# Higher-Order Commutators of Parametric Marcinkiewicz Integrals on Herz Spaces with Variable Exponent

Authors: Hongbin Wang; Dunyan Yan
Journal: Journal of Function Spaces
 (2018)

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## Abstract

LetΩ∈Ls(Sn-1) for s⩾1 be a homogeneous function of degree zero and b be BMO functions. In this paper, we obtain some boundedness of the parametric Marcinkiewicz integral operator μΩρ and its higher-order commutator [bm,μΩρ] on Herz spaces with variable exponent.


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## Body

## 1. Introduction

Function spaces with variable exponent are being concerned with strong interest not only in harmonic analysis but also in applied mathematics. In the past 27 years, the theory of function spaces with variable exponent has made great progress since some elementary properties were given by Kováčik and Rákosník [1] in 1991. In [2–6], the authors proved the boundedness of some integral operators on variable Lp spaces, respectively. Lebesgue and Sobolev spaces with integrability exponent have been widely studied; see [3, 5] and the references therein. Many applications of these spaces were given, for example, in the modeling of electrorheological fluids, in the study of image processing, and in differential equations with nonstandard growth.On the other hand, a class of function spaces called Herz-type spaces onRn has attracted considerable attention in recent years because the interesting norm includes explicitly both local and global information of the function. In 2011, Izuki [7] studied the Herz spaces with variable exponent and proved the boundedness of some sublinear operators on the spaces. In addition, Wang and Liu [8] introduced a certain Herz-type Hardy spaces with variable exponent in 2012.Suppose thatSn-1 denotes the unit sphere in Rn(n⩾2) equipped with normalized Lebesgue measure. Let Ω∈Lipβ(Rn) for 0<β⩽1 be a homogeneous function of degree zero and(1)∫Sn-1Ωx′dσx′=0,where x′=x/x for any x≠0. In 1958, Stein [9] introduced the Marcinkiewicz integral related to the Littlewood-Paley g function on Rn as follows: (2)μΩfx=∫0∞FΩ,tfx2dtt31/2, where (3)FΩ,tfx=∫x-y⩽tΩx-yx-yn-1fydy.It was shown that μΩ is of type (p,p) for 1<p⩽2 and of weak type (1,1).The parametric Marcinkiewicz integral is defined by(4)μΩρfx=∫0∞FΩ,tρfx2dtt2ρ+11/2, where (5)FΩ,tρfx=∫x-y⩽tΩx-yx-yn-ρfydy,ρ>0,t>0.Letm∈N and b∈Lloc⁡1(Rn), the higher-order commutator generated by the parametric Marcinkiewicz integral μΩρ and b is defined by (6)bm,μΩρfx=∫0∞∫x-y⩽tΩx-yx-yn-ρbx-bymfydy2dtt2ρ+11/2.Motivated by [10, 11], we will study the boundedness for the parametric Marcinkiewicz integral operator μΩρ and its commutator [bm,μΩρ] on the Herz spaces with variable exponent, where Ω∈Ls(Sn-1) for s⩾1.Throughout this paper, we denote the Lebesgue measure and the characteristic function of a measurable setA⊂Rn by A and χA, respectively. The notation f≈g means that there exist constants C1,C2>0 such that C1g⩽f⩽C2g. Let Bk={x∈Rn:x⩽2k} and Ak=Bk∖Bk-1 for k∈Z. Denote Z+ and N as the sets of all positive and nonnegative integers, χk=χAk for k∈Z, χ~k=χk if k∈Z+ and χ~0=χB0. In addition, δ1 and δ2 are the same as in Lemma 4.


## 2. Preliminaries

Firstly we give some notation and basic definitions on variable Lebesgue spaces. Given an open setE⊂Rn, and a measurable function p(·):E→[1,∞). p′(·) is the conjugate exponent defined by p′(·)=p(·)/(p(·)-1).DefineP0(E) to be set of p(·):E→(0,∞) such that (7)p-=essinfpx:x∈E>0,p+=esssuppx:x∈E<∞.The set P(E) consists of all p(·):E→[1,∞) satisfying (8)p-=essinfpx:x∈E>1,p+=esssuppx:x∈E<∞.ByLp(·)(E) we denote the space of all measurable functions f on E such that for some λ>0, (9)∫Efxλpxdx<∞. This is a Banach function space with respect to the Luxemburg-Nakano norm (10)fLp·E=inf⁡λ>0:∫Efxλpxdx⩽1.The spaceLloc⁡p(·)(Ω) is defined by (11)Lloc⁡p·Ω≔f:f∈Lp·Efor  all  compact  subsetsE⊂Ω.Letf∈Lloc⁡1(Rn); the Hardy-Littlewood maximal operator is defined by (12)Mfx=supr>01Brx∫Brxfydy, where Br(x)={y∈Rn:x-y<r}. The set B(Rn) consists of p(·)∈P(Rn) satisfying the condition that M is bounded on Lp(·)(Rn).In variableLp spaces there are some important lemmas as follows.Lemma 1 (see [2]).
Ifp(·)∈P(Rn) and satisfies(13)px-py⩽C-log⁡x-y,x-y⩽12and(14)px-py⩽Clog⁡x+e,y⩾x,then p(·)∈B(Rn); that is, the Hardy-Littlewood maximal operator M is bounded on Lp(·)(Rn).Lemma 2 (see [1] generalized Hölder inequality).
Letp(·)∈P(Rn). If f∈Lp(·)(Rn) and g∈Lp′(·)(Rn), then fg is integrable on Rn and (15)∫Rnfxgxdx⩽rpfLp·RngLp′·Rn, where (16)rp=1+1p--1p+.Lemma 3 (see [7]).
Supposeq(·)∈B(Rn). Then there exists a constant C>0 such that for all balls B in Rn, (17)1BχBLq·RnχBLq′·Rn⩽C.Lemma 4 (see [7]).
Letq(·)∈B(Rn). Then there exists a positive constant C such that for all balls B in Rn and all measurable subsets S⊂B,(18)χSLq·RnχBLq·Rn⩽CSBδ1andχSLq′·RnχBLq′·Rn⩽CSBδ2,where δ1,δ2 are constants with 0<δ1,δ2<1.Next we recall the definition of the Herz spaces with variable exponent.Definition 5 (see [7]).
Letα∈R,0<p⩽∞, and q(·)∈P(Rn). The homogeneous Herz space K˙q(·)α,p(Rn) consists of all f∈Lloc⁡q(·)(Rn∖{0}) such that (19)fK˙q·α,pRn=∑k=-∞∞2kαpfχkLq·Rnp1/p<∞. The nonhomogeneous Herz space Kq(·)α,p(Rn) is defined as the set of all f∈Lloc⁡q(·)(Rn) such that (20)fKq·α,pRn=∑k=0∞2kαpfχ~kLq·Rnp1/p<∞.


## 3. Boundedness of the Parametric Marcinkiewicz Integral Operator

In this section we will prove the boundedness of the parametric Marcinkiewicz integral operatorsμΩρ on Herz spaces with variable exponent.A nonnegative locally integrable functionω on Rn is said to belong to Ap(1<p<∞), if (21)supQ1Q∫Qωxdx1Q∫Qωx1-p′dxp-1<∞, where p′=p/(p-1), Q denotes a cube in Rn with its sides parallel to the coordinate axes, and Q denotes the Lebesgue measure of Q.The weighted(Lp,Lp) boundedness of μΩρ has been proved by Shi and Jiang [12].Lemma 6 (see [12]).
Suppose thatΩ∈Ls(Sn-1)(s>1) satisfying (1). If ω∈Ap/s′,s′<p<∞, then for any f∈Lp(ω), there is a constant C, independent of f, such that (22)∫RnμΩρfxpωxdx⩽C∫Rnfxpωxdx.Lemma 7 (see [4]).
Given a familyF and an open set E⊂Rn, assume that for some p0, 0<p0<∞ and for every ω∈A∞, (23)∫Efxp0ωxdx⩽C0∫Egxp0ωxdx,f,g∈F. Given p(·)∈P0(E) such that p(·) satisfies (13) and (14) in Lemma 1. Then for all (f,g)∈F such that f∈Lp(·)(E), (24)fLp·E⩽CgLp·E.SinceAp/s′⊂A∞, by Lemmas 6 and 7 it is easy to get the (Lp(·)(Rn), Lp(·)(Rn))-boundedness of the parametric Marcinkiewicz integral operators μΩρ.To obtain the Theorem11, we need the following lemmas.Lemma 8 (see [13]).
Ifa>0,1⩽s⩽∞,0<d⩽s and -n+(n-1)d/s<ν<∞, then (25)∫y⩽axyνΩx-yddy1/d⩽Cxν+n/dΩLsSn-1.Lemma 9 (see [14]).
Define a variable exponentq~(·) by 1/p(x)=1/q~(x)+1/q(x∈Rn). Then we have (26)fgLp·Rn⩽CfLq~·RngLqRnfor all measurable functions f and g.Lemma 10 (see [5]).
Letp(·)∈P(Rn) satisfy conditions (13) and (14) in Lemma 1. Then (27)χQLp·Rn≈Q1/pxifQ⩽2nandx∈Q,Q1/p∞ifQ⩾1for every cube (or ball) Q⊂Rn, where p(∞)=limx→∞p(x).Theorem 11.
Suppose that0<ν⩽1, 0<p⩽∞, q(·)∈P(Rn) satisfies conditions (13) and (14) in Lemma 1; Ω∈Ls(Sn-1)(s>q′-) and -nδ1-ν-n/s<α<nδ2-ν-n/s. Then μΩρ is bounded on K˙q(·)α,p(Rn) and Kq(·)α,p(Rn).Proof.
We only prove homogeneous case. The nonhomogeneous case can be proved in the same way. We suppose0<p<∞, since the proof of the case p=∞ is easier. Let f∈K˙q(·)α,p(Rn). Denote fj=fχj for each j∈Z, then we have f(x)=∑j=-∞∞fj(x). Then we have(28)μΩρfK˙q·α,pRn=∑k=-∞∞2kαpμΩρfχkLq·Rnp1/p⩽C∑k=-∞∞2kαp∑j=-∞k-2μΩρfjχkLq·Rnp1/p+C∑k=-∞∞2kαp∑j=k-1k+1μΩρfjχkLq·Rnp1/p+C∑k=-∞∞2kαp∑j=k+2∞μΩρfjχkLq·Rnp1/p≕CI1+CI2+CI3.
We first estimateI2, by the (Lq(·)(Rn),Lq(·)(Rn))-boundedness of the commutator μΩρ; we have(29)I2⩽C∑k=-∞∞2kαpfkLq·Rnp1/p=CfK˙q·α,pRn.
Now we estimateI1. We consider (30)μΩρfjx⩽∫0x∫x-y⩽tΩx-yx-yn-ρfjydy2dtt2ρ+11/2+∫x∞∫x-y⩽tΩx-yx-yn-ρfjydy2dtt2ρ+11/2≕I11+I12.Note that x∈Ak,y∈Aj, and j⩽k-2. So we know that x-y~x, and by mean value theorem we have(31)1x-y2ρ-1x2ρ⩽yx-y2ρ+1.By (31), the Minkowski inequality, and the generalized Hölder inequality we have (32)I11⩽C∫RnΩx-yx-yn-ρfjy∫x-yxdtt2ρ+11/2dy⩽C∫RnΩx-yx-yn-ρfjy1x-y2ρ-1x2ρ1/2dy⩽C∫RnΩx-yx-yn-ρfjyy1/2x-yρ+1/2dy⩽C2j/2xn+1/2∫AjΩx-yfydy⩽C2j-k/22-knfjLq·RnΩx-·χj·Lq′·Rn. Similarly, we consider I12. Noting that x-y~x, by the Minkowski inequality and the generalized Hölder inequality, we have (33)I12⩽C∫RnΩx-yx-yn-ρfjy∫x∞dtt2ρ+11/2dy⩽C∫RnΩx-yx-ynfjydy⩽C2-knfjLq·RnΩx-·χj·Lq′·Rn. So we have (34)μΩρfjx⩽C2-knfjLq·RnΩx-·χj·Lq′·Rn. Noting s>q′-, we denote q′~(·)>1 and 1/q′(x)=1/q′~(x)+1/s. By Lemmas 8 and 9 we have (35)Ωx-·χj·Lq′·Rn⩽Ωx-·χj·LsRnχj·Lq′~·Rn⩽Ωx-·χj·LsRnχBjLq′~·Rn⩽C2-jν∫AjΩx-ysysνdy1/sχBjLq′~·Rn⩽C2-jν2kν+n/sΩLsSn-1χBjLq′~·Rn. When Bj⩽2n and xj∈Bj, by Lemma 10 we have (36)χBjLq′~·Rn≈Bj1/q′~xj≈χBjLq′·RnBj-1/s. When Bj⩾1 we have (37)χBjLq′~·Rn≈Bj1/q′~∞≈χBjLq′·RnBj-1/s. So we obtain χBjLq′~(·)(Rn)≈χBjLq′(·)(Rn)Bj-1/s.
By Lemmas3 and 4 we have (38)μΩρfjχkLq·Rn⩽C2-knfjLq·RnΩx-·χj·Lq′·RnχkLq·Rn⩽C2-kn2-jν2kν+n/sΩLsSn-1fjLq·RnχBjLq′~·RnχBkLq·Rn⩽C2-kn2-jν2kν+n/sΩLsSn-1fjLq·Rn×χBjLq′·RnBj-1/sχBkLq·Rn⩽C2k-jν+n/sΩLsSn-1fjLq·RnχBjLq′·RnχBkLq′·Rn⩽C2j-knδ2-ν-n/sΩLsSn-1fjLq·Rn.
Thus we obtain(39)I1⩽CΩLsSn-1∑k=-∞∞2kαp∑j=-∞k-22j-knδ2-ν-n/sfjLq·Rnp1/p=CΩLsSn-1∑k=-∞∞∑j=-∞k-22jα2j-knδ2-ν-n/s-αfjLq·Rnp1/p. If 1<p<∞, take 1/p+1/p′=1. Since nδ2-ν-n/s-α>0, by the Hölder inequality we have(40)I1⩽CΩLsSn-1∑k=-∞∞∑j=-∞k-22jαpfjLq·Rnp2j-knδ2-ν-n/s-αp/2×∑j=-∞k-22j-knδ2-ν-n/s-αp′/2p/p′1/p⩽CΩLsSn-1∑k=-∞∞∑j=-∞k-22jαpfjLq·Rnp×2j-knδ2-ν-n/s-αp/21/p=CΩLsSn-1∑j=-∞∞2jαpfjLq·Rnp×∑k=j+2∞2j-knδ2-ν-n/s-αp/21/p⩽C∑j=-∞∞2jαpfjLq·Rnp1/p=CfK˙q·α,pRn.If 0<p⩽1, then we have(41)I1⩽CΩLsSn-1∑k=-∞∞∑j=-∞k-22jαp2j-knδ2-ν-n/s-αpfjLq·Rnp1/p=CΩLsSn-1∑j=-∞∞2jαpfjLq·Rnp∑k=j+2∞2j-knδ2-ν-n/s-αp1/p⩽CfK˙q·α,pRn.
Let us now estimateI3. Note that x∈Ak, y∈Aj, and j⩾k+2, so we have x-y~y. We consider (42)μΩρfjx⩽∫0y∫x-y⩽tΩx-yx-yn-ρfjydy2dtt2ρ+11/2+∫y∞∫x-y⩽tΩx-yx-yn-ρfjydy2dtt2ρ+11/2≕I31+I32. Similar to the estimate for I11, we get (43)I31⩽C2k-j/22-jnfjLq·RnΩx-·χj·Lq′·Rn. Similar to the estimate for I12, we get (44)I32⩽C2-jnfjLq·RnΩx-·χj·Lq′·Rn. So we have (45)μΩρfjx⩽C2-jnfjLq·RnΩx-·χj·Lq′·Rn. By Lemmas 3 and 4 we have (46)μΩρfjχkLq·Rn⩽C2-jnfjLq·RnΩx-·χj·Lq′·RnχkLq·Rn⩽C2-jn2-jν2kν+n/sΩLsSn-1fjLq·RnχBjLq′~·RnχBkLq·Rn⩽C2-jn2-jν2kν+n/sΩLsSn-1fjLq·Rn×χBjLq′·RnBj-1/sχBkLq·Rn⩽C2k-jν+n/sΩLsSn-1fjLq·RnχBkLq·RnχBjLq·Rn⩽C2k-jnδ1+ν+n/sΩLsSn-1fjLq·Rn. Thus we obtain (47)I3⩽CΩLsSn-1∑k=-∞∞2kαp∑j=k+2∞2k-jnδ1+ν+n/sfjLq·Rnp1/p=CΩLsSn-1∑k=-∞∞∑j=k+2∞2jα2k-jnδ1+ν+n/s+αfjLq·Rnp1/p. If 1<p<∞, take 1/p+1/p′=1. Since nδ1+ν+n/s+α>0, by the Hölder inequality we have(48)I3⩽CΩLsSn-1∑k=-∞∞∑j=k+2∞2jαpfjLq·Rnp2k-jnδ1+ν+n/s+αp/2×∑j=k+2∞2k-jnδ1+ν+n/s+αp′/2p/p′1/p⩽CΩLsSn-1∑k=-∞∞∑j=k+2∞2jαpfjLq·Rnp×2k-jnδ1+ν+n/s+αp/21/p=CΩLsSn-1∑j=-∞∞2jαpfjLq·Rnp×∑k=-∞j-22k-jnδ1+ν+n/s+αp/21/p⩽C∑j=-∞∞2jαpfjLq·Rnp1/p=CfK˙q·α,pRn.If 0<p⩽1, then we have(49)I3⩽CΩLsSn-1∑k=-∞∞∑j=k+2∞2jαp2k-jnδ1+ν+n/s+αpfjLq·Rnp1/p=CΩLsSn-1∑j=-∞∞2jαpfjLq·Rnp∑k=-∞j-22k-jnδ1+ν+n/s+αp1/p⩽CfK˙q·α,pRn.
Thus by (28), (29) and (40), (41), (48), and (49) we complete the proof of Theorem 11.


## 4. BMO Estimate for the Higher-Order Commutator of Parametric Marcinkiewicz Integral Operator

Let us first recall that the spaceBMO(Rn) consists of all locally integrable functions f such that (50)f∗=supQ⁡1Q∫Qfx-fQdx<∞, where fQ=Q-1∫Qf(y)dy, the supremum is taken over all cubes Q⊂Rn with sides parallel to the coordinate axes, and Q denotes the Lebesgue measure of Q.Letb∈BMO(Rn). The weighted (Lp,Lp) boundedness of [bm,μΩρ] has been proved by Shi and Jiang [12].Lemma 12 (see [12]).
Suppose thatΩ∈Ls(Sn-1)(s>1) satisfying (1). If b(x)∈BMO(Rn) and ω∈Ap/s′,s′<p<∞, then for any f∈Lp(ω), there is a constant C, independent of f, such that (51)∫Rnbm,μΩρfxpωxdx⩽C∫Rnfxpωxdx.By Lemmas12 and 7 it is easy to get the (Lp(·)(Rn),Lp(·)(Rn))-boundedness of the commutator [bm,μΩρ].Next, we will give the corresponding result about the commutator[bm,μΩρ] on Herz spaces with variable exponent.Theorem 13.
Suppose thatb∈BMO(Rn),0<ν⩽1,0<p⩽∞,q(·)∈P(Rn) satisfies conditions (13) and (14) in Lemma 1; Ω∈Ls(Sn-1)(s>q′-) and -nδ1-ν-n/s<α<nδ2-ν-n/s. Then [bm,μΩρ] is bounded on K˙q(·)α,p(Rn) and Kq(·)α,p(Rn).In the proof of Theorem13, we also need the following lemma.Lemma 14 (see [15]).
Letp(·)∈B(Rn), m be a positive integer, and B be a ball in Rn. Then we have that, for all b∈BMO(Rn) and all j,i∈Z with j>i, (52)1Cb∗m⩽supB⁡1χBLp·Rnb-bBmχBLp·Rn⩽Cb∗m,b-bBimχBjLp·Rn⩽Cj-imb∗mχBjLp·Rn, where Bi={x∈Rn:x⩽2i} and Bj={x∈Rn:x⩽2j}.Proof of Theorem13.
Similar to Theorem11, we only prove homogeneous case and still suppose 0<p<∞. Let f∈K˙q(·)α,p(Rn). Denote fj=fχj for each j∈Z, then we have f(x)=∑j=-∞∞fj(x). Then(53)bm,μΩρfK˙q·α,pRn=∑k=-∞∞2kαpbm,μΩρfχkLq·Rnp1/p⩽C∑k=-∞∞2kαp∑j=-∞k-2bm,μΩρfjχkLq·Rnp1/p+C∑k=-∞∞2kαp∑j=k-1k+1bm,μΩρfjχkLq·Rnp1/p+C∑k=-∞∞2kαp∑j=k+2∞bm,μΩρfjχkLq·Rnp1/p≕CJ1+CJ2+CJ3.
Noting[bm,μΩρ] is bounded on Lq(·)(Rn), so we have(54)J2⩽C∑k=-∞∞2kαpfkLq·Rnp1/p=CfK˙q·α,pRn.
Now we estimateJ1. We consider (55)bm,μΩρfjx⩽∫0x∫x-y⩽tΩx-yx-yn-ρbx-bymfjydy2dtt2ρ+11/2+∫x∞∫x-y⩽tΩx-yx-yn-ρbx-bymfjydy2dtt2ρ+11/2≕J11+J12. Note that x∈Ak,y∈Aj, and j⩽k-2, and we know that x-y~x. By (31), the Minkowski inequality, and the generalized Hölder inequality we have (56)J11⩽C∫RnΩx-yx-yn-ρbx-bymfjy∫x-yxdtt2ρ+11/2dy⩽C∫RnΩx-yx-yn-ρbx-bymfjy1x-y2ρ-1x2ρ1/2dy⩽C∫RnΩx-ybx-bymfjyx-yn-ρy1/2x-yρ+1/2dy⩽C2j/2xn+1/2∫AjΩx-ybx-bymfydy⩽C2j-k/22-knbx-bBjm∫AjΩx-yfjydy+∫AjΩx-ybBj-bymfjydy⩽C2j-k/22-knfjLq·Rnbx-bBjmΩx-·χj·Lq′·Rn+Ωx-·bBj-b·mχj·Lq′·Rn.Similarly, we consider J12. Noting that x-y~x, by the Minkowski inequality and the generalized Hölder inequality, we have (57)J12⩽C∫RnΩx-yx-yn-ρbx-bymfjy∫x∞dtt2ρ+11/2dy⩽C∫RnΩx-yx-ynbx-bymfjydy⩽C2-knfjLq·Rnbx-bBjmΩx-·χj·Lq′·Rn+Ωx-·bBj-b·mχj·Lq′·Rn. So we have (58)bm,μΩρfjx⩽C2-knfjLq·Rnbx-bBjmΩx-·χj·Lq′·Rn+Ωx-·bBj-b·mχj·Lq′·Rn.
Notings>q′-, we denote q′~(·)>1 and 1/q′(x)=1/q′~(x)+1/s. By Lemmas 8 and 9 we have (59)Ωx-·χj·Lq′·Rn⩽Ωx-·χj·LsRnχj·Lq′~·Rn⩽Ωx-·χj·LsRnχBjLq′~·Rn⩽C2-jν∫AjΩx-ysysνdy1/sχBjLq′~·Rn⩽C2-jν2kν+n/sΩLsSn-1χBjLq′~·Rn. When Bj⩽2n and xj∈Bj, by Lemma 10 we have (60)χBjLq′~·Rn≈Bj1/q′~xj≈χBjLq′·RnBj-1/s. When Bj⩾1 we have (61)χBjLq′~·Rn≈Bj1/q′~∞≈χBjLq′·RnBj-1/s. So we obtain χBjLq′~(·)(Rn)≈χBjLq′(·)(Rn)Bj-1/s.
So we have(62)Ωx-·χj·Lq′·Rn⩽C2k-jν+n/sΩLsSn-1χBjLq′·Rn.Similarly, by Lemma 14 we have(63)Ωx-·bBj-b·mχj·Lq′·Rn⩽Ωx-·χj·LsRnbBj-b·mχj·Lq′~·Rn⩽Cb∗mχBjLq′~·RnΩx-·χj·LsRn⩽Cb∗m2k-jν+n/sΩLsSn-1χBjLq′·Rn.By (62), (63), Lemmas 3, 4, and 14 we have (64)bm,μΩρfjχkLq·Rn⩽C2-knfjLq·Rn2k-jν+n/sΩLsSn-1χBjLq′·Rn×b·-bBjmχBk·Lq·Rn+b∗m2k-jν+n/sΩLsSn-1χBjLq′·RnχkLq·Rn⩽C2-knfjLq·Rnk-jmb∗m2k-jν+n/sΩLsSn-1×χBjLq′·RnχBkLq·Rn+b∗m2k-jν+n/sΩLsSn-1χBjLq′·RnχBkLq·Rn⩽Ck-jmb∗mfjLq·Rn2-kn2k-jν+n/sΩLsSn-1×χBjLq′·RnχBkLq·Rn⩽Ck-jmb∗mfjLq·Rn2k-jν+n/sΩLsSn-1χBjLq′·RnχBkLq′·Rn⩽C2j-knδ2-ν-n/sk-jmb∗mΩLsSn-1fjLq·Rn.Thus we obtain (65)J1⩽C∑k=-∞∞2kαp∑j=-∞k-22j-knδ2-ν-n/sk-jm×b∗mΩLsSn-1fjLq·Rnp1/p=Cb∗mΩLsSn-1∑k=-∞∞∑j=-∞k-22jα2j-knδ2-ν-n/s-αk-jm×fjLq·Rnp1/p. If 1<p<∞, take 1/p+1/p′=1. Since nδ2-ν-n/s-α>0, by the Hölder inequality we have(66)J1⩽Cb∗mΩLsSn-1×∑k=-∞∞∑j=-∞k-22jαpfjLq·Rnp2j-knδ2-ν-n/s-αp/2×∑j=-∞k-22j-knδ2-ν-n/s-αp′/2k-jmp′p/p′1/p⩽Cb∗mΩLsSn-1×∑k=-∞∞∑j=-∞k-22jαpfjLq·Rnp2j-knδ2-ν-n/s-αp/21/p=Cb∗mΩLsSn-1×∑j=-∞∞2jαpfjLq·Rnp∑k=j+2∞2j-knδ2-ν-n/s-αp/21/p⩽Cb∗m∑j=-∞∞2jαpfjLq·Rnp1/p=Cb∗mfK˙q·α,pRn.
If0<p⩽1, then we have(67)J1⩽Cb∗mΩLsSn-1×∑k=-∞∞∑j=-∞k-22jαp2j-knδ2-ν-n/s-αpk-jmpfjLq·Rnp1/p=Cb∗mΩLsSn-1×∑j=-∞∞2jαpfjLq·Rnp∑k=j+2∞2j-knδ2-ν-n/s-αpk-jmp1/p⩽Cb∗mfK˙q·α,pRn.
Let us now estimateJ3. Note that x∈Ak, y∈Aj, and j⩾k+2, so we have x-y~y. We consider (68)bm,μΩρfjx⩽∫0y∫x-y⩽tΩx-yx-yn-ρbx-bymfjydy2dtt2ρ+11/2+∫y∞∫x-y⩽tΩx-yx-yn-ρbx-bymfjydy2dtt2ρ+11/2≕J31+J32.Similar to the estimate for J11, we get (69)J31⩽C2k-j/22-jnfjLq·Rnbx-bBkmΩx-·χj·Lq′·Rn+Ωx-·bBk-b·mχj·Lq′·Rn.Similar to the estimate for J12, we get (70)J32⩽C2-jnfjLq·Rnbx-bBkmΩx-·χj·Lq′·Rn+Ωx-·bBk-b·mχj·Lq′·Rn. So we have (71)bm,μΩρfjx⩽C2-jnfjLq·Rnbx-bBkmΩx-·χj·Lq′·Rn+Ωx-·bBk-b·mχj·Lq′·Rn. By (62), (63), Lemmas 3, 4, and 14 we have (72)bm,μΩρfjχkLq·Rn⩽C2-jnfjLq·Rnb∗m2k-jν+n/s×ΩLsSn-1χjLq′·RnχBkLq·Rn+2k-jν+n/sΩLsSn-1bBk-b·mχBj·Lq′·RnχkLq·Rn⩽C2-jnfjLq·Rnb∗m2k-jν+n/s×ΩLsSn-1χBjLq′·RnχBkLq·Rn+j-kmb∗m2k-jν+n/sΩLsSn-1χBjLq′·RnχBkLq·Rn⩽Cj-kmb∗mfjLq·Rn2k-jν+n/sΩLsSn-1χBkLq·RnχBjLq·Rn⩽C2k-jnδ1+ν+n/sj-kmb∗mΩLsSn-1fjLq·Rn.Thus we obtain (73)J3⩽Cb∗mΩLsSn-1×∑k=-∞∞2kαp∑j=k+2∞2k-jnδ1+ν+n/sj-kmfjLq·Rnp1/p=Cb∗mΩLsSn-1×∑k=-∞∞∑j=k+2∞2jα2k-jnδ1+ν+n/s+αj-kmfjLq·Rnp1/p.If 1<p<∞, take 1/p+1/p′=1. Since nδ1+ν+n/s+α>0, by the Hölder inequality we have(74)J3⩽Cb∗mΩLsSn-1×∑k=-∞∞∑j=k+2∞2jαpfjLq·Rnp2k-jnδ1+ν+n/s+αp/2×∑j=k+2∞2k-jnδ1+ν+n/s+αp′/2j-kmp′p/p′1/p⩽Cb∗mΩLsSn-1×∑j=-∞∞2jαpfjLq·Rnp∑k=-∞j-22k-jnδ1+ν+n/s+αp/21/p⩽Cb∗m∑j=-∞∞2jαpfjLq·Rnp1/p=Cb∗mfK˙q·α,pRn.If 0<p⩽1, then we have(75)J3⩽Cb∗mΩLsSn-1×∑j=-∞∞2jαpfjLq·Rnp∑k=-∞j-22k-jnδ1+ν+n/s+αpj-kmp1/p⩽Cb∗mfK˙q·α,pRn.
Thus by (53), (54) and (66), (67), (74), and (75) we complete the proof of Theorem 13.



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*Source: 7319093-2018-11-01.xml*