≤CΩLsSn-1Bx0,2j+1l1/s.We also note that if x∈B(x0,l),y∈B(x0,2l)c, then |y-x|~|y-x0|. Consequently(55)∫Δifys′x-yn-αs′dy1/s′≤1Bx0,2j+1l1-α/n∫Bx0,2j+1lfys′dy1/s′.Then (56)I1≤C∏j=12∑γ=mj-1DγAj∗·∑j=1∞j22j+1lα-n/s′∫Bx0,2j+1lfys′dy1/s′.Since s′<p, it follows from Hölder’s inequality that (57)∫Bx0,2j+1lfys′dy1/s′≤CfLpωp,Bx0,2j+1lω-1Ls′p/p-s′Bx0,2j+1l.Then(58)∑j=1∞j22j+1lα-n/s′∫Bx0,2j+1sfys′dy1/s′≤C∑j=1∞1+ln⁡2j+1ll22j+1lα-n/s′·fLpωp,Bx0,2j+1lω-1Ls′p/p-s′Bx0,2j+1l≤C∑j=1∞∫2j+1l2j+2l1+ln⁡rl2fLpωp,Bx0,r·ω-1Ls′p/p-s′Bx0,r1r1-α+n/s′dr≤C∫2l∞1+ln⁡rl2fLpωp,Bx0,rω-1Ls′p/p-s′Bx0,r·1r1-α+n/s′dr.From (50) we know(59)ω-1Ls′p/p-s′Bx0,r≤Crn/s′-αωqBx0,r-1/q.Then (60)I1≤C∏j=12∑γ=mj-1DγAj∗∫2l∞1+ln⁡rl2·fLpωp,Bx0,rωqBx0,r-1/q1rdr.
By Hölder’s inequality and (54), (55) we have(61)∫ΔiΩx-yfyx-yn-αDγA2y-DγA2Bx0,ldy≤∫ΔiΩx-ysdy1/s·∫ΔiDγA2y-DγA2Bx0,lfys′x-yn-αs′dy1/s′≤C∑j=1∞2j+1lα-n/s′·∫Bx0,2j+1lDγA2y-DγA2Bx0,ls′·fys′dy1/s′.Applying Hölder’s inequality again, we get (62)∫Bx0,2j+1lDγA2y-DγA2Bx0,ls′·fys′dy1/s′≤CfLpωp,Bx0,2j+1l·DγA2y-DγA2Bx0,l·ω·-1Lps′/p-s′Bx0,2j+1l.Consequently, (63)I2≤C∑γ=m1-1DγA1∗∑γ=m2-1∑j=1∞∫2j+1l2j+2l1+ln⁡2j+1ll·2j+1lα-n/s′fLpωp,Bx0,r·DγA2y-DγA2Bx0,lω·-1Lps′/p-s′Bx0,rdr≤C∑γ=m1-1DγA1∗∑γ=m2-1∫2l∞fLpωp,Bx0,r·DγA2y-DγA2Bx0,lω·-1Lps′/p-s′Bx0,r·1r1-α+n/s′dr.By ωs′∈A(p/s′,q/s′) and (ii) of Lemma 11 we know ω-s′p/(p-s′)∈A1+ps′/(p-s′)q. Then it follows from (31) and (59) that(64)DγA2y-DγA2Bx0,lω·-1Lps′/p-s′Bx0,r≤∫Bx0,rDγA2y-DγA2Bx0,lps′/p-s′·ω-ps′/p-s′ydyp-s′/ps′≤CDγA2∗1+ln⁡rl·ω-ps′/p-s′Bx0,rp-s′/ps′=CDγA2∗1+ln⁡rlω-1Lps′/p-s′Bx0,r≤CDγA2∗1+ln⁡rl·rn/s′-αωqBx0,r-1/q.Hence (65)I2≤C∏j=12∑γ=mj-1DγAj∗∫2l∞1+ln⁡rl·fLpωp,Bx0,rωqBx0,r-1/q1rdr.
Similar to the estimates forI2, we have (66)I3≤C∏j=12∑γ=mj-1DγAj∗∫2l∞1+ln⁡rl·fLpωp,Bx0,rωqBx0,r-1/q1rdr.
Finally, we come to estimateI4.
Similar to (59), we get (67)∫ΔiΩx-yfyx-yn-α∏k=12DγAky-DγAkBx0,ldy≤C2j+1lα-n/s′·∫Bx0,2j+1l∏k=12DγAky-DγAkBx0,ls′·fys′dy1/s′.By Hölder’s inequality we get (68)∫Bx0,2j+1l∏k=12DγAky-DγAkBx0,ls′·fys′dy1/s′≤CfLpωp,Bx0,2j+1l·∏k=12DγAky-DγAkBx0,l·ω·-1Lps′/p-s′Bx0,2j+1l≤CfLpωp,Bx0,2j+1l·∏k=12DγAky-DγAkBx0,l·ω·-1/2L2ps′/p-s′Bx0,2j+1l.Then (69)∑j=1∞∫ΔiΩx-yfyx-yn-α∏k=12DγAky-DγAkBx0,ldy≤C∫2l∞fLpωp,Bx0,r∏k=12DγAky-DγAkBx0,l·ω·-1/2L2ps′/p-s′Bx0,r1r1-α+n/s′dr.Since ω-s′p/(p-s′)∈A1+ps′/p-s′q, similar to (64), we can deduce(70)DγAky-DγAkBx0,l·ω·-1/2L2ps′/p-s′Bx0,2j+1l≤CDγAk∗1+ln⁡rlω-ps′/p-s′Bx0,rp-s′/2ps′.Then by (70) we get (71)∑j=1∞∫ΔiΩx-yfyx-yn-α∏k=12DγAky-DγAkBx0,ldy≤C∏k=12DγAk∗∫2l∞1+ln⁡rl2fLpωp,Bx0,r·ωqBx0,r-1/q1rdr.Thus (72)I4≤C∏j=12∑γ=mj-1DγAj∗∫2l∞1+ln⁡rl·fLpωp,Bx0,rωqBx0,r-1/q1rdr.
Combining the estimates forI1, I2, I3, and I4, we get (73)TΩ,αA1,A2f2x≤C∏j=12∑γ=mj-1DγAj∗·∫2l∞1+ln⁡rlfLpωp,Bx0,rωqBx0,r-1/q·1rdr.Then(74)TΩ,αA1,A2f2Lpωp,Bx0,l≤C∏j=12∑γ=mj-1DγAj∗·ωqBx0,l1/q·∫2l∞fLpωp,Bx0,rωqBx0,r-1/q1rdr.
This completes the proof of Theorem17.


## 4. Proof of Theorem3

Sincef∈Mφ1p(ωp,Rn), then by Lemma 16 and the fact that fLp(ωp,B(x0,r)) is a nondecreasing function of r, we get (75)fLpωp,Bx0,ressinf0<r<t<∞⁡φ1x0,tωpBx0,t1/p≤esssup0<r<t<∞⁡fLpωp,Bx0,rφ1x0,tωpBx0,t1/p≤supt>0,x0∈Rn⁡fLpωp,Bx0,tφ1x0,tωpBx0,t1/p≤fMφ1pωp,Rn.Since (φ1,φ2) satisfies (15), we have (76)∫l∞1+ln⁡rlkfLpωp,Bx0,rωqBx0,r-1/q·1rdr≤∫l∞fLpωp,Bx0,ressinfr<t<∞⁡φ1x0,tωpBx0,t1/p1+ln⁡rlk·essinfr<t<∞⁡φ1x0,tωpBx0,t1/pωqBx0,r1/q1rdr≤CfMφ1pωp,Rn∫l∞1+ln⁡rlk·essinfr<t<∞⁡φ1x0,tωpBx0,t1/pωqBx0,r1/q1rdr≤CfMφ1pωp,Rnφ2x0,l.Then by Theorem 17 we get (77)TΩ,αA1,…,AkfMφ2qωq,Rn≤Csupx0∈Rn,l>0⁡1φ2x0,l·1ωqBx0,l·∫Bx0,lTΩ,αA1,…,Akfyqωqydy1/q≤C∏j=1k∑γ=mj-1DγA∗supx0∈Rn,l>0⁡1φ2x0,l·∫l∞1+ln⁡rlkfLpωp,Bx0,rωqBx0,r-1/q·1rdr≤C∏j=1k∑γ=mj-1DγA∗fMφ1pωp,Rn.



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*Source: 731487-2015-08-03.xml*