# The Higher Order Riesz Transform andBMO Type Space Associated with Schrödinger Operators on Stratified Lie Groups

Authors: Yu Liu; Jianfeng Dong
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Assume thatG is a stratified Lie group and Q is the homogeneous dimension of G. Let -Δ be the sub-Laplacian on G and W≢0 a nonnegative potential belonging to certain reverse Hölder class Bs for s≥Q/2. Let L=-Δ+W be a Schrödinger operator on the stratified Lie group G. In this paper, we prove the boundedness of some integral operators related to L, such as L-1∇2, L-1W, and L-1(-Δ) on the space BMOL(G).


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## Body

## 1. Introduction

In recent years, some problems related to Schrödinger operators on the Euclidean spaceℝn with nonnegative potentials have been investigated by a number of scholars (cf. [1–12], etc.). Later, more scholars want to generalize the above results related to Schrödinger operators to a more general setting, such as Heisenberg group, nilpotent Lie groups, and spaces of homogeneous type (cf. [13–24], etc.). The auxiliary function plays an important role in the Harmonic analysis problems related to Schrödinger operators. Recently, Yang et al. introduced the admissible function. It is known that the auxiliary function is a special case of the admissible function. Accordingly, they investigated function spaces, such as BMO, BLO, and Hardy space, related to the admissible function in [22, 24]. Among the above problems, Riesz transforms and higher order Riesz transforms related to Schrödinger operators are one of hottest issues. Their Lp boundedness has been obtained by Shen [13] and Li [4] in the different settings. Dziubański and Zienkiewicz proved that Riesz transforms related to Schrödinger operators are bounded from Hardy spaces associated with Schrödinger operators into L1 in [1]. Endpoint boundedness of Riesz transforms related to Schrödinger operators had been investigated in [11, 25]. Dong and Liu established the BMO spaces associated with Schrödinger operators for the Riesz transform related to Schrödinger operators in [26]. Lin et al. obtained the corresponding results on the Heisenberg group in [14, 15]. Just now, Dong and Liu established the BMO estimates for the higher order Riesz transform in [27]. The aim of this paper is to obtain the BMO estimates for the higher order transform on stratified Lie groups.Firstly, we recall some basic facts of stratified Lie groups (cf. [28]). A Lie group G is called stratified if it is nilpotent, connected, and simple connected, and its Lie algebra 𝔤 admits a vector space decomposition 𝔤=V1⊕⋯⊕Vm such that [V1,Vk]=Vk+1 for 1≤k<m and [V1,Vm]=0. If G is stratified, its Lie algebra admits a family of dilations, namely,
(1)δr(X1+X2+⋯+Xm)=rX1+r2X2+⋯+rmXm×(Xj∈Vj,j∈{1,…,m}).
Assume that G is a Lie group with underlying manifold ℝn for some positive integer n. G inherits dilations from 𝔤: if x∈G and r>0, we write
(2)δrx=(rd1x1,…,rdnxn),
where 1≤d1≤⋯≤dn. The map x→δrx is an automorphism of G. The left (or right) Haar measure on G is simply dx1⋯dxn, which is the Lebesgue measure on 𝔤. For any measurable set E⊆G, denote by |E| the measure of E. The inverse of any x∈G is simply x-1=-x. The group law has the following form:
(3)xy=(p1(x,y),…,pn(x,y))
for some polynomials p1,…,pn in x1,…,xn,y1,…,yn.The numberQ=∑j=1m‍j(dimVj) is called the homogeneous dimension of G. We fix a homogeneous norm function |·| on G, which is smooth away from e, where e is the unit element of G. Thus, |δrx|=r|x| for all x∈G,r>0,|x-1|=|x| for all x∈G, and |x|>0 if x≠0. The homogeneous norm induces a quasi-metric d which is defined by d(x,y):=|x-1y|. In particular, d(e,x)=|x| and d(x,y)=d(e,x-1y). The ball of radius r centered at x is written by
(4)B(x,r)={y∈G∣d(x,y)<r}.
The measure of B(x,r) is
(5)|B(x,r)|=brQ,
where b is a constant. In particular set V(r)=|B(x,r)| for r>0 and x∈G.LetX={X1,…,Xl} be a basis for V1 (viewed as left-invariant vector fields on G). Following [29], one can define a left invariant metric dc associated with X which is called the Carnot-Caratheodory metric: let x,y∈G, and for every δ>0 define
(6)A(δ)={∑i=1l‍γ:[0,δ]⟶G∣γ(0)=x,γ(δ)=y,γ′(s)=∑i=1l‍ai(s)Xi(γ(s))a.e.with∑i=1l‍|ai|≤1}.
Let us define
(7)dc(x,y)=infA(δ)not  empty{δ}.The Carnot-Caratheodory metricdc is equivalent to the quasi-metric d. From the results of Nagel et al. in [29], we deduce that there exists a constant a=a(G)>1 such that, for any g,h∈G,
(8)a-1dc(x,y)≤d(x,y)≤adc(x,y).It follows from [28] that Xj, j=1,2,…,l, are skew adjoint; that is, Xj*=-Xj. Let Δ=-∑i=1l‍Xi2 be the sub-Laplacian on G. This operator (which is hypoelliptic by Hörmander’s theorem in [30]) plays the same fundamental role on G as the ordinary does on ℝn. The gradient operator ∇ is denoted by ∇=(X1,…,Xl).Definition 1.
A nonnegative locallyLs integrable function W on G is said to belong to the reverse Hölder class Bs(1<s<∞) if there exists C>0 such that the reverse Hölder inequality
(9)(1|B(x,r)|∫B(x,r)‍W(y)sdy)1/s≤C|B(x,r)|∫B(x,r)‍W(y)dy
holds for every ball B(x,r) in G.
Moreover, a locally bounded nonnegative functionW∈B∞ if there exists a positive constant C such that
(10)∥W∥L∞(B(x,r))≤C|B(x,r)|∫B(x,r)‍W(y)dy
holds for every ball B(x,r) in G.Furthermore, it is easy to see thatB∞⊆Bs for any 1<s<∞.LetL=-Δ+W be a Schrödinger operator on the stratified Lie group G, where W≢0 is a nonnegative potential belonging to the reverse Hölder class Bs for some s≥Q/2. Denote by ℛL=∇2L-1 the higher order Riesz transform. Accordingly, denote by ℛ~L=L-1∇2 its dual operator.It follows from [13] that the integral operators WL-1 and (-Δ)L-1 are bounded on Lp(G) for 1≤p≤s and ℛL is bounded on Lp(G) for 1<p≤s. Lin et al. introduced the Hardy type space HL1(G) related to the Schrödinger operator L on the Heisenberg group G in [14]. The dual space of HL1(G) is the BMO type space BMOL(G) investigated by Lin and Liu in [15]. HL1(G) and BMOL(G) were also introduced as applications of results in [11, 22].Next, we recall the definition ofHL1(G) and BMOL(G). Since W≥0 and W∈Llocs(G), the Schrödinger operator L=-Δ+W generates a (C0) semigroup {TtL}t>0={e-tL}t>0. The maximal function with respect to the semigroup {TtL}t>0 is given by
(11)MLf(x)=supt>0|TtLf(x)|.
The Hardy space HL1(G) associated with the Schrödinger operator L is defined as follows in terms of the maximal function mentioned above.Definition 2.
A functionf∈L1(G) is said to be in HL1(G) if the semigroup maximal function MLf belongs to L1(G). The norm of such a function is defined by
(12)∥f∥HL1=∥MLf∥L1.AssumeW∈Bs for s>Q/2. The auxiliary function m(x,W) is defined by
(13)ρ(x)=1m(x,W)≐supr>0{r:1rQ-2∫B(x,r)‍W(y)dy≤1},x∈G.
It follows from Lemma 9 in Section 2 that 0<m(x,W)<∞ for any x∈G.The dual space ofHL1(G) is the BMO type space BMOL(G) (cf. [22]). Let f be a locally integrable function on G and B=B(x,r) be a ball. Set
(14)fB=1|B(x,r)|∫B(x,r)‍f(y)dy,f(B,W)={fB,ifr<ρ(x),0,ifr≥ρ(x).Definition 3.
Letf be a locally integrable function on G. One says f∈BMOL(G) if
(15)∥f∥BMOL=supB1|B|∫B‍|f(y)-f(B,W)|dy<∞.It is clear thatL∞(G)⊂BMOL(G)⊂BMO(G) and ∥f∥BMO≤2∥f∥BMOL. Some remarks are given as follows.Remark 4.
Let1≤p<∞. If f∈BMOL(G), then there exists a positive constant C:
(16)supB(1|B|∫B|f(y)-f(B,W)|pdy‍)1/p≤C∥f∥BMOL.
The above inequality can be easily deduced by Lemma  3.1 in [11].
Similar to Remark  1 in [15], we conclude that a function f∈BMOL(G) if and only if there exist some suitable constants CB and C depending on B=B(x,r) and satisfying CB=0 whenever r≥ρ(x) such that
(17)supB(1|B|∫B‍|f(y)-CB|pdy)1/p<∞,∥f∥BMOL≤CsupB(1|B|∫B‍|f(y)-CB|pdy)1/p.Our main results are given as follows.Theorem 5.
SupposeW∈Bs for some s≥Q/2. Then the operators L-1W and L-1(-Δ) are bounded on the space BMOL(G).Theorem 6.
SupposeW∈Bs for some s≥Q, |∇W|∈Bs1 for some s1≥Q/2, and ρ(x)≲ρ1(x)=1/m(x,|∇W|) and ρ1(x)≤M for some positive constant M. Then operator L-1∇2 is bounded on the space BMOL(G).It shoud be noted that because the left invariant vector fields inV1⊆𝔤 are skew-adjoint and they interact with convolution (see (41) for the details), we generalized the main results in [27] to the stratified Lie groups instead of nilpotent Lie groups.This paper is organized as follows. In Section2, we collect some known facts about the auxiliary function m(x,W). Section 3 gives some estimates of kernel for some operators in this paper. Section 4 gives the proof of the boundedness of L-1W,L-1(-Δ) on the space BMOL(G). In Section 5, we establish the BMOL boundedness of L-1∇2. Finally, we give some examples for the potentials which satisfy the assumptions in Theorem 6 in different settings.Throughout this paper, we will useC to denote the positive constant, which is not necessarily the same at each occurrence and may depend on the dimension Q, and the constant in (9). By A~B and A≲B, we mean that there exist some constants C,C′ such that 1/C≤A/B≤C and A≤C′B, respectively.


## 2. Some Lemmas about the Auxiliary Function

In this section, we collect some known results about auxiliary functionm(x,W). We refer to [13] for the details. Throughout this section, unless otherwise indicated, we always assume that 0≢W∈Bs for some s>Q/2.Lemma 7.
W
∈
B
s is a doubling measure; that is, there exists a constant C>0 such that
(18)∫B(x,2r)‍W(y)dy≤C∫B(x,r)‍W(y)dy.Lemma 8.
There exist constantsC,k0>0 such that
(19)1C(1+d(x,y)m(x,W))-k0≤m(x,W)m(y,W)≤C(1+d(x,y)m(x,W))k0/(k0+1).
In particular, m(x,W)~m(y,W) if d(x,y)≤C/m(x,W).Lemma 9.
There existsC>0 such that, for 0<r<R<∞,
(20)1rQ-2∫B(x,r)‍W(y)dy≤C(rR)2-(Q/s)1RQ-2∫B(x,R)‍W(y)dy.Lemma 10.
Ifr=1/m(x,W), then
(21)1rQ-2∫B(x,r)‍W(y)dy=1.
Moreover,
(22)1rQ-2∫B(x,r)‍W(y)dy~1iffr~1m(x,W).Lemma 11.
There existC>0 and l1>0 such that
(23)∫B(x,r)‍W(y)d(x,y)Q-2dy≤CrQ-2∫B(x,r)‍W(y)dy≤C(1+rm(x,W))l1.
Moreover, if W∈BQ, then there exists C>0 such that
(24)∫B(x,r)‍W(y)dyd(x,y)Q-1≤C1rQ-1∫B(x,r)‍W(y)dy.


## 3. Estimates for the Kernels

In this section we will investigate some necessary estimates about the kernel of the operators in the paper.Letpt(x,y)=ht(y-1x) be the heat kernel of the semigroup e-tΔ,  t≥0, associated with -Δ. Via Theorem 4.2 of [31], the following estimates hold true; that is, there exist positive constants C and C′ such that
(25)|∇2ht(x)|≤Ct-1V(t)-1exp{-C′d(e,x)t},∀x∈G,∀t>0,|∇3ht(x)|≤Ct-3/2V(t)-1exp{-C′d(e,x)t},∀x∈G,∀t>0,
where e is the unit element of G. Moreover, for any 1≤i,j,k≤m and x,y∈G, by using (3.5) in [13] we obtain
(26)|Xi(x)Xj(x)Xk(y)pt(x,y)|≤Ct-3/2V-1(t)exp{-C′d2(x,y)t}.
Let
(27)Htf(x)≐∫G‍f(y)ht(y-1x)dy,∀t>0.
Then for λ≥0,
(28)(λ-Δ)-1f(x)=∫0∞‍e-λtHtf(x)dt.LetΓ0(x,y,λ) be the fundamental solution of the operator -Δ+λ for λ∈[0,∞). In particular, we denote by Γ0(x,y,0)=Γ0(x,y). Then we have the following.Proposition 12.
There exists a positive constantCk such that
(29)|Γ0(x,y,λ)|≤Ck(1+λ1/2d(x,y))k1dQ-2(x,y),(30)|∇xΓ0(x,y,λ)|≤Ck(1+λ1/2d(x,y))k1dQ-1(x,y),(31)|∇x2Γ0(x,y,λ)|≤Ck(1+λ1/2d(x,y))k1dQ(x,y),(32)|∇x2∇yΓ0(x,y,λ)|≤Ck(1+λ1/2d(x,y))k1dQ+1(x,y),(33)|∇x3Γ0(x,y,λ)|≤Ck(1+λ1/2d(x,y))k1dQ+2(x,y)
for x≠y.Proof.
Equations (29) and (30) have been proved by Li in [13]. We only need to show that (32) holds true, because (31) and (33) can be proved similarly.
By (26) and (28),
(34)|Xi2XjyΓ0(x,y,λ)|=|∫0∞‍e-λtXi2Xjypt(x,y)dt|≤|∫0d2(x,y)‍e-λtXi2Xjypt(x,y)dt|+|∫d2(x,y)∞‍e-λtXi2Xjypt(x,y)dt|≐I1+I2.
Firstly, for I1, we have
(35)I1≤C|∫0d2(x,y)‍e-λtt-3/2V-1(t)exp{-C′d2(x,y)t}dt|≤C|∫0d2(x,y)‍e-λtt-3/2(1t)Qexp{-C′d2(x,y)t}dt|≤C|∫0d2(x,y)‍e-λtt-3/2(1t)Q(td(x,y))Q+2dt|≤CdQ+1(x,y).
In addition, for any positive integer k,
(36)I1≤C|∫0d2(x,y)‍e-λtt-3/2V-1(t)exp{-C′d2(x,y)t}dt|≤C|∫0d2(x,y)‍e-λtt-3/2(1t)Qexp{-C′d2(x,y)t}dt|≤C|∫0d2(x,y)‍e-λtt-3/2(1t)Q(td(x,y))Q+k+1dt|≤C|∫0d2(x,y)‍e-λtt(k/2)-1dt|1dQ+k+1(x,y)=C|∫0λd2(x,y)‍e-tt(k/2)-1dt|1(λ1/2d(x,y))kdQ+1(x,y)≤CkdQ+1(x,y)(λ1/2d(x,y))k.
Therefore,
(37)I1≤Ck(1+λ1/2d(x,y))k1dQ+1(x,y).
Secondly, we have
(38)I2≤C|∫d2(x,y)∞‍e-λtt-3/2V-1(t)exp{-C′d2(x,y)t}dt|≤C|∫d2(x,y)∞‍e-λtt-3/2dt|V-1(d(x,y))≤CCk|∫d2(x,y)∞‍t-3/21(1+λ1/2t1/2)kdt|V-1(d(x,y))≤CCk|∫d2(x,y)∞‍t-3/2dt|1(1+λ1/2d(x,y))kV-1(d(x,y))≤CCk1(1+λ1/2d(x,y))k1dQ+1(x,y).
Therefore, (32) holds true.Moreover, we need some other basic facts of fundamental solutions for sub-Laplacian on the stratified Lie groupG (see [32]).In the first place, we use the standard notations𝒟, ℰ, and 𝒟′ for the spaces of C∞ functions with compact support, C∞ functions, and distributions on 𝒟.A measurable functionf on G will be called homogeneous of degree λ(λ∈ℂ) if f∘δr=rλf for all r>0. Likewise, a distribution τ∈𝒟′ will be called homogeneous of degree λ if 〈τ,u∘δr〉=r-Q-λ〈τ,u〉 for all u∈𝒟 and r>0. A distribution which is C∞ away from e and homogeneous of degree α-Q will be called a kernel of type α.A differential operatorD will be called homogeneous of degree λ if D(u∘δr)=rλ(Du)∘δr for all u∈𝒟 and r>0. Since G is stratified, X∈𝔤 is homogeneous of degree j if and only if X∈Vj. In particular, sub-Laplacian Δ is homogeneous of degree 2. It follows from [32] that if K is a kernel of type α and D is homogeneous of degree λ, then DK is a kernel of type α-λ.For sub-LaplacianΔ, XiXjΓ0 is homogeneous of -Q due to the fact that Γ0 is homogeneous of 2-Q, where Xi,Xj∈V1. In addition, by using Proposition  1.7 in [28], we have
(39)|Γ0(xy)-Γ0(x)|≤C|y||x|Q+1whenever|y|≤|x|2.By [28], the left-invariant fields X∈V1 are formally skew-adjoint; that is,
(40)∫G‍Xf(y)g(y)dy=-∫G‍f(y)Xg(y)dy.
Moreover, X interacts with convolution in the following way:
(41)X(f*g)=f*Xg.LetΓ(x,y,λ) be the fundamental solution of L+λ for λ∈[0,∞). Then, Theorem  3.6 in [13] implies that
(42)|Γ(x,y,λ)|≤1(1+λ1/2d(x,y))k×Ck(1+m(x,W)d(x,y))k1dQ-2(x,y).
In particular, Γ(x,y)=Γ(x,y,0)=Γ(y,x,0) is the fundamental solution of Schrödinger operator L=-Δ+W, which satisfies the following.(i)
For eachk∈ℕ there exists Ck>0 such that
(43)|Γ(x,y)|≤Ck(1+m(x,W)d(x,y))k1dQ-2(x,y).(ii)
IfW∈BQ, then for each k∈ℕ there exists Ck>0 such that
(44)|∇Γ(x,y)|≤Ck(1+m(x,W)d(x,y))k1dQ-1(x,y),
where the above estimate can be deduced by Lemma  5.1 in [13].The operator∇2(-Δ+W)-1 is defined by
(45)∇2(-Δ+W)-1f(x)=∫G‍K(x,y)f(y)dy,
where the kernel K(x,y)=∇x2Γ(x,y). Also, its adjoint operator (-Δ+W)-1∇2 is defined by
(46)(-Δ+W)-1∇2f(x)=∫G‍K~(x,y)f(y)dy,
where the kernel K~(x,y)=∇y2Γ(x,y). Since Γ(x,y)=Γ(y,x), then ∇y2Γ(x,y)=∇y2Γ(y,x).Moreover, we also need other estimates for the kernelK(x,y) and K~(x,y) in order to prove the main results.Lemma 13.
AssumeW∈Bs for s>Q. Let x0,y0∈G and R=(1/2)d(x0,y0). Then for any k∈ℕ there exists Ck>0 such that
(47)∥∇2Γ(x,y0)∥Ls(B(x0,R))≤CkV(R)1/s(1+Rm(x0,W))(k/(k0+1))-l11V(R),
where k0 is the constant appearing in Lemma 8 and l1 is the constant appearing in Lemma 11.Proof.
Letu be the solution of -Δ+W=0 in the ball B(x0,2R). By Lemma  3.2 in [13], we choose η∈C0∞(B(x0,2R)) such that 0≤η≤1,η=1 on B(x0,R/C2), |∇η|≤C*/R, and |∇2η|≤C*/R2, where C≥1 and C* are fixed constants, which are independent of x0 and R.
Forx∈B(x0,R/2C2),
(48)u(x)η(x)=∫G‍Γ0(x,z,λ)(-Δ+λ)(uη)(z)dz=∫G‍Γ0(x,z,λ)(-Wuη-2∇u·∇η-uΔη)dz=∫G‍Γ0(x,z,λ)(-Wuη+uΔη)dz+2∫G‍[∇zΓ0(x,z,λ)·∇η]udz.
By (31) and (32) we have
(49)|∇x2u(x)|=|∫G‍∇x2Γ0(x,z,λ)Wuηdz|+|∫G‍∇x2Γ0(x,z,λ)uΔηdz|+2|∫G‍[∇x2∇zΓ0(x,z,λ)·∇η]udz|≤|∫G‍∇x2Γ0(x,z,λ)Wuηdz|+∫G‍|∇x2Γ0(x,z,λ)||uΔη|dz+2∫G‍|∇x2∇zΓ0(x,z,λ)||∇η||u|dz≤|∫B(x0,2R)‍∇x2Γ0(x,z,λ)Wuηdz|+2∫B(x0,2R)‍|∇x2∇zΓ0(x,z,λ)||∇η||u|dz≤|∫B(x0,2R)‍∇x2Γ0(x,z,λ)Wuηdz|+CR2V(R)∫B(x0,2R)‍|u(z)|dz≤∫B(x0,2R)‍|∇x2Γ0(x,z,λ)W||u|dz+CR2V(R)∫B(x0,2R)‍|u(z)|dz.
By (9), the Calderón-Zygmund estimates, and Lemma 11,
(50)∥∇2u∥Ls(B(x0,R))≤C∥Wuη∥s+CV(R)1/sR-2supB(x0,2R)|u|≤CV(R)1/s×{(1V(R)∫B(x0,R)‍Ws(z)dz)1/s+R-2}supB(x0,2R)|u|≤CV(R)1/s{(1V(R)∫B(x0,R)‍W(z)dz)+R-2}supB(x0,2R)|u|≤CV(R)1/sR-2{(R2V(R)∫B(x0,R)‍W(z)dz)+1}supB(x0,2R)|u|≤CV(R)1/sR-2(1+Rm(x0,W))l1·supB(x0,2R)|u|.
Let u(x)=Γ(x,y0) and R=(1/2)d(x0,y0). Then u(x)=Γ(x,y0) is a solution of -Δ+W=0 in B(x0,R). By the above inequality and Lemma 8, we immediately have
(51)∥∇2Γ(x,y0)∥Ls(B(x0,R))≤CV(R)1/sR-2(1+Rm(x0,W))l1·supB(x0,2r)|u|≤CkV(R)1/s(1+Rm(x0,W))(k/(k0+1))-l11V(R).
This finishes the proof of Lemma13.Lemma 14.
SupposeW∈Bs for some s>Q and |∇W|∈Bs1 for some s1>Q/2. Then
(52)|∇2Γ(x,y)-∇2Γ0(x,y)|≤CdQ-2(x,y)∫B(x,d(x,y))‍|∇W(z)|dzdQ-1(x,z)+CdQ-1(x,y)(d(x,y)m(y,|∇W|))δ1+CdQ(x,y)(d(x,y)m(y,W))δ,
where δ1=2-Q/s1 and δ=2-Q/s, if d(x,y)≤A/m(x,|∇W|) for some positive constant A.Proof.
Note that(-Δ+W)Γ(x,y)=0. Let Γ0(x,y) be the fundamental solution of -Δ. Then, we have
(53)-Δ(Γ(x,y)-Γ0(x,y))=-W(x)Γ(x,y).
It follows that
(54)Γ(x,y)-Γ0(x,y)=(-Δ)-1(-W(x)Γ(x,y)).
By (41), we have
(55)∇Γ(x,y)-∇Γ0(x,y)=-∫G‍∇Γ0(x,z)W(z)Γ(z,y)dz=-∫G‍Γ0(x,z)∇(W(z)Γ(z,y))dz=-∫G‍Γ0(x,z)[∇W(z)Γ(z,y)+W(z)∇Γ(z,y)]dz.
Set R=(1/8)d(x,y). Thus, we have
(56)|∇2Γ(x,y)-∇2Γ0(x,y)|≤∫G‍|∇Γ0(x,z)||∇W(z)||Γ(z,y)|dz+∫G‍|∇Γ0(x,z)||W(z)||∇Γ(z,y)|dz≐I1+I2.
ForI1,
(57)I1≤(∫d(x,z)<R‍+∫d(y,z)<R‍+∫d(x,z)≥R,d(y,z)≥R‍)=I11+I12+I13.
Firstly, by (43) and (30), it holds that
(58)I11=∫d(x,z)<R‍|∇Γ0(x,z)||∇W(z)||Γ(z,y)|dz≤CdQ-2(x,y)∫B(x,R)‍|∇W(z)|dQ-1(x,z)dz.
Secondly, via Lemmas 9 and 11, we similarly have
(59)I12=∫d(y,z)<R‍|∇Γ0(x,z)|·|∇W(z)|·|Γ(z,y)|dz≤CdQ-1(x,y)∫B(y,R)‍|∇W(z)|dQ-2(y,z)dz≤CdQ-1(x,y)1dQ-2(x,y)∫B(y,R)‍|∇W(z)|dz≤CdQ-1(x,y)(d(x,y)m(x,|∇W|))δ1,
where δ1=2-(Q/s1).
Now, we turn to estimatingI13. By Lemma 7, (43), and (30), we have that
(60)I13=∫d(x,z)≥R,d(y,z)≥R‍|∇Γ0(x,z)|·|∇W(z)|·|Γ(z,y)|dz≤CdQ-1(x,y)×∑j=1∞‍∫2j-1R≤d(y,z)<2jR|∇W(z)|‍00000000×(dQ-2(y,z)(1+d(y,z)m(y,W))k)-1dz≤CdQ-1(x,y)×∑j=1∞‍Cj2kj1(2jcR)Q-2∫d(y,z)≤R‍|∇W(z)|dz≤CdQ-1(x,y)×∑j=1∞‍(C2k+n-2)j1RQ-2∫d(y,z)≤R‍|∇W(z)|dz≤CdQ-1(x,y)(d(x,y)m(y,|∇W|))δ1
if we choose k large enough.
Then,(61)I1≤CdQ-2(x,y)∫B(x,R)‍|∇W(z)|dQ-1(x,z)dz+CdQ-1(x,y)(d(x,y)m(y,|∇W|))δ1.
A similar argument implies that(62)I2≤CdQ(x,y)(d(x,y)m(y,W))δ.
The proof is completed.Lemma 15.
SupposeW∈Bs for some s≥Q/2. Let s′ be the conjugate index of s. (1)
L-1(-Δ) and L-1W are bounded on the space Lp(G), where s′≤p≤∞.(2)
L-1∇2 is bounded on the space Lp(G) for s′≤p<∞.The above lemmas hold true due to Theorem 4.1 and TheoremA in [13], respectively.


## 4.The Boundedness ofL-1WandL-1(-Δ)onBMOL(G)

Proof of Theorem5.
To prove Theorem5, we adopt the method used in the proof of Theorem  1.6 in [27].
Supposef∈BMOL(G) and B=B(x0,r). Firstly, we suppose r≥ρ(x0). Set B*=B(x0,2r). Then, we have
(63)f=fχB*+fχ(B*)c=f1+f2,
where χA denotes the characteristic function of the set A. Since L-1W is bounded on L2(G), by Remark 4, we have
(64)1|B|∫B‍|(L-1W)f1(x)|dx≤(1|B|∫B‍|(L-1W)f1(x)|2dx)1/2≤(C|B|∫B*‍|f(x)|2dx)1/2≤C∥f∥BMOL.
Letx∈B. By Lemma 8, 1/m(x,W)≤Cr. Set Tj=B(x0,2jr). By using (43) and Lemma 11, we obtain
(65)|(L-1W)f2(x)|≤∫(B*)c‍|Γ(x,y)|W(y)|f(y)|dy≤m(x0,W)-k∑j=0∞‍Ck(2jr)Q-2+k×∫Tj+1∖Tj‍W(y)|f(y)|dy≤m(x0,W)-k×∑j=0∞‍Ck(2jr)k-2(1|Tj+1|∫Tj+1‍W(y)sdy)1/s∥f∥BMOL≤Ck∑j=0∞‍2-j(k-l1)×(1rm(x0,W))k-l1∥f∥BMOL≤C∥f∥BMOL,
where we choose k large enough. Thus
(66)1|B|∫B‍|(L-1W)f2(x)|dx≤C∥f∥BMOL.
The above argument also shows that L-1W is well defined on BMOL(G) without the ambiguity of an additive constant.
Supposer<ρ(x0). Set B♮=B(x0,2ρ(x0)). Then, we can write
(67)f=fχB♮+fχ(B♮)c=f1+f2.
Via Lemma 8, m(x,W)~m(x0,W) for any x∈B. Similar to (66), we have
(68)1|B|∫B‍|(L-1W)f2(x)|dx≤C∥f∥BMOL.
Note that
(69)|f(B,W)-f(2B,W)|≤C|f(2B,W)|≤C|2B|∫2B‍|f(y)|dy,|f(B♮,W)|≤∥f∥BMOL.
Set Tj=B(x,2jρ(x0))(j∈ℤ-∪{0,1}). By Hölder inequality and (9), we have, for any x∈B,
(70)|(L-1W)f1(x)|≤∫B♮‍|Γ(x,y)|W(y)|f(y)|dy≤∑j=-∞1‍Ck(2jρ(x0))Q-2∫Tj∖Tj+1‍W(y)|f(y)|dy≤C∑j=-∞1‍1(2jρ(x0))-2×(1|Tj+1|∫Tj+1‍W(y)sdy)1/s|j+1|∥f∥BMOL≤C∑j=-∞1‍2jδ|j+1|∥f∥BMOL≤C∥f∥BMOL,
where δ=2-(Q/s). Then, we have
(71)1|B|∫B‍|(L-1W)f1(x)-f1(B♮,W)|dx≤C∥f∥BMOL.
Therefore, we prove that(L-1W)f∈BMOL(G) and ∥(L-1W)f∥BMOL≤C∥f∥BMOL, where C is an absolute constant independent of f.
SinceL-1W+L-1(-Δ)=Id, as an immediate consequence, L-1(-Δ) is a bounded operator on BMOL(G).
The proof is completed.


## 5.The Boundedness ofL-1∇2onBMOL(G)

Proof of Theorem6.
Similar to the proof of Theorem  1.7 in [27], we show that Theorem 6 holds true.
Supposef∈BMOL(G) and B=B(x0,r). Firstly, we suppose r≥ρ(x0). Set B*=B(x0,2r). Then, we decompose f as
(72)f=fχB*+fχ(B*)c=f1+f2.
Due to Lemma15, we conclude that R~L is bounded on L2(G). By Remark 4, we have
(73)1|B|∫B‍|R~Lf1(x)|dx≤(1|B|∫B‍|R~Lf1(x)|2dx)1/2≤(C|B|∫B*‍|f(x)|2dx)1/2≤C∥f∥BMOL.
Letx∈B. By Lemma 8, 1/m(x,W)≤Cr. Set Tj=B(x0,2jr). Then, by Hölder inequality and Lemma 13, we have
(74)|R~Lf2(x)|≤∫(B*)c‍|∇2Γ(y,x)||f(y)|dy=∑j=1∞‍∫Tj+1∖Tj‍|∇2Γ(y,x)||f(y)|dy≤∑j=1∞‍(∫Tj+1∖Tj‍|∇2Γ(y,x)|sdy)1/s(∫Tj+1∖Tj‍|f(y)|s′dy)1/s′≤∑j=1∞‍C(1+2j)(k/(k0+1))-l1(1|Tj|∫Tj‍|f(y)|s′dy)1/s′≤C∥f∥BMOL,
where we choose k sufficiently large.
Thus(75)1|B|∫B‍|R~Lf2(x)|dx≤C∥f∥BMOL.
The above argument also shows that R~L are well defined on BMOL(G) without the ambiguity of an additive constant.
Supposer<ρ(x0). Set B♮=B(x0,2ρ(x0)). Then, we can write f as follows:
(76)f=fχB♮+fχ(B♮)c=f1+f2.
Note that m(x,W)~m(x0,W) for any x∈B. Similar to (66), we have
(77)1|B|∫B‍|R~Lf2(x)|dx≤C∥f∥BMOL.
To complete the proof of the theorem, by Remark 4, we only need to prove that there exists a constant CB such that
(78)1|B|∫B‍|R~Lf1(x)-CB|dx≤C∥f∥BMOL.
The left side of (78) is bounded by
(79)1|B|∫B‍|R~Lf1(x)-R~f1(x)|dx+1|B|∫B‍|R~f1(x)-CB|dx,
where R~=(-Δ)-1∇2 is the dual operators of classical higher order Riesz transform R=∇2(-Δ)-1. Let x∈B and Bx,k=B(x,22-kρ(x0)),k=0,1,…. Note that m(x,W)~m(x0,W). It is clear that |f(Bx,0)|≤C∥f∥BMOL,
(80)|f(Bx,k)-f(Bx,k-1)|≤C∥f∥BMO
(cf. [28, Page 148]); therefore,
(81)|f(Bx,k)|≤C(k+1)∥f∥BMOL.
It follows that
(82)∫Bx,k‍|f(x)|dx≤C(k+1)|Bx,k|∥f∥BMOL.
By Lemma 14, we get
(83)|R~Lf1(x)-R~f1(x)|≤∫B♮‍|∇y2Γ(y,x)-∇y2Γ0(y,x)||f(y)|dy≤∑k=0∞‍∫Bx,k∖Bx,k+1‍|∇y2Γ(y,x)-∇y2Γ0(y,x)||f(y)|dy≤C∑k=0∞‍∫Bx,k∖Bx,k+1‍{1dQ-2(x,y)∫B(x,d(x,y))‍|∇W(z)|dzdQ-1(x,y)+1dQ-1(x,y)(d(x,y)m(y,|∇W|))δ1+CdQ(x,y)(d(x,y)m(y,W))δ1dQ-2(x,y)∫B(x,d(x,y))‍|∇W(z)|dzdQ-1(x,y)}×|f(y)|dy=I1+I2+I3.
Sincey∈B♮=B(x0,2ρ(x0)), m(x0,W)~m(y,W). It is easy to see that
(84)I3≤C∑k=0∞‍2-kδ1(2-kρ(x0))Q∫Bx,k‍|f(y)|dy≤C∑k=0∞‍(k+1)2-kδ∥f∥BMOL≤C∥f∥BMOL.
By the same argument and noting that M≤m(x0,|∇W|)≤Cm(x0,W)≤1/r, that is, r≤ρ(x0)≤Cρ1(x0)≤M,
(85)I2≤C∑k=0∞‍2-k(1+δ1)1(2-kρ(x0))Q∫Bx,k‍|f(y)|dy≤C∑k=0∞‍(k+1)2-k(1+δ1)∥f∥BMOL≤C∥f∥BMOL.
Because |∇W|∈Bs1 for s1≥Q/2, then δ1≥0. Thus, the last series converges. Using the fractional integral and the condition ρ(x0)≤Cρ1(x0)≤M, we have
(86)I1≤C∑k=0∞‍(2-kρ(x0))2(2-kρ(x0))Q(∫Bx,k-1‍|∇W|s1dy)1/s1×(∫Bx,k-1‍|f(y)|p′dy)1/p′≤Cρ(x0)∑k=0∞‍(k+1)2-k(1+δ1)∥f∥BMOL≤C∥f∥BMOL,
where (1/p)+(1/p′)=1 and 1/p=(1/s1)-(1/Q). Thus
(87)1|B|∫B‍|R~Lf1(x)-R~f1(x)|dx≤C∥f∥BMOL.
It remains to show
(88)1|B|∫B‍|R~f1(x)-CB|dx≤C∥f∥BMOL.
Let Bk=B(x0,21-kρ(x0)),k=0,1,…,k0, where k0 satisfies 2-k0-1ρ(x0)≤r<2-k0ρ(x0). Note that R~f1=R~(f1-f(Bk0)). Set
(89)f1-f(Bk0)=(f1-f(Bk0))χBk0+(f1-f(Bk0))χB0∖Bk0-f(Bk0)χ(B0)c=f1,1+f1,2+f1,3.
Since R~ are bounded on L2(G), by Remark 4, we have
(90)1|B|∫B‍|R~f1,1(x)|dx≤(1|B|∫B‍|R~f1,1(x)|2dx)1/2≤(C|Bk0|∫Bk0‍|f1-f(Bk0)|2dx)1/2≤C∥f∥BMOL.
Since ∇2Γ0 is homogeneous of degree -Q, then by (39),
(91)|∇y2Γ0(y,x)-∇y2Γ0(y,x0)|≤Cd(x,x0)d(y,x0)Q+1ford(x,x0)≤d(y,x0)2.
Then, for x∈B, we have
(92)|R~f1,2(x)-R~f1,2(x0)|≤∑k=0k0-1‍∫Bk∖Bk+1‍|∇y2Γ0(y,x)-∇y2Γ0(y,x0)|×|f1(y)-f(Bk0)|dy≤C∑k=0k0-1‍2k-k0|Bk|∫Bk‍(|f(Bk)-f(Bk0)||f(y)-f(Bk)|+|f(Bk)-f(Bk0)|)dy≤C∑k=0k0-1‍(k0-k+1)2k-k0∥f∥BMO≤C∥f∥BMO.
Thus
(93)1|B|∫B‍|R~f1,2(x)-R~f1,2(x0)|dx≤C∥f∥BMOL.
For the third term, we have
(94)|R~f1,3(x)-R~f1,3(x0)|≤|f(Bk0)|∫(B0)c‍|∇2Γ0(y,x)-∇2Γ0(y,x0)|dy≤C(k0+1)2-k0∥f∥BMOL≤C∥f∥BMOL.
Then
(95)1|B|∫B‍|R~f1,3(x)-R~f1,3(x0)|dx≤C∥f∥BMOL.
The proof of Theorem 6 is completed.


## 6. Example

In this section, we give some examples for the potentials which satisfy the assumption in Theorem6.Case 1.
AssumeG=ℝn. At this time, the homogeneous norm |·| on G is defined as
(96)|x|=(x12+⋯+xn2)1/2
for anyx∈ℝn.Example 1.
LetW(x)=A+|x|=A+(x12+⋯+xn2)1/2, where A is a positive constant.
Following from [33], we know that if P(x) is a polynomial and α>0, then W(x)=|P(x)|α belongs to B∞(ℝn). For 1<p<∞, it is easy to see B∞(ℝn)⊆Bp(ℝn).
Therefore,W(x)=A+|x|=A+(x12+⋯+xn2)1/2∈B∞(ℝn).
IfW1(x)≤C1W2(x), then
(97)1m(x,W1)≐supr>0{r:1rn-2∫B(x,r)‍W1(y)dy≤1}≥C2m(x,W2)≐supr>0{r:1rn-2∫B(x,r)‍W2(y)dy≤1},
where C1 and C2 are positive constants. Thus,
(98)∇W(x)=(x1(x12+⋯+xn2)1/2,…,xn(x12+⋯+xn2)1/2).
Therefore,|∇W(x)|=1. Clearly, |∇W(x)|∈B∞(ℝn). So W(x)≥A|∇W(x)|. Therefore, ρ(x)≤B1ρ1(x) and ρ1(x)=M1, where M1 and B1 are positive constants. Then the potential W(x)=A+|x|=A+(x12+⋯+xn2)1/2 satisfies the assumption of Theorem 6.Example 2.
LetW(x)=A+|x|3=A+(x12+⋯+xn2)3/2, where A is a positive constant. By the above argument, we conclude that W∈B∞(ℝn).
Then(99)∇W(x)=(3(x12+⋯+xn2)1/2x1,…,3(x12+⋯+xn2)1/2xn).
Thus,|∇W(x)|=3|x|2=3(x12+⋯+xn2). Clearly, |∇W(x)|∈B∞(ℝn). From [5], we know that m(x,|∇W|)~1+|x|. Furthermore, ρ(x)≤B2ρ1(x) and ρ1(x)≤M2, where B2 and M2 are positive constants. Then the potential W(x)=A+|x|=A+(x12+⋯+xn2)3/2 satisfies the assumption of Theorem 6.Case 2.
Assume thatG=ℍn. ℍn is an nonabelian stratified Lie group. To begin with, we recall some basic facts on the Heisenberg group, which are found in [34]. The Heisenberg group ℍn is a Lie group with the underlying manifold ℝ2n×ℝ and the multiplication
(100)(x,t)(y,s)=(x+y,t+s+2∑j=1n‍(xn+jyj-xjyn+j)).
A basis for the Lie algebra of left-invariant vector fields on ℍn is given by
(101)Xj=∂∂xj+2xn+j∂∂t,Xn+j=∂∂xn+j-2xj∂∂t,j=1,…,n,X2n+1=∂∂t.
The gradient ∇ℍn is defined by
(102)∇ℍn=(X1,…,X2n).Q=2n+2 is the homogeneous dimension of ℍn. We define a homogeneous norm function on ℍn by
(103)|g|=(|x|2+|t|2)1/4,g=(x,t)∈ℍn.
By the equivalence of two quasi-norm in the finite dimension quasi-normed linear space, we conclude that, for any polynomialP(g) on ℍn and α>0, W(g)=|P(g)|α belongs to B∞. For 1<p<∞, we also have B∞(ℍn)⊆Bp(ℍn).Example 3.
For anyg=(x1,…,x2n,t)∈ℍn, let W(g)=A+(x12+⋯+x2n2)1/2, where A is a positive constant. Then W(g)∈B∞(ℍn). Then
(104)∇W(g)=(x1(x12+⋯+x2n2)1/2,…,x2n(x12+⋯+x2n2)1/2)
and |∇W(g)| = 1. Clearly, |∇W(g)|∈B∞(ℍn). So W(g)≥A|∇W(g)|. Therefore, ρ(x)≤B3ρ1(x) and ρ1(x)=M3, where M3 and B3 are positive constants. Then the potential W(g)=A+(x12+⋯+x2n2)1/2 satisfies the assumption of Theorem 6.Case 3.
Assume thatG is a group of Heisenberg type with Lie algebra 𝔤=V1⊕V2. Denote by {X1,…,Xl} and {Y1,…,Yk} two orthonormal basis of V1 and V2, respectively. The group of Heisenberg type G is a Lie group of step two with the underlying manifold ℝl×ℝk and the multiplication
(105)(x,y)(x′,y′)=(x+x′,y+y′+S(x,x′)),
where x,x′∈ℝl,  y,y′∈ℝk, and S(x,x′) is a skew-symmetric bilinear function from ℝl×ℝk to ℝk with integer coefficients when expressed in the standard bases of ℝl and ℝk.
By [35], we know that there exist real constants cijm with i,j=1,…,l,  m=1,…,k, such that
(106)Xi=∂∂xi+∑m=1l‍∑j=1k‍cijmxm∂∂yj,i=1,…,l.
The gradient∇G is defined by
(107)∇G=(X1,…,Xl).Q=l+2k is the homogeneous dimension of G. We define a homogeneous norm function on G by
(108)|g|=(|x|4+|y|2)1/4,g=(x,y)∈G.
By the equivalence of two quasi-norm in the finite dimension quasi-normed linear space, we also conclude that, for any polynomialP(g) on G and α>0, W(g)=|P(g)|α belongs to B∞. For 1<p<∞, we also have B∞(G)⊆Bp(G).Example 4.
For anyg=(x,y)∈G, let W(g)=A+(x12+⋯+xl2)1/2, where A is a positive constant. Then W(g)∈B∞(G). Then
(109)∇W(g)=(x1(x12+⋯+xl2)1/2,…,xl(x12+⋯+xl2)1/2)
and |∇W(g)|=1. Clearly, |∇W(g)|∈B∞(G). So W(g)≥A|∇W(g)|. Therefore, ρ(x)≤B4ρ1(x) and ρ1(x)=M4, where M4 and B4 are positive constants. Then the potential W(g)=A+(x12+⋯+xl2)1/2 satisfies the assumption of Theorem 6.



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*Source: 483951-2013-12-22.xml*