# Precompact Sets, Boundedness, and Compactness of Commutators for Singular Integrals in Variable Morrey Spaces

Authors: Wei Wang; Jingshi Xu
Journal: Journal of Function Spaces
 (2017)

---

## Abstract

We give sufficient conditions for subsets to be precompact sets in variable Morrey spaces. Then we obtain the boundedness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces. Finally, we discuss the compactness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces.


---

## Body

## 1. Introduction

Let the Calderón-Zygmund singular integral operatorT be defined by(1)Tfx≔p.v.∫RnΩx-yx-ynfydy,∀x∈Rn,where Ω is a measurable function on Rn and satisfies the following conditions: (i)
Ω is a homogeneous function of degree zero on Rn∖{0}; that is,(2)Ωμx=Ωxfor  anyμ>0,x∈Rn∖0.(ii)
Ω has mean zero on Sn-1; that is,(3)∫Sn-1Ωx′dσx′=0.Here Sn-1≔x∈Rn:x=1 is the unit sphere in Rn and dσ is the area measure on it.For a functionb∈LlocRn (the set of all locally integrable functions on Rn), let Mb be the corresponding multiplication operator defined by Mbf=bf for a measurable function f. Then the commutator between T and Mb is denoted by (4)b,T≔MbT-TMb=p.v.∫RnΩx-yx-ynbx-byfydy,for suitable functions f. Denote the bounded mean oscillation function space by(5)BMO⁡Rn≔b∈LlocRn:b∗≔supcube:Q⊂Rn⁡Mb,Q<∞,here and in the sequel(6)Mb,Q≔1Q∫Qbx-bQdx,bQ≔1Q∫Qbydy.It is well known that commutators play a very important role in harmonic analysis and PDEs. Indeed, Coifman et al. [1] characterized the Lp-boundedness of b,Rj, where Rjj=1,…,n are the Reisz transforms and b∈BMO⁡Rn. Using this characterization, the authors of [1] obtained a decomposition theorem of the real Hardy spaces H1Rn. Uchiyama [2] and Janson [3] showed that the Riesz transform Rj may be replaced by the Calderón-Zygmund singular integral operator T as in (1). Coifman et al. generalized the boundedness results of b,T to Hardy spaces and gave important applications to some nonlinear PDEs in [4]. The characterization of Lp-compactness of b,T was obtained by Uchiyama [2]. We remark that the interest in the compactness of b,T in complex analysis is from the connection between the commutators and the Hankel-type operators; see [5]. In recent years, Chen et al. have considered the compactness of commutators in [6–8]. Specially, the results in [2] were generalized to Morrey spaces in [8]. The Morrey space Lp,λRn was introduced by Morrey in 1938 and it is connected to certain problems in elliptic PDE [9]. After that the Morrey spaces were found to have many important applications to the Navier-Stokes equations (see [10]), the Schrödinger equations (see [11]), and potential theory (see [12–14]).During last three decades, the theory of variable function spaces has developed quickly; see [15–40]. We claim that the list is not exhaust. The boundedness in variable function spaces of many classical operators from harmonic analysis has been obtained; see [18, 19, 41–44]. Motivated by these works, we will consider analogous results in [8] to variable exponent situation. The structure of this paper is as follows. In Section 2, we give sufficient conditions for a set to be a precompact set in a variable Morrey space. In Section 3, we obtain the boundedness of singular integrals and their commutator in variable Morrey spaces. In Section 4, we discuss compactness of commutators in variable Morrey spaces. The remainder of this section is some notions.LetE be a measurable subset in Rn with E>0, where as usual E is the Lebesgue measure of E. Let p· be a measurable function on E with range in 1,∞. The variable exponent modulus is defined for measurable functions f on E by(7)ρp·f=∫Efxpxdx<∞.Lp·E denotes the set of measurable functions f on E such that ρp·λf<∞ for some λ>0. The set becomes a Banach function space when equipped with the norm(8)fLp·Ω≔inf⁡λ>0:ρp·fλ≤1.These spaces are the so-called variable Lebesgue spaces. Denote by P1E the set of measurable functions p· on E with range in 1,∞ such that(9)1<p-≔essinfx∈E⁡px,esssupx∈E⁡px≕p+<∞.Forp·∈P1Rn and 0<λx<n for x∈Rn, the variable Morrey space Lp·,λ·Rn is defined as the set of integrable functions f on Rn with the finite norm(10)fLp·,λ·Rn≔supx∈Rn,r>0⁡1v1rλx/pxfχBx,rLp·Rn,where Bx,r denotes a ball centered at x with radius r and v1 is the volume of the unit ball in Rn.


## 2. Precompact Sets in Variable Morrey Spaces

In this section, we give a compactness criterion in variable Morrey spaces. We remark here that a compactness criterion for variable exponent Lebesgue spaces was given in [45].Theorem 1.
Letp·∈P1Rn and 0<λx<n for x∈Rn. Suppose W is a subset in Lp·,λ·Rn satisfying the following conditions:(i)
Norm boundedness uniformly is(11)supf∈W⁡fLp·,λ·<∞.(ii)
Translation continuity uniformly is(12)limy→0⁡f·+y-f·Lp·,λ·=0foranyf∈W.(iii)
Uniformly convergence at infinity is(13)limα→∞⁡fχEαLp·,λ·=0foranyf∈W,where Eα=x∈Rn:x>α.
ThenW is a precompact set in Lp·,λ·Rn.To prove Theorem1, we need the following two lemmas, which are well known; for example, see [46].Lemma 2.
A setE is precompact in a Banach space X if and only if it is totally bounded which means for every positive number ϵ there is a finite subset Nϵ of points of X such that (14)E⊂⋃y∈NϵBϵy,where Bϵy denotes a ball centered at y with radius ϵ. The set Nϵ is called an ϵ-net of E.Lemma 3 (the Ascoli-Arzela theorem).
LetE be a bounded domain in Rn. A subset F of CΩ¯ is precompact in CΩ¯ if the following two conditions hold:(i)
There exists a constantM such that ux⩽M holds for every u∈F and x∈E.(ii)
For everyϵ>0, there exists δ>0 such that ux-uy<ϵ for u∈F, x,y∈E, and x-y<δ.Now there is a position to prove Theorem1.Proof of Theorem1.
Leth>0, we denote the mean of f on Bx,h by(15)Mhfx≔1v1hn∫y∈Bx,hfx+ydy,forx∈Rn.For f∈Lp·,λ·Rn and x∈Rn(16)Mhfx-fx=1v1hn∫y∈Bx,hfx+y-fxdy.Then by Hölder’s inequality and Fubini’s Theorem, for any t∈Rn and r>0,(17)Mhf-fLp·Bt,r⩽CsupgLp′·Bt,r⩽1⁡∫x∈Bt,r∫y∈Bx,h1v1hnfx+y-fxdygxdx=CsupgLp′·Bt,r⩽1⁡∫y∈Bx,h1v1hn∫x∈Bt,rfx+y-fxgxdxdy⩽Csupy⩽h⁡f·+y-f·Lp·Bt,r.Thus,(18)Mhf-fLp·,λ·⩽Csupy⩽h⁡f·+y-f·Lp·,λ·.Therefore, to prove that W is a precompact set, we need only to prove that MhW is precompact for small h. By Lemma 2, it suffices to show that MhW has finite ϵ-net for any ϵ>0. To do so, firstly, by Lemma 3, we show that MhW is precompact in CEαc for each α>0, where Eαc=x∈Rn:x⩽α. For any x∈Rn, by Hölder’s inequality, (19)Mhfx⩽1v1hn∫y⩽hfx+ydy⩽C1v1hnfLp·Bx,h1Lp′·Bx,h⩽Chλx/pxhn1v1hλx/pxfLp·Bx,h1Lp′·Bx,h⩽Chλx/pxhn1Lp′·Bx,hfLp·,λ·⩽Ch,λ·,p·fLp·,λ·,since 1Lp′·Bx,h⩽max⁡v1h1/p′-,v1h1/p′+, and hλ(x)/p(x)⩽max⁡1,hn/p-. Therefore, MhW are uniformly bounded functions by Condition (i). Then for x1,x2∈Rn,(20)Mhfx1-Mhfx2⩽1v1hn∫y⩽hfx1+y-fx2+ydy⩽C1v1hnfx1-x2+·-f·Lp·Bx2,h1Lp′·Bx,h⩽Ch,λ·,p·fx1-x2+·-f·Lp·,λ·.Thus, by Condition (ii) and Lemma 3, MhW is precompact in CEαc.
Finally, we verify thatMhW has finite 2ϵ-net for each small positive ϵ. For 0<ϵ<1, there exist N>0 and α>0 such that 1<ϵ-N/4<αn/p-<ϵ-N/2 and for each f∈W(21)fEαLp·,λ·<ϵ8.
By Lemma3, there exist f1,f2,…,fm⊂W such that Mhf1,Mhf2,…,Mhfm is a finite ϵN+1-net in MhW in the norm of CEαc. Below we verify that Mhf1,Mhf2,…,Mhfm is a finite ϵ-net in MhW in the norm of Lp·,λ·.
To finish the proof, we only need to show that, forf∈W, there is a fjj∈1,2,…,m such that for r>0, t∈Rn(22)I≔1v1rλt/ptMhf-MhfjLp·Bt,r<ϵ.
Now, we choosefj,j∈1,2,…,m such that(23)supy∈Eαc⁡Mhfy-Mhfjy<ϵN+1.
To show (22), we consider Bt,r into three cases.
Case  1. Bt,r⊂Eαc.
Ifr⩽1,(24)I⩽1v1rλt/ptϵN+1rn/p+⩽ϵN+1rn-λt/p+<ϵ.
Ifr>1,(25)I⩽Mhf-MhfjLp·Eαc⩽ϵN+1αn/p-<ϵ.
Case  2. Bt,r⊂Eα. Thus, we have(26)I=1v1rλt/ptMhf-MhfjχEαLp·Bt,r⩽1v1rλt/ptMhf-fχEαLp·Bt,r+f-fjχEαLp·Bt,r+fj-MhfjχEαLp·Bt,r⩽Mhf-fLp·,λ·+fχEαLp·,λ·+fjχEαLp·,λ·+fj-MhfjLp·,λ·<ϵ.
Case  3. Bt,r∩Eαc≠∅, and Bt,r∩Eα≠∅. Hence,(27)I⩽1v1rλt/ptMhf-MhfjχEαLp·Bt,r+1v1rλt/ptMhf-MhfjχEαcLp·Bt,r≔I1+I2.
Here,I1,I2 can be estimated that I1,I2<ϵ by Cases 1 and 2, respectively.
Therefore,MhW has a finite 2ϵ-net in Lp·,λ·Rn. This completes the proof.


## 3. Boundedness of Singular Integrals and Their Commutators

To consider the boundedness of singular integrals, a fundamental operator is the Hardy-Littlewood maximal operator. Given a functionf, the maximal function Mf is defined by (28)Mfx=supQ∋x⁡1Q∫Qfydy,∀x∈Rn,where the supremum is taken over all cubes containing x. It is well known that M is bounded on Lp, 1<p<∞. However, for any p·∈P1Rn, M need not be bounded in Lp·Rn. Let BRn be the set of p·∈P1Rn such that M is bounded on Lp·Rn. For the set BRn, we refer the reader to [19, 41] for details. If p·∈BRn, we will use the following results.Lemma 4 (see [22]).
Letp·∈BRn. Then there exist 0<δ<1 depending only on p· and n such that for balls B in Rn and all measurable subsets S⊂B(29)χSLp·RnχBLp·Rn⩽CSBδ.Lemma 5 (see [22]).
Letp·∈BRn. Then there exists a positive constant C>0 such that for balls B in Rn,(30)1BχBLp·RnχBLp′·Rn⩽C,where and what follows p′· is the conjugate exponent of p·, which means p′x=px/(px-1).Lemma 6 (see [23, 24]).
Supposep·∈BRn, and then there exists a positive constant C such that for each b∈BMO⁡Rn(31)C-1bBMO⁡Rn≤supQ⁡b-bQχQLp·χQLp·≤CbBMO⁡Rn.Theorem 7.
Let0<λx<n for x∈Rn. Suppose S is a linear or sublinear operator satisfying(32)Sfx⩽C∫Rnfyx-yndy,forx∉supp⁡f.If p·∈BRn, λ+<nδp-, where δ is as in Lemma 4 and the operator S is bounded on Lp·Rn, then S is also bounded on Lp·,λ·Rn. That means (33)SfLp·,λ·⩽CfLp·,λ·,where the constant C is independent of f.Proof.
Letf∈Lp·,λ·, pick any t∈Rn, and write fx=f0x+∑i=1∞fix, where f0=χBt,2rf, fi=χBt,2i+1r∖Bt,2irf.
Firstly, we estimateSf0 on Bt,r. By the boundedness of S on Lp·Rn, we have(34)Sf0Lp·Rn⩽Cf0Lp·Rn⩽Cv1rλt/ptfLp·,λ·.Thus,(35)Sf0Lp·Bt,r⩽Sf0Lp·Rn⩽Cv1rλt/ptfLp·,λ·.Hence, we obtain (36)Sf0Lp·,λ·⩽CfLp·,λ·.It remains to estimate S∑i=1∞fix on Bt,r. By the size estimate of S, we have(37)Sfix⩽C2ir-n∫Rnfiydy,forx∈Bt,r.Thus, by Lemmas 4 and 5 and Hölder’s inequality, we have (38)S∑i=1∞fiLp·Bt,r⩽∑i=1∞SfiLp·Bt,r⩽C∑i=1∞2ir-n∫RnfiydyLp·Bt,r⩽C∑i=1∞2ir-n1Lp·Bt,r∫Bt,2i+1rfydy⩽C∑i=1∞2ir-n1Lp·Bt,r1Lp′·Bt,2i+1rfLp·Bt,2i+1r⩽C∑i=1∞2ir-n1Lp·Bt,r1Lp·Bt,2i+1r1Lp·Bt,2i+1r×1Lp′·Bt,2i+1rv12i+1rλt/ptfLp·,λ·⩽C∑i=1∞2-nδi+1v12i+1rλt/ptfLp·,λ·.Thus,(39)S∑i=1∞fiLp·,λ·⩽C∑i=1∞2-i+1nδ-λ+/p-fLp·,λ·⩽CfLp·,λ·.This finishes the proof of Theorem 7.Next we turn to the boundedness of commutators in variable Morrey spaces. Many authors have studied it; see [44], but they considered that it restricts the underlying space with finite measure.Theorem 8.
SupposeS is a linear operator satisfying(40)Sfx=∫RnΩx-yx-ynfydy,forx∉supp⁡f,where Ω is a bounded measurable function. Let 0<λx<n, and b∈BMO⁡Rn and p·∈BRn. If the commutator b,S is bounded on Lp·Rn, then b,S is also bounded on Lp·,λ·Rn.Proof.
For anyt∈Rn and r>0, let B=Bt,r and write f as in the proof of Theorem 7. By the Lp·-boundedness of b,S, we obtain (41)b,Sf0xχBLp·≤Cf0Lp·≤Cv12rλ·/pxfLp·,λ·Rn.
Fork>0 and x∈B, we write (42)b,Sfkx≤C2krn∫2k+1Bbx-b2rΩx-yfkydy+C2krn∫2k+1Bb2r-b2k+1rΩx-yfkydy+C2krn∫2k+1Bby-b2k+1rΩx-yfkydy≔I1kx+I2kx+I3kx,where and in what follows, for k⩾0, b2k+1r is defined by (43)b2k+1r=1Bt,2k+1r∫Bt,2k+1rbydy.
By the well-known fact that, for anyr>0 and k∈N, b2k+1r-b2r≤Cnkb∗ (see [47, Proposition  7.1.5(i)]), we obtain(44)I2kx≤Ck+1b∗2kr-n∫2k+1Bfkydy≤Ck+12kr-nb∗χ2k+1BLp′·fLp·2k+1B.Hence, for k>0, using Lemmas 4 and 5 we have(45)I2kχBLp·≤Ck+12kr-nb∗χBLp·χ2k+1BLp′·fLp·2k+1B≤Ck+1b∗2-nδk+1v12k+1rλt/ptfLp·,λ·≤Ck+1b∗2-k+1nδ-λ+/p-v12rλt/ptfLp·,λ·.
ForI3kX, by Hölder’s inequality and Lemma 6, we have (46)I3kx≤C12krnb-b2k+1rχ2k+1BLp′·fLp·2k+1B≤C12krnb∗χ2k+1BLp′·fLp·2k+1B.
Therefore, as the argument as forI2k, we have(47)I3kχBLp·≤Cb∗2-k+1nδ-λ+/p-v12rλt/ptfLp·,λ·.
Finally, forI1kx by Hölder’s inequality, we have (48)I1kx≤Cbx-br12krnχ2k+1BLp′·fLp·2k+1B.Thus, as the argument as before, we obtain that(49)I1kχBLp·≤Cb∗12krnχBLp·χ2k+1BLp′·fLp·2k+1B≤Cb∗2-k+1nδ-λ+/p-v12rλt/ptfLp·,λ·.From (45), (47), and (49), we get (50)b,S∑k=1∞fkχBLp·≤C∑k=1∞k+1b∗2-k+1nδ-λ+/p-v12rλt/ptfLp·,λ·≤Cb∗v12rλt/ptfLp·,λ·.Hence, (51)b,SfLp·,λ·Rn≤Cb∗fLp·,λ·Rn.This finishes the proof of Theorem 8.Corollary 9.
Let0<λx<n. Suppose that Ω is a bounded homogeneous function of degree 0 and satisfies conditions (2) and(52)∫01wδδdδ<∞,where(53)wδ=supx′,y′∈Sn-1,x′-y′<δ⁡Ωx′-Ωy′.If p·∈BRn, then the Calderón-Zygmund singular integral operator T defined by (1) and its commutator b,T with b∈BMO⁡Rn are both bounded on Lp·,λ·Rn.Proof.
Corollary9 is the result of the following lemmas. Indeed, the boundedness of T on Lp·,λ·Rn is the direct result of Theorem 7, Lemmas 12 and 13. For the commutator, if w∈Ap, p∈1,∞, then by Corollary  9.2.6 in [47] there exists ϵ>0 such that w∈Ap-ϵ and p-ϵ>1. Then we choose s=p/(p-ϵ)<p in Lemma 11; for w∈Ap by Lemma 10 we obtain (54)∫Rnb,Tfxpwxdx⩽C∫Rnfxpwxdx,for bounded functions b and compactly supported functions f. Finally, using the similar argument for Theorem  7.5.6 in [47], we obtain that the last inequality holds for any b∈BMO⁡Rn. Thus, by Lemma 13, we obtain that b,T is bounded on Lp·Rn. Consequently, by Theorem 8, b,T is bounded on Lp·,λ·Rn.We remark here that the boundedness in variable Lebesgue spacesLp·Rn of commutator b,T has been proved in [44] by another method when Ω is an infinitely differentiable function on Sn-1.Lemma 10 (see Lemma  2.1 in [48]).
Let1<p<∞ and w∈Ap (Muckenhoupt weight); then there exists a positive constant C such that (55)∫RnMfxpwxdx≤C∫RnM#fxpwxdx,for functions f such that the left-hand side is finite.Here we sayw∈Ap, 1<p<∞ if for every cube Q(56)1Q∫Qwxdx1Q∫Qwx1-p′dxp-1≤C<∞.For properties of Ap, we refer the reader to [47].Lemma 11 (see Lemma  2.4.1 in [49]).
Letb∈BMO⁡Rn and p∈1,∞. Then for any s∈1,p, there exists a constant C, independent of b and f, such that(57)M#b,Tfx≤Cb∗MTfsx1/s+Mfsx1/s,∀x∈Rn.Lemma 12 (see Theorem  2.1.6 in [49]).
Suppose thatΩ is a bounded homogeneous function of degree 0 and satisfies conditions (1) and (52). If p∈1,∞ and w∈Ap, then there exists a constant C, independent of f, such that(58)∫RnTfxpwxdx≤C∫Rnfxpwxdx.Lemma 13 (see Corollary  1.11 in [18]).
Given thatF denotes a family of ordered pairs of nonnegative, measurable functions f,g on Rn, assume that (59)∫Rnfxp0wxdx⩽C0∫Rngxp0wxdx,f,g∈F,holds for some 1<p0<∞, for every ω∈Ap0 and for all f,g∈F. Let p·∈BRn. Then (60)fLp·⩽CgLp·.


## 4. Compactness of Commutators

Now we obtain sufficient conditions for the commutatorb,T to be a compact operator on Lp·,λ·Rn.Theorem 14.
Let0<λx<n for x∈Rn and p·∈BRn such that λ·/p· is a constant function. Suppose that Ω is a bounded homogeneous function of degree of 0 and satisfies (2) and for some q∈1,p-(61)∫01ωqδδ1+log⁡δdδ<∞,where ωqδ denotes the integral modulus of continuity of order q of Ω defined by (62)ωqδ=supρ<δ⁡∫Sn-1Ωρx′-Ωx′qdσx′1/q,and ρ is a rotation in Rn and ρ=supx′∈Sn-1⁡ρx′-x′. If b∈VMO⁡Rn (the closure of the set of compactly supported infinite differential functions in BMO⁡Rn), then the commutator b,T is a compact operator on Lp·,λ·Rn.Lemma 15.
Let0<λ·<n. Suppose that Ω is a bounded homogeneous function of degree of 0 and satisfies (2) and (52). For η>0, let (63)Tηfx=∫x-y>ηΩx-yx-ynfydy.Then for p·∈BRn, TηLp·,λ·⩽CfLp·,λ·, where C is independent of η and f.Proof.
Lemma15 is a direct consequence of Theorem 7. In fact, by Theorem  2.1.8 in [49], given that 1<p<∞, if ω∈Ap, then (64)∫ΩTηfxpωxdx⩽C0∫Ωfxpωxdxholds uniformly in η. Using Lemma 13, we can get that Tη is bounded on Lp·Rn uniformly in η. Now all conditions in Theorem 7 are fulfilled.Lemma 16 (see Lemma  2.2 in [8]).
Suppose that0<β<n, Ω satisfies (2), and Ω∈LqSn-1, where q>1. Then there exist C>0 such that for an R>0 and x∈Rn with x<R/2(65)∫R<y<2RΩy-xy-xn-β-Ωyyn-βqdy1/q⩽CRn/q-n-βxR+∫x/2Rx/Rwqδδdδ.Proof of Theorem14.
We will use the method in [8]. Let F be the unit ball in Lp·,λ·Rn. By density, we only need to prove that when b∈Cc∞Rn, the set G=b,Tf:f∈F is a precompact in Lp·,λ·Rn. By Theorem 1, it is sufficient to show that (11)–(13) hold uniformly in G.
Notice thatb∈Cc∞Rn. Applying Corollary 9, we have (66)supf∈F⁡b,TfLp·,λ·⩽Cb∗supf∈F⁡fLp·,λ·⩽Cb∗<∞.This shows that (11) holds.
Next we show that (13) holds. To do so, we suppose that β>1 taken so large that supp⁡b⊂x:x≤β. For any 0<ϵ<1, we take α>β such that α-βn1-q<ϵq. Below we show that for every t∈Rn and r>0,  q>1(67)1v1rλt/ptb,TfχEαLp·Bt,r<CϵΩLqSn-1.
In fact, for anyx∈Eα=x∈Rn:x>α and every f∈F, by Hölder’s inequality we have (68)b,Tfx=∫RnΩx-yx-ynbx-byfydy≤Cb∞∫y<βΩx-yx-ynfydy⩽C∫x-y⩽βΩyqynqfx-yqdy1/q.Then for every t∈Rn and r>0, by the Minkowski inequality and the choice of α, we get (69)1v1rλt/ptb,TfχEαLp·Bt,r⩽C1v1rλt/pt∫·-y≤βΩyqynqf·-yqdy1/qχEαLp·Bt,r⩽C1v1rλt/pt∫·-y≤βΩyqynqf·-yqdyχEαLp·/qBt,r1/q⩽C1rλt/pt-λt-y/pt-ysupx-y⩽β⁡1v1rλt-y/pt-yfLp·Bt-y,r∫y>α-βΩyqynqdy1/q⩽CϵΩLqSn-1fLp·,λ·.Thus, we get (67), which shows that (13) holds for b,T in G uniformly.
Finally, we show that the translation continuity condition (12) holds for the commutator b,T in G uniformly. We need to prove that, for any 0<ϵ<1/2, if z is sufficiently small depending only on ϵ, then for every f∈F(70)b,Tf·-b,Tf·+zLp·,λ·⩽Cϵ.Now for z∈Rn, we write(71)b,Tfx+z-b,Tfx=∫x-y>e1/ϵzΩx-yx-ynbx+z-bxfydy+∫x-y>e1/ϵzΩx-yx-yn-Ωx+z-yx+z-ynby-bx+zfydy+∫x-y⩽e1/ϵzΩx-yx-ynby-bxfydy-∫x-y⩽e1/ϵzΩx+z-yx-ynby-bx+zfydy≔J1+J2+J3-J4.
Sinceb∈Cc∞Rn, we have bx-bx+z⩽C∇b∞z. Since Ω∈LqSn-1 and applying Lemma 15, we get(72)J1Lp·,λ·⩽CzfLp·,λ·<Cz.
As forJ2, for every t∈Rn and r>0, using Lemma 16 and the Minkowski inequality, we get(73)1v1rλt/ptJ2Lp·Bt,r⩽2b∞1v1rλt/ptsupx-y>e1/ϵz⁡fLp·Bt-y,r∫y>e1/ϵzΩyyn-Ωy+zy+zndy⩽C1rλt/pt-λt-y/pt-yfLp·,λ·∑k=0∞z2ke1/ϵz+∫z/2k+1e1/ϵzz/2ke1/ϵzωδδdδ⩽CfLp·,λ·∑k=0∞12ke1/ϵ+11+k+1/ϵ∫z/2k+1e1/ϵzz/2ke1/ϵzωδδ1+log⁡δdδ⩽Ce-1/ϵ+ϵfLp·,λ·.Thus, we have(74)J2Lp·,λ·⩽Cϵ.
RegardingJ3, we have bx-by⩽C∇b∞x-y by b∈Cc∞Rn. Thus, (75)J3⩽C∫x-y⩽e1/ϵzΩx-yx-y-n+1fydy.By the Minkowski inequality, for every t∈Rn and r>0, we have(76)1v1rλt/ptJ3Lp·Bt,r⩽C1v1rλt/ptsupx-y⩽e1/ϵzfLp·Bt-y,r∫y⩽e1/ϵzΩyy-n+1dy⩽C1rλt/pt-λt-y/pt-ye1/ϵzfLp·,λ·⩽Ce1/ϵzfLp·,λ·.Thus,(77)J3Lp·,λ·⩽Ce1/ϵz.
Finally, bybx+z-by⩽C∇b∞x+z-y, we have(78)J4⩽C∫x-y⩽e1/ϵzΩx+z-yx+z-y-n+1dy.Using the same argument for J3, it is easy to check that(79)J4Lp·,λ·⩽Ce1/ϵz+z.
From (72), (74), (77), and (79), and taking z to be sufficiently small, we can get (80)b,Tf·-b,Tf·+zLp·,λ·⩽J1Lp·,λ·+J2Lp·,λ·+J3Lp·,λ·+J4Lp·,λ·⩽Cϵ.Therefore, we show that the translation continuity (12) holds for the commuator b,T in G uniformly and this completes the proof of Theorem 14.We remark that in Theorem14 the condition b∈VMO⁡Rn is necessary by Theorem  1.2 in [8].



---

*Source: 3764142-2017-07-20.xml*