# Commutators of Multilinear Calderón-Zygmund Operator on Weighted Herz-Morrey Spaces with Variable Exponents

Authors: Jin Meng; Shengrong Wang; Jing Zhang
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we acquire the boundedness of commutators generated by multilinear Calderón-Zygmund operator and BMO functions on products of weighted Herz-Morrey spaces with variable exponents.


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## Body

## 1. Introduction

The space of all Schwartz functions onℝn was denoted by Sℝn, and the space of all tempered distributions on ℝn was denoted by S′ℝn. The space of compactly supported bounded functions denoted by LC∞ℝn, and the support set of function f was denoted by suppf. On the m-fold of the Schwartz function space Sℝn, we also set T as an m-linear operator originally defined and m≥2, and its value belongs to S′ℝn:(1)T:Sℝn×Sℝn×⋯×Sℝn⏟m⟶S′ℝn.We say thatT is an m-linear Calderón-Zygmund operator, if for some p1,⋯,pm∈1,∞, it extends to a bounded multilinear operator from Lp1×Lp2×⋯×Lpm to Lp with 1/p1+1/p2+⋯+1/pm=1/p, and for f1,f2,⋯,fm∈LC∞ℝn, x∉∩j=1msuppfj(2)Tf1,f2,⋯,fmx≔∫ℝnmKx,y1,y2,⋯,ym∏i=1mfiyidy1dy2⋯dym,where kernel K is a function in ℝnm+1 away from the diagonal x=y1=y2=⋯=ym and there exist positive constants ε,A satisfies the following:(3)Kx,y1,⋯,ym≤A∑i=1mx−yi−mn,Kx,y1,⋯,ym−Kx′,y1,⋯,ym≤Ax−x′ε∑i=1mx−yimn+ε,whenever x−x′≤1/2maxx−y1,x−y2,⋯,x−ym, and for all 1≤i≤m,(4)Kx,y1,⋯,yi,⋯,ym−Kx,y1,⋯,yi′,⋯,ym≤Ayi−yi′ε∑i=1mx−yimn+ε,where yi−yi′≤1/2maxx−y1,x−y2,⋯,x−ym.Ifb∈Lloc1ℝn, set(5)b∗≔supB1B∫Bbx−bBdx,where bB=1/B∫Bbydy and the supremum is taken over all B⊂ℝn, and what follows ∣B∣ is the Lebesgue measure of measurable set B in ℝn. A function b is called bounded mean oscillation if b∗<∞. Denote by BMOℝn the set of all bounded mean oscillation functions on ℝn.Although our method suits any multilinear operator, only the bilinear Calderón-Zygmund operator will be considered here for the sake of simplicity. Specifically, we will discuss the commutator of a bilinear Calderón-Zygmund operatorT, BMO functions b1 and b2, and suitable functions f1 and f2,(6)b1,b2,Tf1,f2x=b1xb2xTf1,f2x−b1xTf1,b2f2x−b2xTb1f1,f2x+Tb1f1,b2f2.Many analyses of linear commutators have been extended to other fields, such as weighted space, homogeneous space, multiparameter, and multilinear settings. Huang and Xu [1] obtained boundedness of multilinear singular integrals and their commutators from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. Huet al. [2] proved the boundedness of commutators generated by fractional integrals and BMO on generalized Herz spaces with general Muckenhoupt weights. Tang et al. [3] obtained the boundedness of a commutator generated by the multilinear Calderón-Zygmund operator and BMO functions in Herz-Morrey spaces with variable exponents. Chen et al. [4] studied multiple weighted norm inequalities for maximal vector-valued multilinear singular operator and maximal commutators. Wang et al. [5] proved the boundedness for a class of multisublinear singular integral operators on the product central Morrey spaces with variable exponents.Motivated by the mentioned works, we will consider the boundedness of commutators generated by multilinear Calderón-Zygmund operator and BMO functions on products of weighted Herz-Morrey spaces with variable exponents.


## 2. Notations and Main Result

In this section, we recall some notations and definitions; then, we describe our results. Assumep· be a measurable function on ℝn and take values in 1,∞, the Lebesgue space with variable exopnent Lp·ℝn is acquired by(7)Lp·ℝn≔fismeasurable:∫ℝnfxλpxdx<∞forsomeλ>0.The norm is defined by(8)fLp·≔infλ>0:∫ℝnfxλpxdx≤1.On a Banach function space, the Lebesgue spaceLp·ℝn is equipped with the norm fLp· The space Llocp·ℝn is defined by(9)Llocp·ℝn≔f:fχK∈Lp·ℝnforallcompactsubsetsK⊂ℝn,where and what follows, χA denotes the characteristic function of a measurable set A⊂ℝn.Letp·:ℝn⟶0,∞, we denote(10)p−≔essinfx∈ℝnpx,p+≔esssupx∈ℝnpx.The setPℝn consists of all p· satisfying p−>1 and p+<∞; P0ℝn consists of all p· satisfying p−>0 and p+<∞. Lp· can be equally defined as above for p·∈P0ℝn. q′· is the conjugate exponent of p·, defined pointwise by 1/p·+1/p′·=1.Letp·∈Pℝn and w be a weight which is a nonnegative measurable function on ℝn. Then, the weighted variable exponent Lebesgue space Lp·w is the set of all complex-valued measurable function f such that fw∈Lp·. The space Lp·w is a Banach space equipped with the norm(11)fLp·w≔fwLp·.Letf∈Lloc1ℝn. Then, the standard Hardy-Littlewood maximal function of f is defined by(12)Mfx≔supB∋x1B∫Bfydy,∀x∈ℝn,where the supremum is taken over all balls containing x in ℝn. Generally speaking, on weighted variable Lebesgue spaces, the Hardy-Littlewood maximal operator is not bounded. But if it meets certain conditions, it will be established. Namely, let p·∈Pℝn and meet the following global log-Hölder continuous and w∈Ap· such that M is bounded on Lp·w, see [6].Definition 1.
Assumeα· be a real-valued measurable function on ℝn.(i)
We say thatα· satisfies the local log-Hölder continuity condition if there exists a constant C1 such that(13)αx−αy≤C1loge+1/x−y,x,yĜ∈ℝn,x−y<12(ii)
We say thatα· satisfies the log-Hölder continuous at the origin if there exists a constant C2 such that(14)αx−α0≤C2loge+1/x,∀x∈ℝnDenote byP0logℝn the set of all log-Hölder continuous functions at the origin.(iii)
We say thatα· satisfies the log-Hölder continuous at the infinity if there exists α∞∈ℝ and a constant C3 such that(15)αx−α∞≤C3loge+x,∀x∈ℝnDenote byP∞logℝn the set of all log-Hölder continuous functions at infinity.(iv)
We say thatα· satisfies the global log-Hölder continuous if α· is both log-Hölder continuous and locally log-Hölder continuous at infinity. We denote by Plogℝn the set of all global log-Hölder continuous functionsDefinition 2.
Givenp·∈Pℝn and a positive measurable function w, we say that w∈Ap· if there exists a positive constant C for all balls B in ℝn such that(16)1BwχBLp·w−1χBLp′·≤C.Remark 3.
In [7], Cruz-Uribe et al. obtained that if p·∈Pℝn and w∈Ap·, then w−1∈Ap′·.The MuckenhouptAp class with constant exponent p∈1,∞ was firstly proposed by Muckenhoupt in [8]. The variable Muckenhoupt Ap· was considered in [7, 9–12].Lemma 4 (see [7, Theorem 1.5]).
Ifp·∈Plogℝn∩Pℝn and w∈Ap·, then there is a positive constant C such that for each f∈Lp·w,(17)MfwLp·≤CfwLp·.Next, we define the weighted Herz-Morrey space with variable exponents, and we use the following concepts. Letk∈ℤ, we define(18)Bk≔x∈ℝn:x≤2k,Dk≔Bk\Bk−1,χk≔χDk,χ~m=χm,m≥1,χ~0=χB0.Definition 5.
Letq·, p·∈P0ℝn, λ∈0,∞. Let α· be a bounded real-valued measurable function on ℝn. The nonhomogeneous weighted Herz-Morrey space MKq·,λα·,p·w and homogeneous weighted Herz-Morrey space MK̇q·,λα·,p·w are defined, respectively, by(19)MKq·,λα·,p·w≔f∈Llocp·ℝn,w:fMKq·,λα·,p·w<∞,MK̇q·,λα·,p·w≔f∈Llocp·ℝn\0,w:fMK̇q·,λα·,p·w<∞,and where(20)fMK̇q·,λα·,p·w≔supL∈ℤ2−Lλ2α·kfχkk≤Lℓq·Lp·w,fMKq·,λα·,p·w≔supL∈ℕ02−Lλ2α·kfχ~kk=0Lℓq·Lp·w.LetB and C be two real numbers. If there exists a constant K>0 such that B≤KC, we denote B≲C. If B≲C and C≲B, we denote B≈C.Proposition 6 (see [13, Proposition 1]).
Letp·, q·∈P0ℝn, w be a weight, λ∈0,∞, and α·∈L∞ℝn.(i)
Ifα·,q·∈P0logℝn∩P∞logℝn, then for all f∈Llocp·ℝn\0,w(21)fMK̇p·,λα·,q·w≈maxsupL≲0,L∈ℤ2−Lλ2kα0fχkk≤Lℓq0Lp·w,supL>0,L∈ℤ2−Lλ2kα0fχkk<0ℓq0Lp·w+2−Lλ2kα∞fχkk=0Lℓq∞Lp·w,where and hereafter, q0≔q0(ii)
Ifα·,q·∈P∞logℝn, then(22)MKp·,λα·,q·w=MKp·,λα∞,q∞wLemma7 has been proved by Noi and Izuki in [14, 15].Lemma 7.
Ifp·∈Plogℝn∩Pℝn and w∈Ap·, then there exist constants δ1, δ2∈0,1 and C>0 such that for all balls B in ℝn and all measurable subsets S⊂B,(23)χSLp·wχBLp·w≤CSBδ1,χSLp′·w−1χBLp′·w−1≤CSBδ2.The following is the main result.Theorem 8.
LetT be a bilinear Calderón-Zygmund operator and let b1 and b2 be BMO functions. Given p1·,p2·∈Plogℝn∩Pℝn satisfying 1/px=1/p1x+1/p2x for x∈ℝn. Let w1,w2 be weights, w=w1w2, wi∈Api·, i=1,2. Assume that α·∈L∞ℝn∩P0logℝn∩P∞logℝn, α0=α10+α20, α∞=α1∞+α2∞, q·∈P0logℝn∩P∞logℝn,1/q0=1/q10+1/q20, 1/q∞=1/q1∞+1/q2∞, λ=λ1+λ2, 0≤λi<∞, δi1, δi2∈0,1 are the constants in Lemma 7 for exponents pi· and weights wi, i=1,2. If λi−nδi1<αi∞,αi0<nδi2,i=1,2, then(24)b1,b2,Tf1,f2MK̇p·,λα·,q·w≲f1MK̇p1·,λ1α1·,q1·w1f2MK̇p2·,λ2α2·,q2·w2.


## 3. Proof of Theorem8

Before we prove Theorem8, we need to introduce some lemmas.Lemma 9 (see [1, Theorem 2.3]).
Letp·, p1·, p2·∈P0ℝn such that 1/px=1/p1x+1/p2x for x∈ℝn. Then, there exists a constant Cp,p1 independent of functions f and g such that(25)fgLp·≤Cp,p1fLp1·gLp2·,holds for every f∈Lp1·ℝn and g∈Lp2·ℝn.Ifp∈Pℝn, w∈Ap· with w=w1w2, then by the Hölder inequality, we have(26)fgLp·w≤Cp,p1fLp1·w1gLp2·w2.Lemma 10 (see [16, Corollary 3.11]).
Letb∈BMOℝn, p·∈Plogℝn∩Pℝn, w∈Ap·, t∈1,∞, and k,i∈ℕ such that k>i, then one has(27)b−bBitχBkLp·w≤Ck−itb∗tχBkLp·w.Lemma 11 (see [17, Theorem 2.6]).
If0<p<∞ and δ>0, then there exists a positive constant C such that(28)∑l=−∞∞∑k=−∞∞2−k−lδakp1/p≤C∑l=−∞∞alp1/p,for nonnegative sequences all=−∞∞.Lemma 12 (see [10, Theorems 2.23 and 2.24]).
Assume that for somep0, p0∈1,∞, and every w0∈A∞,(29)∫ℝnfxp0w0xdx≤C∫ℝngxp0w0xdx,f,g∈F,where F is a pair of nonnegative functions. Given p·∈P0ℝn, assume that there exists s≤p− such that ws∈Ap·/s and M is bounded on Lp·/s′w−sℝn. Then,(30)fLp·w≤CgLp·w,f,g∈F.Lemma 13.
Letp1·, p2·∈Pℝn, 1<pi−≤pi+<∞, and pi·∈Plogℝn∩Pℝn satisfying 1/px=1/p1x+1/p2x for x∈ℝn, i=1,2. Let b1 and b2 be BMO functions. Let w1∈Ap1·, w2∈Ap2·, and w=w1w2. If T is a bilinear Calderón-Zygmund operator, then(31)b1,b2,Tf1,f2Lp·w≲f1Lp1·w1f2Lp2·w2.Proof.
We assume thatb1 and b2 are bounded functions. Let f1 and f2 be bounded functions with compact support. By the same argument in the proof of [1, Theorem 2.2], we have(32)∫ℝnb1,b2,Tf1,f2xp0w0xdx≲b1∗p0b2∗p0∫ℝnM1f1xM2f2xp0w0xdx.
By Lemma12, we have(33)b1,b2,Tf1,f2Lp·w≲b1∗b2∗M1f1M2f2Lp·w.
By Lemmas9 and 4, we have(34)b1,b2,Tf1,f2Lp·w≲b1∗b2∗f1Lp1·w1f2Lp2·w2.☐Proof of Theorem 8.
Assumef1 and f2 are bounded functions with compact support and write(35)fi=∑l=−∞∞fiχl≕∑l=−∞∞fil,i=1,2.
By Proposition6, we have(36)b1,b2,Tf1,f2MK̇p·,λα·,q·w≈maxsupL≤0,L∈ℤ2−Lλ2kα0b1,b2,Tf1,f2χkk≤Llq0Lp·w,supL>0,L∈ℤ2−Lλ2kα0b1,b2,Tf1,f2χkk<0lq0Lp·w+2−Lλ2kα∞b1,b2,Tf1,f2χkk=0Llq∞Lp·w≔maxE,H,where(37)E≔supL≤0,L∈ℤ2−Lλ2kα0b1,b2,Tf1,f2χkk≤Llq0Lp·w,H≔supL∈ℕF+G,F≔2−Lλ2kα0b1,b2,Tf1,f2χkk<0lq0Lp·w,G≔2−Lλ2kα∞b1,b2,Tf1,f2χkk=0Llq∞Lp·w.
Since the estimates ofE and F are essentially analogical, we only need to obtain E and G bounded in the Herz-Morrey space with variable exponents. It is easy to see that(38)E≤C∑i=i9Ei,G≤C∑i=i9Gi,where(39)E1≔supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=−∞k−2∑j=−∞k−2b1,b2,Tf1l,f2jχkLp·wq01/q0,E2≔supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=−∞k−2∑j=k−1k+1b1,b2,Tf1l,f2jχkLp·wq01/q0,E3≔supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=−∞k−2∑j=k+2∞b1,b2,Tf1l,f2jχkLp·wq01/q0,E4≔supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=k−1k+1∑j=−∞k−2b1,b2,Tf1l,f2jχkLp·wq01/q0,E5≔supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=k−1k+1∑j=k−1k+1b1,b2,Tf1l,f2jχkLp·wq01/q0,E6≔supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=k−1k+1∑j=k+2∞b1,b2,Tf1l,f2jχkLp·wq01/q0,E7≔supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=k+2∞∑j=−∞k−2b1,b2,Tf1l,f2jχkLp·wq01/q0,E8≔supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=k+2∞∑j=k−1k+1b1,b2,Tf1l,f2jχkLp·wq01/q0,E9≔supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=k+2∞∑j=k+2∞b1,b2,Tf1l,f2jχkLp·wq01/q0,G1≔2−Lλ∑k=0L2kα∞q∞∑l=−∞k−2∑j=−∞k−2b1,b2,Tf1l,f2jχkLp·wq∞1/q∞,G2≔2−Lλ∑k=0L2kα∞q∞∑l=−∞k−2∑j=k−1k+1b1,b2,Tf1l,f2jχkLp·wq∞1/q∞,G3≔2−Lλ∑k=0L2kα∞q∞∑l=−∞k−2∑j=k+2∞b1,b2,Tf1l,f2jχkLp·wq∞1/q∞,G4≔2−Lλ∑k=0L2kα∞q∞∑l=k−1k+1∑j=−∞k−2b1,b2,Tf1l,f2jχkLp·wq∞1/q∞,G5≔2−Lλ∑k=0L2kα∞q∞∑l=k−1k+1∑j=k−1k+1b1,b2,Tf1l,f2jχkLp·wq∞1/q∞,G6≔2−Lλ∑k=0L2kα∞q∞∑l=k−1k+1∑j=k+2∞b1,b2,Tf1l,f2jχkLp·wq∞1/q∞,G7≔2−Lλ∑k=0L2kα∞q∞∑l=k+2∞∑j=−∞k−2b1,b2,Tf1l,f2jχkLp·wq∞1/q∞,G8≔2−Lλ∑k=0L2kα∞q∞∑l=k+2∞∑j=k−1k+1b1,b2,Tf1l,f2jχkLp·wq∞1/q∞,G9≔2−Lλ∑k=0L2kα∞q∞∑l=k+2∞∑j=k+2∞b1,b2,Tf1l,f2jχkLp·wq∞1/q∞.
We shall use the following estimates. Ifl≤k−1, then pass Hölder’s inequality, we have(40)∫ℝnbix−biyifildyi≲∫ℝnbix−biBlfildyi+∫ℝnbiyi−biBlfildyi≲bix−biBlfilχlLpi·wiχlLpi′·wi−1+filχlLpi·wibi−biBlχlLpi′·wi−1.
By Lemmas7 and 10, Hölder’s inequality, and Definition 2, we acquire that(41)2−kn∫ℝnbix−biyifildyiχkLpi·wi≲2−knbi−biBl·χkLp·wfilχlLpi·wiχlLpi′·wi−1+2−knfilχlLpi·wibi−biBlχlLpi′·wi−1χkLp·w≲2−knbi∗filχlLpi·wiχBlLpi′·wi−1χBkLp·w+2−knbi∗k−lfilχlLpi·wiχBlLpi′·wi−1χBkLp·w≲k−l2−knbi∗filχlLpi·wiχBlLpi′·wi−1χBkLp·w≲k−l2−knbi∗χBkLpi·wifilwiχlLpi·χlwi−1Lpi′·≲k−l2−knbi∗BkχBkLpi′·wi−1−1χBlLpi′·wi−1filχlLpi·wi≲k−l2l−knδi2bi∗filLpi·wi.
Ifl=k, then(42)2−kn∫ℝnbix−biyifildyiχkLpi·wi≲2−knbi−biBk·χkLp·wfilχlLpi·wiχlLpi′·wi−1+2−knfilχlLpi·wibi−biBlχlLpi′·wi−1χkLp·w≲2−knbi∗χBkLpi·wifilwiχlLpi·χlwi−1Lpi′·≲2−knbi∗χBkLpi·wiχBlLpi′·wi−1filχlLpi·wi≲bi∗filLpi·wi.
Ifl≥k+1, then pass Hölder’s inequality, we have(43)∫ℝnbix−biyifildyi≲∫ℝnbix−biBlfildyi+∫ℝnbiyi−biBlfildyi≲bix−biBkfilχlLPi·wiχlLPi′·wi−1+filχlLPi·wibi−biBkχlLPi′·wi−1.
By Lemmas7 and 10, Hölder’s inequality, and Definition 2, we acquire that(44)2−kn∫ℝnbix−biyifildyiχkLpi·wi≲2−knbi−biBk·χkLp·wfilχlLpi·wiχlLpi′·wi−1+2−knfilχlLpi·wibi−biBkχlLpi′·wi−1χkLp·w≲2−knbi∗filχlLpi·wiχBlLpi′·wi−1χBkLp·w+2−knbi∗l−kfilχlLpi·wiχBlLpi′·wi−1χBkLp·w≲2−knl−kbi∗filχlLp1·w1χBlLp1′·w1−1χBkLp·w≲2−knl−kbi∗χBkLpi·wifilwiχlLpi·χlwi−1Lpi′·≲2−knl−kbi∗χBkLpi·wiχBlLpi·wiχBlLpi·wi−1×χBlLpi′·wi−1filχlLpi·wi≲2l−kn1−δ1il−kbi∗filLpi·wi.
By the interchange off1 and f2, we see that the estimates of E2, E3, and E6 are similar to E4, E7, and E8, respectively. Thus, we only to estimate E1, E2, E3, E5, E6, and E9.
To estimateE1, due to l, j≤k−2, we infer that for i=1,2,(45)x−yi≥x−yi>2k−1−2minl,j≥2k−2,x∈Dk,y1∈Dl,y2∈Dj.
Therefore, forx∈Dk, we have(46)Kx,y1,y2≤Cx−y1+x−y2−2n≤C2−2kn.
Thus,∀x∈Dk and l,j≤k−2, we have(47)b1,b2,Tf1l,f2jx≲∫ℝ2n∏i=12bix−biyiKx,y1,y2f1ly1f2jy2dy1dy2≲2−2kn∫ℝ2n∏i=12bix−biyif1ly1∣f2jy2dy1dy2.
Thus, according to Hölder’s inequality, we have(48)∑l=−∞k−2∑j=−∞k−2b1,b2,Tf1l,f2jχkLp·w≲∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1×∑j=−∞k−22−kn∫ℝnb2x−b2y1f2jy2dy2χkLp·w≲∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1×∑j=−∞k−22−kn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2.
Since1/q0=1/q10+1/q20, λ=λ1+λ2, according to Hölder’s inequality, we will have(49)E1≲supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q0×∑j=−∞k−22−kn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q01/q0≲supL≤0,L∈ℤ2−Lλ1×∑k=−∞L2kα10q10∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q101/q10×supL≤0,L∈ℤ2−Lλ2×∑k=−∞L2kα20q20∑j=−∞k−22−kn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q201/q20≔E1,1×E1,2.where(50)E1,i≔supL≤0,L∈ℤ2−Lλi×∑k=−∞L2kαi0qi0∑l=−∞k−22−kn∫ℝnbix−biyifilyidyiχkLpi·wiqi01/qi0.
Sincenδi2−αi0>0, by (41) and Lemma 11, we acquire that(51)E1,i≲supL≤0,L∈ℤ2−Lλi∑k=−∞L2kαi0qi0bi∗∑l=−∞k−22l−knδi2filLpi·wiqi01/qi0=bi∗supL≤0,L∈ℤ2−Lλi×∑k=−∞L∑l=−∞k−22lαi0filLpi·wi2l−knδi2−αi0qi01/qi0≲bi∗supL≤0,L∈ℤ2−Lλi∑l=−∞L−22lαi0qi0filLpi·wiqi01/qi0≲bi∗fiMK̇pi·,λiαi·,qi·wi.
Thus, we obtain that(52)E1≲f1MK̇p1·,λ1α1·,q1·w1f2MK̇p2·,λ2α2·,q2·w2.
To estimateE2, due to l≤k−2,k−1≤j≤k+1 for i=1,2, then, we have(53)x−y1≥x−y1≥2k−2,x∈Dk,y1∈Dl.
So, according to Hölder’s inequality, we have(54)∑l=−∞k−2∑j=k−1k+1b1,b2,Tf1l,f2jχkLp·w≲∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1×∑j=k−1k+12−kn∫ℝnb2x−b2y2f2jy2dy2χkLp·w≲∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1×∑j=k−1k+12−kn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2.
Since1/q0=1/q10+1/q20, λ=λ1+λ2, by Hölder’s inequality, we acquire that(55)E2≲supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q0×∑j=k−1k+12−kn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q01/q0≲supL≤0,L∈ℤ2−Lλ1∑k=−∞L2kα10q10×∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q101/q10×supL≤0,L∈ℤ2−Lλ2∑k=−∞L2kα20q20×∑j=k−1k+12−kn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q201/q20≔E2,1×E2,2.
It is obviously that(56)E2,1=E1,1≲b1∗f1MK̇p1·,λ1α1·,q1·w1.
Now we estimateE2,2. Taking (41), (42), and (44) together, we have(57)E2,2≲b2∗supL≤0,L∈ℤ2−Lλ2∑k=−∞L2kα20q20∑j=k−1k+12j−knq20f2jχjLp2·w2q201/q20≲b2∗supL≤0,L∈ℤ2−Lλ2∑k=−∞L2kα20q20f2jχkLp2·w2q201/q20≲b2∗f2MK̇p2·,λ2α2·,q2·w2,where we used 2−nδ22<1 and 2j−kn1−δ12<2j−kn<22n, j∈k−1,k,k+1 for (41) and (44), respectively. Therefore, we acquire that(58)E2≲b1∗b2∗f1MK̇p1·,λ1α1·,q1·w1f2MK̇p2·,λ2α2·,q2·w2.
To estimateE3, since l≤k−2,j≥k+2, then, we have(59)x−y1≥x−y1≥2k−2,x−y2≥y2−x>2j−2,x∈Dk,y1∈Dl,y2∈Dj.
Therefore,∀x∈Dk,l≤k−2,j≥k+2, we get(60)b1,b2,Tf1l,f2jx≲∫ℝ2n∏i=12bix−biyiKx,y1,y2f1ly1f2jy2dy1dy2≲2−kn2−jn∫ℝ2n∏i=12bix−biyif1ly1∣f2jy2dy1dy2.
So, according to Hölder’s inequality, we acquire that(61)∑l=−∞k−2∑j=k+2∞b1,b2,Tf1l,f2jχkLp·w≲∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp·w≲∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2.
Since1/q0=1/q10+1/q20,λ=λ1+λ2, according to Hölder’s inequality, we acquire that(62)E3≲supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q0×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q01/q0≲supL≤0,L∈ℤ2−Lλ1∑k=−∞L2kα10q10×∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q101/q10×supL≤0,L∈ℤ2−Lλ2∑k=−∞L2kα20q20×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q201/q20≔E3,1×E3,2.
It is obviously that(63)E3,1=E1,1≲f1MK̇p1·,λ1α1·,q1·w1.
By (44), we acquire that(64)E3,2≲b2∗supL≤0,L∈ℤ2−Lλ2×∑k=−∞L2kα20q20∑j=k+2∞2k−jnδ21f2jχjLp2·w2q201/q20≲b2∗supL≤0,L∈ℤ2−Lλ2×∑k=−∞L∑j=k+2L2jα20f2jχjLp2·w22k−jnδ21+α20q201/q20+b2∗supL≤0,L∈ℤ2−Lλ2×∑k=−∞L2kα20∑j=L+10f2jχjLp2·w22k−jnδ21q201/q20+b2∗supL≤0,L∈ℤ2−Lλ2×∑k=−∞L2kα20∑j=1∞f2jχjLp2·w22k−jnδ21q201/q20≔I1+I2+I3.
We considerI1. By Lemma 11, we have(65)I1≲b2∗supL≤0,L∈ℤ2−Lλ2×∑k=−∞L∑j=k+2L2jα20f2jχjLp2·w22k−jnδ21+α20q201/q20≲b2∗supL≤0,L∈ℤ2−Lλ2∑j=−∞L+22jα20q20f2jχjLp2·w2q201/q20≲b2∗f2MK̇p2·,λ2α2·,q2·w2,where 2−k−jnδ21+α20=2−k−jα2 for α2=nδ21+α20>0.
We considerI2. Since nδ21+α20−λ2>0, we obtain that(66)I2≲b2∗supL≤0,L∈ℤ2−Lλ2∑k=−∞L2knδ21+α20∑j=L+102jα20f2jχjLp2·w2×2−jnδ21+α20q201/q20≲b2∗supL≤0,L∈ℤsupj≤02−jλ22jα20f2jχjLp2·w2×2−Lλ2∑k=−∞L2knδ21+α20∑j=L+102−jnδ21+α20−λ2q201/q20≲b2∗f2MK̇p2·,λ2α2·,q2·w2supL≤0,L∈ℤ2L−nδ21−α20∑k=−∞L2knδ21+α20q201/q20≲b2∗f2MK̇p2·,λ2α2·,q2·w2.
We considerI3. Since nδ21+α2∞−λ2>0 and nδ21+α20−λ2>0, we obtain(67)I3≲b2∗supL≤0,L∈ℤ2−Lλ2∑k=−∞L2knδ21+α20×∑j=1∞2jα2∞f2jχjLp2·w22−jnδ21+α2∞q201/q20≲b2∗supL≤0,L∈ℤsupj≥12−jλ22jα2∞f2jχjLp2·w2×2−Lλ2∑k=−∞L2knδ21+α20∑j=1∞2−jnδ21+α2∞−λ2q201/q20≲b2∗f2MK̇p2·,λ2α2·,q2·w2supL≤0,L∈ℤ2−Lλ2∑k=−∞L2knδ21+α20q201/q20≲b2∗f2MK̇p2·,λ2α2·,q2·w2supL≤0,L∈ℤ2L−λ2+nδ21+α20≲b2∗f2MK̇p2·,λ2α2·,q2·w2.
Thus, we have(68)E3≲b1∗b2∗f1MK̇p1·,λ1α1·,q1·w1f2MK̇p2·,λ2α2·,q2·w2.
To estimateE5, according to Lemma 13 and Hölder’s inequality, we have(69)E5≲supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=k−1k+1∑j=k−1k+1b1,b2,Tf1l,f2jχkLp·wq01/q0≲b1∗b2∗supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0f1Lp1·w1f2Lp2·w2q01/q0≲b1∗b2∗supL≤0,L∈ℤ2−Lλ1∑k=−∞L2kα10q10f1Lp1·w1q101/q10×2−Lλ2∑k=−∞L2kα20q20f2Lp2·w2q201/q20≲b1∗b2∗f1MK̇p1·,λ1α1·,q1·w1f2MK̇p2·,λ2α2·,q2·w2.
To estimateE6, due to k−1≤l≤k+1 and j≥k+2, then we have(70)x−y1>2k−2,x−y2>2j−2,x∈Dk,y1∈Dl,y2∈Dj.
Thus,∀x∈Dk,k−1≤l≤k+1 and j≥k+2, we obtain that(71)b1,b2,Tf1l,f2jx≲∫ℝ2n∏i=12bix−biyiKx,y1,y2f1ly1f2jy2dy1dy2≲2−kn2−jn∫ℝ2n∏i=12bix−biyif1ly1f2jy2dy1dy2.
So, according to Hölder’s inequality, we acquire that(72)∑l=k−1k+1∑j=k+2∞b1,b2,Tf1l,f2jχkLp·w≲∑l=k−1k+12−kn∫ℝnb1x−b1y1f1ly1dy1×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp·w≲∑l=k−1k+12−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2.
Since1/q0=1/q10+1/q20,λ=λ1+λ2, according to Hölder’s inequality, we have(73)E6≲supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=k−1k+12−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q0×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q01/q0≲supL≤0,L∈ℤ2−Lλ1∑k=−∞L2kα10q10×∑l=k−1k+12−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q101/q10×supL≤0,L∈ℤ2−Lλ2∑k=−∞L2kα20q20×∑j=k+2∞2−jn∫ℝnf2jy2dy2χkLp2·w2q201/q20≔E6,1×E6,2.
By the interchange off1 and f2, we acquire that the estimate of E6,1 is analogical to the estimate of E2,2 and E6,2=E3,2.
To estimateE9, since l,j≥k+2, then. we get(74)x−yi>2k−2,x∈Dk,y1∈Dl,y2∈Dj.
Therefore,∀x∈Dk,l,j≥k+2, we have(75)b1,b2,Tf1l,f2jx≲∫ℝ2n∏i=12bix−biyiKx,y1,y2f1ly1f2jy2dy1dy2≲2−ln2−jn∫ℝ2n∏i=12bix−biyif1ly1f2jy2dy1dy2.
So, according to Hölder’s inequality, we obtain that(76)∑l=k+2∞∑j=k+2∞b1,b2,Tf1l,f2jχkLp·w≲∑l=k+2∞2−ln∫ℝnb1x−b1y1f1ly1dy1×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp·w≲∑l=k+2∞2−ln∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2.
Since1/q0=1/q10+1/q20, λ=λ1+λ2, according to Hölder’s inequality, we have(77)E9≲supL≤0,L∈ℤ2−Lλ∑k=−∞L2kα0q0∑l=k+2∞2−ln∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q0×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q01/q0≲supL≤0,L∈ℤ2−Lλ1∑k=−∞L2kα10q10×∑l=k+2∞2−ln∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q101/q10×supL≤0,L∈ℤ2−Lλ2∑k=−∞L2kα20q20×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q201/q20≔E9,1×E9,2.clearly, the estimate E9,i is analogical to the estimated E3,2 for i=1,2.
Combining all the estimates ofEi, i=1,2,⋯,9, we obtain that(78)E≲b1∗b2∗f1MK̇p1·,λ1α1·,q1·w1f2MK̇p2·,λ2α2·,q2·w2.
In order to continue, we need further preparation. Ifl<0, since Proposition 6, we obtain that(79)filχlLpi·wi=2−lαi02lαi0qi0filχlLpi·wiqi01/qi0≲2−lαi0∑t=−∞l2tαi0qi0fitχtLpi·wiqi01/qi0≲2lλi−αi02−lλi∑t=−∞l2tαi0fitχtLpi·wiqi01/qi0≲2lλi−αi0fiMK̇pi·,λiαi·,qi·wi.conclusively, we estimate G. according to the interchange of f1 and f2, we see that the estimates of G2, G3, and G6 are similar to G4, G7, and G8, respectively. Thus, it was only necessary to estimate G1,G2,G3,G5,G6, and G9.
To estimateG1, due to l,j≤k−2, 1/q∞=1/q1∞+1/q2∞, λ=λ1+λ2, by (48) and Hölder’s inequality, we obtain that(80)G1≲2−Lλ∑k=0L2kα∞q∞∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q∞×∑j=−∞k−22−kn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q∞1/q∞≲2−Lλ1∑k=0L2kα1∞q1∞∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q1∞1/q1∞×2−Lλ2∑k=0L2kα2∞q2∞∑j=−∞k−22−kn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q2∞1/q2∞≔G1,1×G1,2.where(81)G1,i≔2−Lλi∑k=0L2kαi∞qi∞∑l=−∞k−22−kn∫ℝnbix−biyifilyidyiχkLpi·wiqi∞1/qi∞.
We considerG1,i. By (41), we obtain that(82)G1,i≲bi∗2−Lλi∑k=0L2kαi∞qi∞∑l=−∞−1filχlLpi·wi2l−knδi2+∑l=0kfilχlLpi·wi2l−knδi2qi∞1/qi∞≲bi∗2−Lλi∑k=0L2kαi∞qi∞∑l=−∞−1filχlLpi·wi2l−knδi2qi∞1/qi∞+bi∗2−Lλi∑k=0L2kαi∞qi∞∑l=0kfilχlLpi·wi2l−knδi2qi∞1/qi∞≔I4+I5.
Ifqi∞≥1, since nδi2−αi∞>0 and nδi2−αi0>0, then by the Minkowski inequality and (79), we have(83)I4=bi∗2−Lλi∑k=0L2kαi∞qi∞∑l=−∞−1filχlLpi·wi2l−knδi2qi∞1/qi∞≲bi∗2−Lλi∑l=−∞−1filχlLpi·wi∑k=0L2kαi∞2l−knδi2qi∞1/qi∞≲bi∗2−Lλi∑l=−∞−12lnδi2filχlLpi·wi∑k=0L2−knδi2−α1∞qi∞1/qi∞≲bi∗fiMK̇pi·,λiαi·,qi·wi2−Lλi∑l=−∞−12lnδi2+λi−αi0≲bi∗fiMK̇pi·,λiαi·,qi·wi,
Ifqi∞<1, since nδi2−αi∞>0 and nδi2−αi0>0, then by (79), we have(84)I4≲bi∗2−Lλi∑k=0L2kαi∞qi∞∑l=−∞−1filχlLpi·w1qi∞2l−knδi2qi∞1/qi∞=bi∗2−Lλi∑l=−∞−1filχlLpi·wiqi∞2lnδi2qi∞∑k=0L2kαi∞qi∞2−knδi2q1∞1/qi∞=bi∗2−Lλi∑l=−∞−1filχlLpi·wiqi∞2lnδi2qi∞∑k=0L2−knδi2−αi∞qi∞1/qi∞≲bi∗2−Lλi∑l=−∞−1filχlLpi·wiqi∞2lnδi2qi∞1/qi∞≲bi∗fiMK̇pi·,λiαi·,qi·2−Lλi∑l=−∞−12lnδi2+λi−αi0qi∞1/qi∞≲bi∗fiMK̇pi·,λiαi·,qi·,
We considerI5. Since nδi2−αi∞>0, then by Lemma 11, we have(85)I5=bi∗2−Lλi∑k=0L2kαi∞qi∞∑l=0kfilχlLpi·wi2l−knδi2qi∞1/qi∞=bi∗2−Lλi∑k=0L∑l=0k2lαi∞filχlLpi·wi2l−knδi2−αi∞qi∞1/qi∞≲bi∗2−Lλi∑l=0k2lαi∞qi∞filχlLpi·wiqi∞1/qi∞≲bi∗fiMK̇pi·,λiαi·,qi·wi,where we wrote 2−∣k−l∣nδi2−αi∞≲2−∣k−l∣αi for αi=nδi2−αi∞.
Thus, we get(86)G1≲bi∗b2∗f1MK̇p1·,λ1α1·,q1·w1f2MK̇p2·,λ2α2·,q2·w2.
To estimateG2, due to l≤k−2, k−1≤j≤k+1, 1/q∞=1/q1∞+1/q2∞, λ=λ1+λ2, by (54) and Hölder’s inequality, we obtain that(87)G2≲2−Lλ∑k=0L2kα∞q∞∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q∞×∑j=k−1k+12−kn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q∞1/q∞≲2−Lλ1∑k=0L2kα1∞q1∞∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q1∞1/q1∞×2−Lλ2∑k=0L2kα2∞q2∞∑j=k−1k+12−kn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q2∞1/q2∞≔G2,1×G2,2.
It is obviously that(88)G2,1=G1,1≲b1∗f1MK̇p1·,λ1α1·,q1·w1.
Now, we estimateG2,2. Combing (41), (42), and (44), we have(89)G2,2≲b2∗2−Lλ2∑k=0L2kα2∞q2∞∑j=k−1k+12j−knq2∞f2jχjLp2·w2q2∞1/q2∞≲b2∗2−Lλ2∑k=0L2kα2∞q2∞f2χkLp2·w2q2∞1/q2∞≲b2∗f2MK̇p2·,λ2α2·,q2·w2,where we used 2−nδ22<1 and 2j−kn1−δ12<2j−kn for (41) and (44), respectively.
Thus, we acquire that(90)G2≲b1∗b2∗f1MK̇p1·,λ1α1·,q1·w1f2MK̇p2·,λ2α2·,q2·w2.
To estimateG3, since l≤k−2,j≥k+2, 1/q∞=1/q1∞+1/q2∞,λ=λ1+λ2, using (61) and Hölder’s inequality, we obtain that(91)G3≲2−Lλ∑k=0L2kα∞q∞∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q∞×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q∞1/q∞≲2−Lλ1∑k=0L2kα1∞q1∞∑l=−∞k−22−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q1∞1/q1∞×2−Lλ2∑k=0L2kα2∞q2∞∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q2∞1/q2∞≔G3,1×G3,2.
It is obviously that(92)G3,1=G1,1≲b1∗f1MK̇p1·,λ1α1·,q1·w1.
Sincenδ12+α2∞>0, by (44) and Lemma 11, we obtain that(93)G3,2≲b2∗2−Lλ2∑k=0L2kα2∞q2∞∑j=k+2∞2k−jnδ21f2jχjLp2·w2q2∞1/q2∞≲b2∗2−Lλ2∑k=0L∑j=k+2L+22jα2∞f2jχjLp2·w22k−jnδ21+α2∞q2∞1/q2∞+b2∗2−Lλ2∑k=0L2kα2∞∑j=L+3∞f2jχjLp2·w22k−jnδ21q2∞1/q2∞≔I6+I7.
We considerI6. By Lemma 11, we obtain that(94)I6≲b2∗2−Lλ2∑k=0L∑j=k+2L+22jα2∞f2jχjLp2·w22k−jnδ21+α2∞q2∞1/q2∞≲b2∗2−Lλ2∑j=0L+22jα2∞q2∞f2jχjLp2·w2q2∞1/q2∞≲b2∗f2MK̇p2·,λ2α2·,q2·w2,where wrote 2−k−jnδ21+α2∞=2−∣k−j∣ζ2 for ζ2=nδ21+α2∞>0.
We considerI7. Since nδ21+α2∞−λ2>0, we have(95)I7≲b2∗⋅2-Lλ2∑k=0L2knδ21+α2∞∑j=L+3∞2jα2∞f2χjLP2·w22-jnδ21+α2∞q2∞1/q2∞≲b2∗supj≥12-jλ22jα2∞f2jχjLP2·w2×2-Lλ2∑k=0L2knδ21+α2∞∑j=L+3∞2-jnδ21+α2∞-λ2q2∞1/q2∞≲b2∗f2MK˙p2·,λ2α2·,q2·w2⋅2-Lλ2∑k=0L2knδ21+α2∞q2∞2-Lnδ21+α2∞-λ21/q2∞≲b2∗f2MK˙p2·,λ2α2·,q2·w2⋅2-Lλ22nδ21+α2∞L2-Lnδ21+α2∞-λ2≲b2∗f2MK˙p2·,λ2α2·,q2·w2
Thus, we get(96)G3≲b1∗b2∗f1MK̇p1·,λ1α1·,q1·w1f2MK̇p2·,λ2α2·,q2·w2.
To estimateG5, according to Lemma 13 and Hölder’s inequality, we have(97)G5≲2−Lλ∑k=0L2kα∞q∞∑l=k−1k+1∑j=k−1k+1b1,b2,Tf1l,f2jχkLp·wq∞1/q∞≲2−Lλ∑k=0L2kα∞q∞b1∗b2∗f1Lp1·w1f2Lp2·w2q∞1/q∞≲b1∗2−Lλ1∑k=0L2kα1∞q1∞f1Lp1·w1q1∞1/q1∞×b2∗2−Lλ2∑k=0L2kα2∞q2∞f2Lp2·w2q2∞1/q2∞≲b1∗b2∗f1MK̇p1·,λ1α1·,q1·w1f2MK̇p2·,λ2α2·,q2·w2.
To estimateG6, due to k−1≤l≤k+1 and j≥k+2, 1/q∞=1/q1∞+1/q2∞,λ=λ1+λ2, using (72) and Hölder’s inequality, we obtain that(98)G6≲2−Lλ∑k=0L2kα∞q∞∑l=k−1k+12−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q∞×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q∞1/q∞≲2−Lλ1∑k=0L2kα1∞q1∞∑l=k−1k+12−kn∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q1∞1/q1∞×2−Lλ2∑k=0L2kα2∞q2∞∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q2∞1/q2∞≔G6,1×G6,2.
Since the symmetry off1 and f2, we can know that the estimate G6,1 is analogical to the estimated G1,2 and G6,2=G3,2.
To estimateG9, due to l,j≥k+2, 1/q∞=1/q1∞+1/q2∞,λ=λ1+λ2, using (76) and Hölder’s inequality, we obtain that(99)G9≲2−Lλ∑k=0L2kα∞q∞∑l=k+2∞2−ln∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q∞×∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q∞1/q∞≲2−Lλ1∑k=0L2kα1∞q1∞∑l=k+2∞2−ln∫ℝnb1x−b1y1f1ly1dy1χkLp1·w1q1∞1/q1∞×2−Lλ2∑k=0L2kα2∞q2∞∑j=k+2∞2−jn∫ℝnb2x−b2y2f2jy2dy2χkLp2·w2q2∞1/q2∞≔G9,1×G9,2.clearly, the estimate G9,i is analogical to the estimated G3,2 for i=1,2.
Combining all the estimates ofrGi together, i=1,2,⋯,9, we obtain that(100)G≲f1MK̇p1·,λ1α1·,q1·w1f2MK̇p2·,λ2α2·,q2·w2.
Combining the above estimates forE, F, and G, the proof of Theorem 8 is completed.



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*Source: 9947489-2021-07-29.xml*