# On the Compactness of the Commutator of the Parabolic Marcinkiewicz Integral with Variable Kernel

Authors: Yanping Chen; Kai Zhu; Yong Ding
Journal: Journal of Function Spaces
 (2014)

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## Abstract

The authors prove that the commutator of the parabolic Marcinkiewicz integralgΩ with variable kernel is a compact operator on Lp(Rn)(1<p<∞) if and only if b∈CMO(Rn,ρ). The result is substantial improvement and extension of some known results.


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## Body

## 1. Introduction and Main Results

LetRn be the n-dimension Euclidean space. Let Sn-1={x∈Rn:|x|=1} be the unit sphere on Rn equipped with the Lebesgue measure dσ(x′). For b∈Lloc⁡(Rn), denote by mb the multiplication operator defined by mbf=bf for the measurable function f. Suppose that A is a linear operator on some measurable function space; then the commutator formed by A and mb is defined by
(1)b,Afx∶=mbAfx-Ambfx=bxAfx-Abfx.
A famous theorem of Coifman et al. [1] characterized the Lp-boundedness of [b,Rj], where Rj  (j=1,…,n) are the Riesz transforms and b∈BMO(Rn). Using this characterization, the authors of [1] got a decomposition theorem of the real Hardy space H1(Rn). Uchiyama [2] and Janson [3] showed that the Riesz transform Rj may be replaced by the Calderón-Zygmund singular integral operator TΩ. Moreover, in 1978, Uchiyama [2] gave a characterization of the compactness for the commutator [b,TΩ] defined by
(2)b,TΩfx=p.v.∫Rn‍bx-byΩx-yx-ynfydy,
where Ω is homogeneous function of degree zero on Rn∖{0}, that is,
(3)Ωtx=Ωxforanyt>0,x∈Rn∖0,
and Ω has mean zero on Sn-1, the unit sphere in Rn, that is,
(4)∫Sn-1‍Ωx′dσx′=0.Theorem A (see [2]).
Suppose thatΩ∈Lip(Sn-1) satisfies (3) and (4). Let b∈Lloc⁡(Rn).(i)
If[b,TΩ] is a compact operator on Lp(Rn) for some p, 1<p<∞, then b∈CMO(Rn).(ii)
Ifb∈CMO(Rn), then [b,TΩ] is a compact operator on Lp(Rn) for 1<p<∞, where CMO(Rn) is the BMO-closure of Cc1(Rn) and Cc1(Rn) denotes the set of C1-functions with compact support in Rn.The interest in the compactness of[b,TΩ] in complex analysis is from the connection between the commutators and the Hankel-type operators. In fact, Beatrous and Li [4] proved the boundedness and compactness of [b,TΩ] on Lp over some spaces of homogeneous type. Krantz and Li (see [5]) have applied the characterization of Lp-compactness of commutator to give a compactness of Hankel operators on holomorphic Hardy spaces H2(D), where D is a bounded, strictly pseudoconvex domain in Cn.In this paper, we will study the compactness of the commutator of the parabolic Marcinkiewicz integral with variable kernel. Let us recall the definition of parabolic metric. Letα1,…,αn be fixed real numbers, αi≥1. For fixed x=(x1,…,xn)∈Rn, the function F(x,ρ)=∑i=1n‍(xi2/ρ2αi) is a decreasing function in ρ (ρ>0). We denote by ρ(x) the unique solution of the equation F(x,ρ)=1. In 1966, Fabes and Rivière [6] proved that ρ(x) is a metric on Rn. For μ>0 and x∈Rn, define the dilation on Rn by
(5)Aμ:x1,x2,…,xn⟶μα1x1,μα2x2,…,μαnxn.
Then it is immediate to see that ρ(Aμx)=μρ(x) and ρ(tx)≤ρ(x) when t≤1. One has the polar decomposition x=Aρx′ with x′=(x1/ρ(x)α1,…,xn/ρ(x)αn)∈Sn-1,ρ=ρ(x) and dx=ρα-1J(x′)dρdσ(x′), where α=∑i=1n‍αi,J(x′) is a C∞ function on Sn-1 and bounded below uniformly by 1.In 1974, Madych [7] gave the Lp (1<p<∞) boundedness of the parabolic Marcinkiewicz integral gψ with respect to the transform At, where
(6)gψfx=∫0∞‍ψt*fx2dtt1/2
with ψ∈S(Rn) satisfying ∫Rn‍ψ(x)dx=0, and ψt(x)=t-αψ(At-1x) for t>0. Inspired by the works in [6, 7], recently, Xue et al. [8] improved the above result. More precisely, the authors in [8] proved that the parabolic Marcinkiewicz integral is still bounded on Lp if replacing ψ(x) by a kernel function ϕ(x)=Ω(x)ρ(x)-α+1χ{ρ(x)≤1}(x) with Ω∈Lq(Sn-1)  (q>1) satisfying the following conditions:
(7)ΩAμz=Ωz,foranyz∈Rn∖0,μ>0,∫Sn-1‍Ωz′Jz′dσz′=0.
Afterwards, many authors studied the properties of the parabolic Marcinkiewicz integral and its commutators which is defined by
(8)gψ;bfx=∫0∞‍b,ψtfx2dtt1/2
(see, e.g., [9–12]).Notice that ifα1=⋯=αn=1, then ρ(x)=|x| and (Rn,ρ)=(Rn,|·|). In this case, gψ and gψ;b are just the classical Marcinkiewicz integral and it is commutators which play a very important role in the singular integral operators theory, function spaces theory, and PDE (see, e.g., [13–23]).Let us turn to the definitions of the parabolic Marcinkiewicz integral with variable kernel and its commutator. Suppose thatΩ(x,z) satisfies the following conditions:
(9)Ωx,Aμz=Ωx,z,foranyx∈Rn,z∈Rn∖0,μ>0,(10)∫Sn-1‍Ωx,z′Jz′dσz′=0,foranyx∈Rn.
Then the parabolic Marcinkiewicz integral with variable kernel is defined by
(11)gΩfx=∫0∞‍∫ρ(x-y)≤t‍Ω(x,x-y)ρ(x-y)α-1f(y)dy2dtt31/2.
Let us now give the definition of parabolic BMO space. Denote by E(x,r) the ellipsoid center at x and radius r. More precisely, E(x,r)={y:ρ(x-y)<r}. For j>0, we denote jE(x,r) to be the j-times extension of E(x,r) with the same center. That is, jE(x,r)={y:ρ(x-y)<jr}. Moreover, |E(x,r)| stands the Lebesgue measure of E(x,r), which is comparable to rα. Ec denotes the complement of E.For a measurable and locally integrable functionb:Rn→R set
(12)ηbθ=sup⁡r≤θ1Er∫E(r)‍by-bErdyforθ>0,
where E(r) is any ellipsoid in Rn of radius r and bE(r)=|E(r)|-1∫E(r)‍b(y)dy is the average of b on E(r). We say b∈BMO(Rn,ρ) if ∥b∥*,ρ:=sup⁡θηb(θ)<∞ (see [24]).Letb∈Lloc⁡(Rn); the commutator of the parabolic Marcinkiewicz integral with variable kernel is defined by
(13)gΩ;bfx=∫0∞‍∫ρ(x-y)≤t‍Ω(x,x-y)ρ(x-y)α-1(b(x)-b(y))f(y)dy2dtt31/2.Notice that ifα1=⋯=αn=1, then ρ(x)=|x|; gΩ and gΩ;b are the classical Marcinkiewicz integral with variable kernel and the classical commutator of Marcinkiewicz integral with variable kernel which was studied by many authors (see, e.g., [25–28]).In 2009, Chen and Ding [29] studied the parabolic Marcinkiewicz integral with variable kernel and its commutator on the Lebesgue space Lp(Rn), which is stated as follows.Theorem B.
Let1<p<∞. Suppose that Ω(x,z) satisfies (9), (10) and for every multi-index β,
(14)sup⁡x∈Rnξ′∈Sn-1DξβΩx,ξ′≤Cβ.
Let b∈BMO(Rn,ρ). Then there exists a constant C>0 such that, for any f∈Lp(Rn),
(15)gΩ;bfp≤Cb*,ρfp.In this paper, we will give a characterization of the commutator of the parabolic Marcinkiewicz integral with variable kernel which is compact operator onLp(Rn) (1<p<∞). First, we will prove that the commutator of the parabolic Marcinkiewicz integral with variable kernel is bounded on Lp(Rn) (1<p<∞). To show our result, we give the following definition.Definition 1 (see [30]).
LetX and Y be Banach spaces and let U be a subset of X. Then T:U↦Y is compact if T is continuous and maps bounded subsets of U into strongly precompact subsets of Y.Our first result can be stated as follows.Theorem 2.
LetΩ(x,z) satisfy (9), (10), (14) and for any multi-index β(16)sup⁡ξ′∈Sn-1DξβΩx,ξ′-DξβΩy,ξ′≤Cρx-y.
If b∈CMO(Rn,ρ), where CMO(Rn,ρ) is the BMO(Rn,ρ)-closure of Cc1(Rn) and Cc1(Rn) denotes the set of C1-functions with compact support in Rn, then gΩ;b is a compact operator on Lp(Rn) (1<p<∞).On the other hand, we will give the converse part of the compactness of the commutator of the Marcinkiewicz integral with variable kernel.Theorem 3.
Suppose thatΩ satisfies (9) and (10),
(17)sup⁡z∈RnΩz,x′-Ωz,y′≤ρx′-y′,forx′,y′∈Sn-1,
and there exists Θ⊂Sn-1,0<δ<1 such that, for any x∈Rn,
(18)σΘ=σy′∈Sn-1:inf⁡x∈RnΩx,y′≥2δ>0,
where σ is the measure on Sn-1 which is induced from the Lebesgue measure on Rn. If [b,gΩ] is a bounded operator on Lp(Rn) for some 1<p<∞, then b∈BMO(Rn,ρ).Theorem 4.
LetΩ(x,z) satisfy the same conditions as in Theorem 2. If gΩ;b is a compact operator on Lp(Rn) for some p (1<p<∞), then b∈CMO(Rn,ρ).Remark 5.
Our results are still new even for the caseρ(x)=|x|.This paper is organized as follows. First, in Section2, we give some important notations and lemmas, which will be used in the proofs of the main results. In Section 3, we prove Theorem 2. The proof of Theorem 3 is given in Section 4. In Section 5, we will give the proof of Theorem 4.The letterC in the paper denotes positive constants independent of essential variables. For p≥1, p′ denotes the dual exponent of p, that is, p′=p/(p-1).


## 2.  Notations and Lemmas

For a given measurable functionf∈Lloc⁡(Rn), the Hardy-Littlewood maximal operator M and the sharp maximal operator M# related to ellipsoids are defined by
(19)Mfx=sup⁡E∋x1E∫E‍fydy,M#fx=sup⁡E∋x1E∫E‍fy-fEdy,
respectively. Define also the operator Mqf(x):=(M(|f|q)(x))1/q for 1≤q<∞.Denote byHm the space of all n-dimensional spherical harmonic of degree m with its dimension Dm=dim⁡Hm. By [24, 31], we have
(20)Dm≤Cnmn-2,m≥1.Furthermore, let{Ym,s(x′)}s=1Dm be an orthonormal base of Hm. Then {Ym,s(x′)}s=1Dm (m=0,1,…) is a complete orthonormal system in L2(Sn-1) and for any multi-index β (see [24, 31]),
(21)sup⁡x′∈Sn-1Dx′βYm,sx′≤Cnm|β|+(n-2)/2,m=1,2,….
If, for instance, ϕ∈C∞(Sn-1), then ∑m‍∑s‍am,sYm,s(x′) is the Fourier series expansion of ϕ(x′) with respect to {Ym,s(x′)}m,s, where
(22)am,s=∫Sn-1‍ϕy′Ym,sy′dσy′,
and (see [24, 31])
(23)am,s=-1lm-lm+n-2-l×∫Sn-1‍LlϕYm,sx′dσx′,m≥1,
where L(ϕ)=|x|2Δϕ(x), Ll(ϕ)=L(Ll-1(ϕ)) for any integer l. In particular, the expansion of ϕ into spherical harmonics converges uniformly to ϕ. The next results will be employed in the forthcoming considerations.Lemma 6 (see [29]).
Suppose thatE=E(x0,r). Let γ>0. Then
(24)Ym,sx-y′Jx-y′ρx-yγ-Ym,sx0-y′Jx0-y′ρx0-yγ≤Cmn/2ρx-x0ρy-x0γ+1,
for each x∈E and y∈(2E)c, where x′=(x1/ρ(x)α1,…,xn/ρ(x)αn).Lemma 7 (Frechet-Kolmogorov, see [32, 33]).
A subsetG of Lp(Rn), 1<p<∞, is strongly precompact if and only if it satisfies the conditions
(25)sup⁡f∈Gfp<∞;(26)lim⁡ρ(y)→0f·+y-f·p=0uniformlyinf∈G;(27)lim⁡β→∞fχEβp=0uniformlyinf∈G,whereEβ=x∈Rn:ρx>β.Lemma 8 (see [29]).
Suppose thatE=E(x0,r). Let γ>0. Then
(28)1ρx-yγ-1ρx0-yγ≤22γ-1ρx-x0ρy-x0γ+1,
for each x∈E and y∈(2E)c. (Here and below, for a>0 and E(x0,r), aE denotes the a-times extension of E with the same center, that is, aE={y:ρ(y-x0)<ar}.)Lemma 9 (see [32]).
Ifρ(x)≥4ρ(y), then
(29)ρx-y′-x′≤3ρyρx,
where x′=(x1/ρ(x)α1,…,xn/ρ(x)αn)∈Sn-1.Lemma 10 (see [2, 32]).
Ifb∈BMO(Rn,ρ), C5>C4>2, E is an ellipsoid centered at x0 and radius r, then there exist positive constants C6, C7, and C8 (depending on C4, C5, and b), such that
(30)C4r<ρx-x0<C5r:bx-bE>ν+C6≤C7Ee-C8ν,0<ν<∞.Lemma 11 (see [34]).
Suppose thatf(x) is a measurable function, λf(w)=|{x∈Rn:|f(x)|>w>0}|, and E is a measurable set. Define f*(t)=inf⁡{w:λf(w)≤t} for t>0; then
(31)∫E‍f(x)pdx≤∫0|E|‍f*tpdt,1≤p<∞.Lemma 12 (see [2, 32]).
Letb∈BMO(Rn,ρ). Then b∈CMO(Rn,ρ) if and only if b satisfies the following three conditions:(i)
lim⁡a→0sup⁡|E|=aM(b,E)=0,(ii)
lim⁡a→∞sup⁡|E|=aM(b,E)=0,(iii)
lim⁡x→∞M(b,E+x)=0,for eachE,whereM(b,E)=(1/E)∫E‍|b(y)-bE|dy.It is easy to get the following lemma.Lemma 13.
IfΩ satisfies conditions (9) and (17), then for β>0,
(32)Ωz,x-yρx-yβ-Ωz,xρxβ≤Cρyρxβ+1
for ρ(x)≥4ρ(y).


## 3. Proof of Theorem2

Suppose thatb∈CMO(Rn,ρ). Then, by Theorem A, we can get
(33)gΩ;bfp≤Cb*,ρfp,foranyf∈LpRn.
Then we can easily get gΩ;b is continuous on Lp(Rn), so by Definition 1, it suffices to prove that, for any bounded set F in Lp(Rn), G={gΩ;bf:f∈F} is strongly precompact in Lp(Rn). Since for any ɛ>0 there exists bɛ∈C0∞ such that ∥b-bɛ∥*,ρ<ɛ, then by (33),
(34)gΩ;b-gΩ;bɛLp↦Lp≤gΩ;b-bɛLp↦Lp<Cɛ.
Thus, it suffices to prove that G is strongly precompact in Lp(Rn) for b∈Cc∞(Rn). By Lemma 7, we need only to verify that (25)–(27) hold uniformly in G.Suppose thatfp≤D for every f∈F. Notice that b∈Cc∞(Rn) and applying (33), we have
(35)sup⁡f∈FgΩ;bfp≤Cb*,ρsup⁡f∈Ffp≤C′D<∞.
On the other hand, for any ɛ>0, we may take B>0 large enough such that
(36)∫ρ(x)>B‍ρx-αpdx1/p<ɛ.
Assume supp⁡(b)⊂{y:ρ(y)<r} for some r>0. Thus, for any x satisfying ρ(x)>max⁡{B,4r} and every f∈F, then by Ω∈L∞(Rn)×L∞(Sn-1) and ρ(x-y)~ρ(x), we have
(37)gΩ;bfx=∫0∞‍∫ρx-y≤t‍Ωx,x-yρx-yα-1bx-byf(y)dy2dtt31/2≤C∫supp⁡b‍byρx-yα-1fy∫ρ(x-y)≤t‍dtt31/2dy≤Cρx-αfp.
Applying (36), we have
(38)∫ρ(x)>B‍gΩ;bf(x)pdx1/p≤Cɛ.
Equation (38) shows that (27) holds uniformly in G. Finally, to finish the proof of Theorem 2, it remains to show (26) holds uniformly in G. We need to prove that, for any ɛ>0, if ρ(z) is sufficiently small, then for every f∈F,
(39)gΩ;bf·-gΩ;bf·+zp≤Cɛ.
To do this, for x,y∈Rn, by (10), we have
(40)Jy′Ωx,y′=∑m=1∞‍∑s=1Dm‍am,sxYm,sy′,
where
(41)am,sx=∫Sn-1‍Ωx,y′Ym,sy′dσy′.
So we get
(42)Ωx,y′=∑m=1∞‍∑s=1Dm‍am,sxYm,sy′Jy′.
In this way, (14) and (23) imply for any x∈Rn(43)am,sx≤Cn,lm-2l,
for any integer l>1. We fix hereafter l=n. Let
(44)gm,sfx=∫0∞‍∫ρx-y≤t‍Ym,sx-y′Jx-y′ρx-yα-1fydy2dtt31/2,gm,s;bfx=∫0∞‍∫ρx-y≤t‍Ym,sx-y′Jx-y′ρx-yα-1×bx-byf(y)dy∫ρx-y≤t‍Ym,sx-y′Jx-y′ρx-yα-12dtt31/2.It is easy to see thatΩm,s(x)=(Ym,s/J)(x′) has the following properties:
(45)∫Sn-1‍Ωm,sx′Jx′dσx′=0,Ωm,sAλx=Ωm,sx,∀λ>0,x∈Rn∖0.
Moreover, Ωm,s∈L2(Sn-1)⊂L(log⁡+L)2(Sn-1)⊂Llog⁡+L(Sn-1). Then by the result in [11, 12], we know for 1<p<∞,
(46)gm,sfp≤Cfp,(47)gm,s;bfp≤Cb*,ρfp,
where C is independent of f,m,s.For anyz∈Rn, by the Minkowski inequality, the Hölder inequality, we get
(48)gΩ;bfx-gΩ;bfx+z≤×bx+z-byfydy∫ρ(x-y)≤t‍Ym,sx-y′Jx-y′ρx-yα-1∑m=1∞‍∑s=1Dm‍2∫0∞‍×bx+z-byfydy∫ρ(x-y)≤t‍Ym,sx-y′Jx-y′ρx-yα-1∑m=1∞‍∑s=1Dm‍2dtt3∑m=1∞‍∑s=1Dm‍am,s(x)×∫ρ(x-y)≤t‍Ym,sx-y′Jx-y′ρx-yα-1×bx-byfydy-∫ρ(x+z-y)≤t‍ρx+z-yα-1-1Ym,sx+z-y′×Jx+z-y′×ρx+z-yα-1-1Ym,sx+z-y′×bx+z-byfydy∫ρ(x-y)≤t‍Ym,sx-y′Jx-y′ρx-yα-1∑m=1∞‍∑s=1Dm‍2dtt31/2+×bx+z-byfydy∑m=1∞‍∑s=1Dm‍2∫0∞‍∑m=1∞‍∑s=1Dm‍am,sx-am,sx+z×∫ρ(x+z-y)≤t‍Ym,sx+z-y′Jx+z-y′ρx+z-yα-1×bx+z-byfydy∑m=1∞‍∑s=1Dm‍2dtt31/2=∫0∞‍∑m=1∞‍∑s=1Dm‍am,sxIm,sx,t,z2dtt31/2+×IIm,s(x,t,z)∑m=1∞‍∑s=1Dm‍2∫0∞‍∑m=1∞‍∑s=1Dm‍am,sx-am,sx+z×IIm,s(x,t,z)∑m=1∞‍∑s=1Dm‍2dtt31/2≤×x,t,z∑s=1Dm‍2∫0∞‍∑m=1∞‍m-2∑m=1∞‍m2∑s=1Dm‍am,sxIm,s×x,t,z∑s=1Dm‍2dtt31/2+IIm,sx,t,z∑s=1Dm‍2∫0∞‍∑m=1∞‍m-2∑m=1∞‍m2∑s=1Dm‍am,sx-am,sx+z×IIm,sx,t,z∑s=1Dm‍2dtt31/2≤C∫0∞‍∑m=1∞‍m2∑s=1Dm‍am,sx2×∑s=1Dm‍Im,s(x,t,z)2dtt31/2+C∫0∞‍∑m=1∞‍m2∑s=1Dm‍am,s(x)-am,sx+z2×∑s=1Dm‍IIm,s(x,t,z)2dtt31/2,
where
(49)Im,sx,t,z=∫ρx-y≤t‍Ym,sx-y′Jx-y′ρx-yα-1×bx-byfydy-∫ρ(x+z-y)≤t‍Ym,sx+z-y′Jx+z-y′ρx+z-yα-1×bx+z-byfydy∫ρx-y≤t‍Ym,sx-y′Jx-y′ρx-yα-1,IIm,sx,t,z=∫ρ(x+z-y)≤t‍Ym,sx+z-y′Jx+z-y′ρx+z-yα-1×bx+z-byfydy.
Then by (43) and the Minkowski inequality, we get
(50)gΩ;bfx-gΩ;bfx+z≤C∑m=1∞‍m-3n/2∑s=1Dm‍∫0∞Im,sx,t,z2‍dtt31/2+C∑m=1∞‍m∑s=1Dm‍am,sx-am,sx+z21/2×∑s=1Dm‍∫0∞‍IIm,sx,t,z2dtt31/2∶=K1+K2.
We first give the Lp norm of K2. By (16), we have
(51)am,sx+z-am,sx≤Cm-2nρz.
Moreover, by (47), we get
(52)∫0∞‍IIm,s·,t,z2dtt31/2p=gm,s;bf·+zLp≤Cfp.
Therefore, using Dm≃mn-2 and the Minkowski inequality, we get
(53)K2p≤C∑m=1∞‍m-2ρzfp≤Cρ(z)fp.
Now, we give the Lp norm of K1. Take an arbitrary 4-1>ε2>0 and z∈Rn such that ε-2ρ(z)≤1. Decompose Im,s as
(54)Im,sx,t,z=∫ρ(x-y)>ε-2ρ(z)ρ(x-y)≤t,ρ(x+z-y)≥tYm,sx-y′Jx-y′ρx-yα-1×bx+z-byfydy+∫ρ(x-y)>ε-2ρ(z)ρ(x-y)≥t,ρ(x+z-y)≤tYm,sx+z-y′Jx+z-y′ρx+z-yα-1×by-bx+zfydy+∫ρ(x-y)>ε-2ρ(z)ρ(x-y)≤t,ρ(x+z-y)≤tYm,sx-y′Jx-y′ρx-yα-1-ρx+z-yα-1-1Ym,sx+z-y′×Jx+z-y′×ρx+z-yα-1-1Ym,sx-y′Jx-y′ρx-yα-1×bx+z-byfydy+∫ρ(x-y)>ε-2ρ(z)ρ(x-y)≤tYm,sx-y′Jx-y′ρx-yα-1×bx-bx+zfydy+∫ρ(x-y)>ε-2ρ(z)ρ(x-y)≤tYm,sx-y′Jx-y′ρx-yα-1×bx-byfydy-∫ρ(x-y)>ε-2ρ(z)ρ(x+z-y)≤tYm,sx+z-y′Jx+z-y′ρx+z-yα-1×bx+z-byfydy∶=J1(x,t)+J2(x,t)+J3(x,t)+J4(x,t)+J5(x,t)-J6(x,t).
We will estimate each term, respectively. For J1, when ρ(x-y)>4ρ(z), by Lemma 8, (21), |b(x+z)-b(y)|<C, J(x′)≥1, and the Minkowski inequality, we have
(55)∫0∞‍J1x,t2dtt31/2≤C∫ρx-y>ε-2ρz‍Ym,sx-y′ρx-yα-1fy×∫ρ(x-y)≤t,ρ(x+z-y)≥t‍dtt31/2dy≤C∫ρx-y>ε-2ρz‍Ym,sx-y′ρx-yα-1fy×1ρx-y2-1ρx-y+z21/2dy≤Cmn/2-1∫ρx-y>ε-2ρz‍ρz1/2ρx-yα+1/2fydy.
Then by the Minkowski inequality, we get
(56)∫0∞‍J1x,t2dtt31/2≤Cmn/2-1∫Rn‍∫ρ(y)>ε-2ρ(z)‍ρz1/2ρyα+1/2×fx-ydy∫ρ(y)>ε-2ρ(z)‍ρ(z)1/2ρ(y)α+1/2pdx∫ρ(y)>ε-2ρ(z)‍ρ(z)1/2ρ(y)α+1/21/p≤Cmn/2-1fp∫ρ(y)>ε-2ρ(z)‍ρz1/2ρyα+1/2dy≤Cmn/2-1ρz1/2fp∫ε-2ρ(z)∞‍drr1+1/2∫Sn-1‍dσy′≤Cmn/2-1ɛfp.
Similar to the estimate of J1(x,t), we can get
(57)∫0∞‍J2x,t2dtt31/2≤Cmn/2-1∫ρ(x-y)>ε-2ρ(z)‍ρz1/2ρx+z-yα+1/2fydy.
Then by the Minkowski inequality, we get
(58)∫0∞‍J2x,t2dtt31/2≤Cmn/2-1×∫Rn∫ρ(y)>(ε-2-1)ρ(z)‍ρz1/2ρyα+1/2×fx+z-ydy∫ρ(y)>(ε-2-1)ρ(z)‍ρz1/2ρyα+1/2pdx1/p≤Cmn/2-1fpρz1/2∫ρ(y)>(ε-2-1)ρ(z)‍1ρyα+1/2dy≤Cmn/2-1ε-2-1-1/2fp≤Cmn/2-1ɛfp.
For J3, since |b(x+z)-b(y)|<C, by Lemma 6, the Minkowski inequality, and Lemma 8, we have
(59)∫0∞‍J3x,t2dtt31/2≤C∫ρ(x-y)>ε-2ρ(z)‍Ym,sx-y′Jx-y′ρx-yα-1-Ym,sx+z-y′Jx+z-y′ρx+z-yα-1×fy∫ρ(x-y)≤tρ(x+z-y)≤tdtt31/2dy≤Cmn/2∫ρ(x-y)>ε-2ρ(z)‍ρzρx-yα+1dy.
Then similar to estimate (56), we get
(60)∫0∞‍J3x,t2dtt31/2p≤Cmn/2ɛfp.
About J4, we have
(61)∫0∞‍J4(x,t)2dtt31/2≤bx-bx+z×fydy∫ρ(x-y)>ε-2ρ(z)ρ(x-y)≤tYm,sx-y′Jx-y′ρx-yα-12dtt3∫0∞‍∫ρ(x-y)>ε-2ρ(z)ρ(x-y)≤tYm,sx-y′Jx-y′ρx-yα-1×fydy∫ρ(x-y)>ε-2ρ(z)ρ(x-y)≤tYm,sx-y′Jx-y′ρx-yα-12dtt31/2=bx-bx+zgm,sε-2ρzfx.
Similar to the estimate of S21/ɛ|v|ρf(x) in [35], we get for any fixed 1<q<p<∞,
(62)gm,sε-2ρzfx≤CMgm,sfx+CMfqx1/q,
where C is independent of m,s,ɛ and ρ(z). So, for any 1<p<∞, taking 1<q<p in (62), then by (43) and the Lp boundedness of (M(|f|q))1/q, we get
(63)gm,sε-2ρ(z)fp≤Cfp.
Since b∈C0∞, we have |b(x)-b(x+z)|≤Cρ(z). Then by (63), we get
(64)∫0∞‍J4x,t2dtt31/2p≤Cmn/2ρ(z)fp.
For J5, since Jx′≥1, and |b(x)-b(y)|≤Cρ(x-y) when ρ(x-y)≤1, then by (21) and the Minkowski inequality, we get
(65)∫0∞‍J5x,t2dtt31/2≤Cmn/2-1∫ρ(x-y)≤ε-2ρ(z)‍fyρx-yα-1dy.
Then we can get
(66)∫0∞‍J5(x,t)2dtt31/2p≤Cmn/2-1ε-2ρz.
Similarly, we can get
(67)∫0∞‍J6(x,t)2dtt31/2p≤CDmn/2-1ε-2ρz+ρz.
Combining the estimate of J1 with J6, by the Minkowski inequality and Dm≃Cm(n-2), we get
(68)K1p≤C∑m=1∞‍m-3n/2∑s=1Dm‍∫0∞‍Im,sx,t2dtt31/2p≤C∑m=1∞‍m-3n/2∑s=1Dm‍mn/2-1ε-2ρz+ρz+ɛfp≤C∑m=1∞‍m-3n/2mn-2+n/2-1ε-2ρz+ρz+ɛfp≤C∑m=1∞‍m-3ε-2ρz+ρz+ɛfp≤Cε-2ρz+ρz+ɛfp,
which combined the Lp norm of K2 (see (53)); then by taking ρ(z) sufficiently small depending on ɛ, we can get
(69)gΩ;bf·-gΩ;bf·+zp≤Cɛ.
Thus we establish the proof of Theorem 2.


## 4.  Proof of Theorem3

In this proof forj=1,2,…,13,Bj is a positive constant depending only on Ω, p, n, δ, Θ and Bi (1≤i<j). Suppose that gΩ;b is a bounded operator from Lp(Rn) to Lp(Rn); we are going to prove that b∈BMO. Without loss of generality, we can assume gΩ;bLp↦Lp=1. To prove that b∈BMO(Rn,ρ), we need to show there exists a constant B>0 such that for any x0∈Rn and r>0,
(70)N=1Ex0,r∫E(x0,r)‍by-bEx0,rdy≤Bp,Ω,Θ,δ,n,
where bE(x0,r)=(1/|E(x0,r)|)∫E(x0,r)‍b(y)dy. Since gΩ;b-bE(x0,r)=gΩ;b, we may assume bE(x0,r)=0. So it suffices to prove that for any fixed x0∈Rn,r∈R+,
(71)N=1Ex0,r∫E(x0,r)‍bydy≤Bp,Ω,Θ,δ,n,
where (1/Ex0,r)∫E(x0,r)‍b(y)dy=0. Let
(72)fy=sgn⁡by-c0χEx0,ry,
where c0=|E(x0,r)|-1∫E(x0,r)‍sgn⁡(b(y))dy. We get |c0|<1. Then f satisfies the following properties:
(73)f∞≤2,(74)supp⁡f⊂Ex0,r,(75)∫Rn‍fydy=0,(76)fyby>0,(77)1Ex0,r∫Rn‍fybydy=N.
Then for x∈G={x∈Rn:ρ(x-x0)>B1r=(3δ-1+1)r and x-x0′∈Θ},
(78)gΩ;bfx≥gΩbfx-bxgΩfx=ρx-y1-αbyfydy∫ρx-y≤t‍Ωx,x-y′2dtt3∫0∞‍∫ρx-y≤t‍Ωx,x-y′×ρx-y1-αbyfydy∫ρx-y≤t‍Ωx,x-y′2dtt31/2-bx×ρx-y1-αfydy∫ρx-y≤t‍2dtt3∫0∞‍∫ρx-y≤t‍Ωx,x-y′×ρx-y1-αfydy∫ρx-y≤t‍2dtt31/2∶=I1-I2.
For I1, by (18), we know
(79)Ωx,x-x0′≥2δ.
Noting that if ρ(y-x0)<r and ρ(x-x0)>B1r, by Lemma 9, we get ρx-y′-x-x0′≤δ. Since
(80)sup⁡x∈RnΩx,x-y′-Ωx,x-x0′≤ρx-y′-x-x0′≤δ,
if Ω(x,x-x0′)≥2δ, we can get Ω(x,x-y′)≥δ. Then by (74), (76), (77), (3/4)ρ(x-x0)≤ρ(x-y)≤(5/4)ρ(x-x0) and the Hölder inequality, we get
(81)I1≥δχ{ρ(x-y)≤t}dy∫Ex0,r‍2dtt3∫ρ(x-x0)∞‍∫Ex0,r‍byfyρx-y1-α×χ{ρ(x-y)≤t}dy∫Ex0,r‍2dtt31/2≥δ∫ρ(x-x0)∞‍∫E(x0,r)‍byfyρx-y1-αχ{ρ(x-y)≤t}dydtt3×∫ρ(x-x0)∞‍dtt3-1/2=Cδρx-x0∫Ex0,r‍ρx-y1-αbyfy×∫ρ(x-x0)≤tρ(x-y)≤tdtt3dy≥B2δρx-x0-α∫Ex0,r‍byfydy=B3Nrαρx-x0-α.
By (73), (74), (75), ρ(x-x0)≃ρ(x-y),ρ(y-x0)<r and Ω∈L∞(Rn)×L∞(Sn-1), then by Lemma 13 and Lemma 8, we have
(82)I2=bx×fydyΩx,x-yρx-yα-1χ{ρ(x-y)≤t}2dtt3∫0∞‍∫Rn‍Ωx,x-yρx-yα-1χ{ρ(x-y)≤t}-Ωx,x-x0ρx-x0α-1χ{ρ(x-x0)≤t}×fydyΩx,x-yρx-yα-1χ{ρ(x-y)≤t}2dtt31/2≤bx∫Ex0,r‍fyΩx,x-yρx-yα-1-Ωx,x-x0ρx-x0α-1×∫ρ(x-y)≤tρ(x-x0)≤tdtt31/2dy+bx∫Ex0,r‍Ωx,x-yρx-yα-1fy×∫ρ(x-y)≤tρ(x-x0)≥tdtt31/2dy+bx∫Ex0,r‍Ωx,x-x0ρx-x0α-1fy×∫ρ(x-y)≥tρ(x-x0)≤tdtt31/2dy≤Cbxr∫E(x0,r)‍fyρx-x0α+1dy+Cbxr1/2∫E(x0,r)‍fyρx-x0α+1/2dy≤B4bxrα+1/2ρx-x0-α-1/2.
Then using the same argument of the proof of Theorem 1.2 in [29], we get N≤B(Ω,n,δ,Θ,p).


## 5. Proof of Theorem4

SincegΩ;b is a compact operator on Lp(Rn), then gΩ;b is bounded on Lp(Rn). Thus from Theorem 3, b∈BMO(Rn,ρ). Thus we may assume ∥b∥*,ρ=1. By Lemma 12, to prove that b∈CMO(Rn,ρ), it suffices to show that b satisfies conditions (i), (ii), and (iii) of Lemma 12.First, suppose thatb does not satisfy (i) of Lemma 12. Then there exists ζ>0 and a sequence of ellipsoids {Ej(yj,rj)}j=1∞ with lim⁡j→∞rj=0, such that for every j(83)Mb,Ej=Ej-1∫Ej‍by-bEjdy>ζ.
Let fj(y)=|Ej|-1/p{[sgn⁡(b(y)-bEj)-c0]χEj(y)}, where c0=|Ej|-1∫Ej‍sgn⁡(b(y)-bEj)dy. So |c0|<1. Then fj has the following properties:
(84)supp⁡fj⊂Ej;(85)fjyby-bEj>0;(86)∫Rn‍fjydy=0;(87)fjy≤2Ej-1/p,
for y∈Ej. It is easy to see that ∥fj∥p≤C, where C is independent of j.LetB1=(3/δ)+1. If ρ(y-yj)<rj, then
(88)x∈B1Ejc∩x:x-yj′∈Θ.
Similar to the proof of I1 in Theorem 3, by (84), (85) and noting that ρ(x-yj)≃ρ(x-y), we get
(89)gΩb-bEjfjx≥δ∫0∞‍∫Ej‍by-bEjfjyρx-yα-1χ{ρ(x-y)≤t}dy2dtt31/2≥δ∫ρ(x-yj)∞‍∫Ej‍by-bEjfjyρx-yα-1χ{ρ(x-y)≤t}dydtt3×∫ρ(x-yj)∞‍dtt3-1/2≥δEj-1/pρx-yj×∫ρ(x-yj)∞‍∫Ej‍ρx-yα-1-1by-bEj×sgn⁡by-bEj-c0χEjy×ρx-yα-1-1by-bEj×χ{ρ(x-y)≤t}dydtt3≥δEj-1/pρx-yj-α×∫Ej‍by-bEj-c0by-bEjdy=δEj-1/pρx-yj-α∫Ej‍by-bEjdy≥B2Ej-1/p′ρx-yj-α.
On the other hand, for x∈(B1Ej)c, by (84), (87), Ω∈L∞(Rn)×L∞(Sn-1), and ρ(x-yj)≃ρ(x-y) when y∈Ej, we have
(90)gΩb-bEjfjx≤∫Ej‍by-bEjfjyΩx,x-yρx-yα-1×∫ρ(x-y)≤t‍dtt31/2dy≤CEj1/p′ρx-yj-α.
On the other hand, for x∈(B1Ej)c, by (84), (86), (87), and ρ(x-yj)≃ρ(x-y), we have
(91)bx-bEjgΩfjx≤bx-bEj×fj(y)dyΩx,x-yρx-yα-1-Ωx,x-yjρx-yjα-12dtt3∫0∞‍∫ρ(x-y)≤tρ(x-yj)≤tΩx,x-yρx-yα-1-Ωx,x-yjρx-yjα-1×fj(y)dyΩx,x-yρx-yα-1-Ωx,x-yjρx-yjα-12dtt31/2+bx-bEj×∫0∞‍∫ρ(x-y)≤tρ(x-yj)>tΩ(x,x-y)ρ(x-y)α-1fj(y)dy2dtt31/2+bx-bEj×∫0∞∫ρ(x-y)>tρ(x-yj)≤tΩx,x-yjρx-yjα-1fjydy2dtt3‍1/2≤CEj1/p′bx-bEjrj1/2ρx-yjα+1/2.
In fact, in the above estimate we have used Lemma 13 for x∈(B1Ej)c. Then similar to the proof of Theorem 1.3 in [29], we show that b satisfies (i)–(iii) of Lemma 12.



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*Source: 693534-2014-11-12.xml*