The Higher Order Riesz Transform and BMO Type Space Associated with Schrödinger Operators on Stratified Lie Groups

Abstract

Assume that 𝐺 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 𝐵_𝑠 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 BMO_L(G).

1. Introduction

In recent years, some problems related to Schrödinger operators on the Euclidean space ℝ^𝑛 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 𝐿^𝑝 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⊕⋯⊕𝑉_𝑚 such that [𝑉₁,𝑉_𝑘]=𝑉_𝑘+1 for 1≤𝑘<𝑚 and [𝑉₁,𝑉_𝑚]=0. If 𝐺 is stratified, its Lie algebra admits a family of dilations, namely, 𝛿_𝑟𝑋₁+𝑋₂+⋯+𝑋_𝑚=𝑟𝑋₁+𝑟²𝑋²+⋯+𝑟^𝑚𝑋^𝑚×𝑋_𝑗∈𝑉_𝑗,𝑗∈{1,…,𝑚}. (1) Assume that 𝐺 is a Lie group with underlying manifold ℝ^𝑛 for some positive integer 𝑛. 𝐺 inherits dilations from 𝔤: if 𝑥∈𝐺 and 𝑟>0, we write 𝛿_𝑟𝑥=𝑟^𝑑1𝑥₁,…,𝑟^𝑑𝑛𝑥_𝑛, (2) where 1≤𝑑₁≤⋯≤𝑑_𝑛. The map 𝑥→𝛿_𝑟𝑥 is an automorphism of 𝐺. The left (or right) Haar measure on 𝐺 is simply 𝑑𝑥₁⋯𝑑𝑥_𝑛, 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: 𝑥𝑦=𝑝₁𝑥,𝑦,…,𝑝_𝑛𝑥,𝑦 (3) for some polynomials 𝑝₁,…,𝑝_𝑛 in 𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛.

The number 𝑄=∑^𝑚_𝑗=1𝑗(dim𝑉_𝑗) 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, |𝛿_𝑟𝑥|=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 𝐵(𝑥,𝑟)=𝑦∈𝐺∣𝑑𝑥,𝑦<𝑟. (4) The measure of 𝐵(𝑥,𝑟) is |𝐵(𝑥,𝑟)|=𝑏𝑟^𝑄, (5) where 𝑏 is a constant. In particular set 𝑉(𝑟)=|𝐵(𝑥,𝑟)| for 𝑟>0 and 𝑥∈𝐺.

Let 𝑋={𝑋₁,…,𝑋_𝑙} be a basis for 𝑉₁ (viewed as left-invariant vector fields on 𝐺). Following [29], one can define a left invariant metric 𝑑_𝑐 associated with 𝑋 which is called the Carnot-Caratheodory metric: let 𝑥,𝑦∈𝐺, and for every 𝛿>0 define 𝐴(𝛿)=[](0)(𝛿)𝛾𝛾∶0,𝛿⟶𝐺∣𝛾=𝑥,𝛾=𝑦,_′(𝑠)=^𝑙_𝑖=1𝑎_𝑖(𝑠)𝑋_𝑖𝛾(𝑠)a.e.with^𝑙_𝑖=1𝑎_𝑖≤1. (6) Let us define 𝑑_𝑐𝑥,𝑦=inf_{𝐴(𝛿)notempty}{𝛿}. (7)

The Carnot-Caratheodory metric 𝑑_𝑐 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, 𝑎⁻¹𝑑_𝑐𝑥,𝑦≤𝑑𝑥,𝑦≤𝑎𝑑_𝑐𝑥,𝑦. (8)

It follows from [28] that 𝑋_𝑗, 𝑗=1,2,…,𝑙, are skew adjoint; that is, 𝑋^∗_𝑗=−𝑋_𝑗. Let Δ=−∑^𝑙_𝑖=1𝑋²_𝑖 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 ℝ^𝑛. The gradient operator ∇ is denoted by ∇=(𝑋₁,…,𝑋_𝑙).

Definition 1. A nonnegative locally 𝐿^𝑠 integrable function 𝑊 on 𝐺 is said to belong to the reverse Hölder class 𝐵_𝑠(1<𝑠<∞) if there exists 𝐶>0 such that the reverse Hölder inequality 1|𝐵(𝑥,𝑟)|_{𝐵(𝑥,𝑟)}𝑊𝑦^𝑠𝑑𝑦^1/𝑠≤𝐶|𝐵(𝑥,𝑟)|_{𝐵(𝑥,𝑟)}𝑊𝑦𝑑𝑦 (9) holds for every ball 𝐵(𝑥,𝑟) in 𝐺.
Moreover, a locally bounded nonnegative function 𝑊∈𝐵_∞ if there exists a positive constant 𝐶 such that ‖𝑊‖_𝐿^∞_{(𝐵(𝑥,𝑟))}≤𝐶|𝐵(𝑥,𝑟)|_{𝐵(𝑥,𝑟)}𝑊𝑦𝑑𝑦 (10) holds for every ball 𝐵(𝑥,𝑟) in 𝐺.

Furthermore, it is easy to see that 𝐵_∞⊆𝐵_𝑠 for any 1<𝑠<∞.

Let 𝐿=−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 𝐵_𝑠 for some 𝑠≥𝑄/2. Denote by ℛ^𝐿=∇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 𝐿^𝑝(𝐺) for 1≤𝑝≤𝑠 and ℛ^𝐿 is bounded on 𝐿^𝑝(𝐺) for 1<𝑝≤𝑠. Lin et al. introduced the Hardy type space 𝐻¹_𝐿(𝐺) related to the Schrödinger operator 𝐿 on the Heisenberg group 𝐺 in [14]. The dual space of 𝐻¹_𝐿(𝐺) is the 𝐵𝑀𝑂 type space 𝐵𝑀𝑂_𝐿(𝐺) investigated by Lin and Liu in [15]. 𝐻¹_𝐿(𝐺) and 𝐵𝑀𝑂_𝐿(𝐺) were also introduced as applications of results in [11, 22].

Next, we recall the definition of 𝐻¹_𝐿(𝐺) and 𝐵𝑀𝑂_𝐿(𝐺). Since 𝑊≥0 and 𝑊∈𝐿^𝑠_loc(𝐺), the Schrödinger operator 𝐿=−Δ+𝑊 generates a (𝐶₀) semigroup {𝑇^𝐿_𝑡}_𝑡>0={𝑒^−𝑡𝐿}_𝑡>0. The maximal function with respect to the semigroup {𝑇^𝐿_𝑡}_𝑡>0 is given by 𝑀^𝐿𝑓(𝑥)=sup_𝑡>0𝑇^𝐿_𝑡𝑓(𝑥). (11) The Hardy space 𝐻¹_𝐿(𝐺) associated with the Schrödinger operator 𝐿 is defined as follows in terms of the maximal function mentioned above.

Definition 2. A function 𝑓∈𝐿¹(𝐺) is said to be in 𝐻¹_𝐿(𝐺) if the semigroup maximal function 𝑀^𝐿𝑓 belongs to 𝐿¹(𝐺). The norm of such a function is defined by 𝑓_𝐻1𝐿=𝑀^𝐿𝑓_𝐿1. (12)

Assume 𝑊∈𝐵_𝑠 for 𝑠>𝑄/2. The auxiliary function 𝑚(𝑥,𝑊) is defined by 𝜌(𝑥)=1𝑚(𝑥,𝑊)≐sup_𝑟>01𝑟∶𝑟^𝑄−2_{𝐵(𝑥,𝑟)}𝑊𝑦𝑑𝑦≤1,𝑥∈𝐺. (13) It follows from Lemma 9 in Section 2 that 0<𝑚(𝑥,𝑊)<∞ for any 𝑥∈𝐺.

The dual space of 𝐻¹_𝐿(𝐺) is the 𝐵𝑀𝑂 type space 𝐵𝑀𝑂_𝐿(𝐺) (cf. [22]). Let 𝑓 be a locally integrable function on 𝐺 and 𝐵=𝐵(𝑥,𝑟) be a ball. Set 𝑓_𝐵=1|𝐵(𝑥,𝑟)|_{𝐵(𝑥,𝑟)}𝑓𝑦𝑓()=𝑓𝑑𝑦,𝐵,𝑊^𝐵,(𝑥),(𝑥).if𝑟<𝜌0,if𝑟≥𝜌 (14)

Definition 3. Let 𝑓 be a locally integrable function on 𝐺. One says 𝑓∈𝐵𝑀𝑂_𝐿(𝐺) if 𝑓_{𝐵𝑀𝑂𝐿}=sup_𝐵1|𝐵|_𝐵𝑓𝑦−𝑓(𝐵,𝑊)𝑑𝑦<∞. (15)

It is clear that 𝐿^∞(𝐺)⊂𝐵𝑀𝑂_𝐿(𝐺)⊂𝐵𝑀𝑂(𝐺) and ‖𝑓‖_𝐵𝑀𝑂≤2‖𝑓‖_{𝐵𝑀𝑂𝐿}. Some remarks are given as follows.

Remark 4. Let 1≤𝑝<∞. If 𝑓∈𝐵𝑀𝑂_𝐿(𝐺), then there exists a positive constant 𝐶: sup_𝐵1|𝐵|_𝐵𝑓𝑦−𝑓(𝐵,𝑊)^𝑝𝑑𝑦^1/𝑝≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (16) 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∈𝐵𝑀𝑂_𝐿(𝐺) if and only if there exist some suitable constants 𝐶_𝐵 and 𝐶 depending on 𝐵=𝐵(𝑥,𝑟) and satisfying 𝐶_𝐵=0 whenever 𝑟≥𝜌(𝑥) such that sup_𝐵1|𝐵|_𝐵𝑓𝑦−𝐶_𝐵^𝑝𝑑𝑦^1/𝑝𝑓<∞,_{𝐵𝑀𝑂𝐿}≤𝐶sup_𝐵1|𝐵|_𝐵𝑓𝑦−𝐶_𝐵^𝑝𝑑𝑦^1/𝑝. (17)

Our main results are given as follows.

Theorem 5. Suppose 𝑊∈𝐵_𝑠 for some 𝑠≥𝑄/2. Then the operators 𝐿⁻¹𝑊 and 𝐿⁻¹(−Δ) are bounded on the space 𝐵𝑀𝑂_𝐿(𝐺).

Theorem 6. Suppose 𝑊∈𝐵_𝑠 for some 𝑠≥𝑄, |∇𝑊|∈𝐵_𝑠1 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 𝐵𝑀𝑂_𝐿(𝐺).

It shoud be noted that because the left invariant vector fields in 𝑉₁⊆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 Section 2, 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 𝐵𝑀𝑂_𝐿(𝐺). In Section 5, we establish the 𝐵𝑀𝑂_𝐿 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 use 𝐶 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 function 𝑚(𝑥,𝑊). We refer to [13] for the details. Throughout this section, unless otherwise indicated, we always assume that 0≢𝑊∈𝐵_𝑠 for some 𝑠>𝑄/2.

Lemma 7. 𝑊∈𝐵_𝑠 is a doubling measure; that is, there exists a constant 𝐶>0 such that _{𝐵(𝑥,2𝑟)}𝑊𝑦𝑑𝑦≤𝐶_{𝐵(𝑥,𝑟)}𝑊𝑦𝑑𝑦. (18)

Lemma 8. There exist constants 𝐶,𝑘₀>0 such that 1𝐶1+𝑑𝑥,𝑦𝑚(𝑥,𝑊)^−𝑘0≤𝑚(𝑥,𝑊)𝑚𝑚()𝑦,𝑊≤𝐶1+𝑑𝑥,𝑦𝑥,𝑊^{𝑘0/(𝑘0+1)}. (19) In particular, 𝑚(𝑥,𝑊)∼𝑚(𝑦,𝑊) if 𝑑(𝑥,𝑦)≤𝐶/𝑚(𝑥,𝑊).

Lemma 9. There exists 𝐶>0 such that, for 0<𝑟<𝑅<∞, 1𝑟^𝑄−2_{𝐵(𝑥,𝑟)}𝑊𝑦𝑟𝑑𝑦≤𝐶𝑅^{2−(𝑄/𝑠)}1𝑅^𝑄−2_{𝐵(𝑥,𝑅)}𝑊𝑦𝑑𝑦. (20)

Lemma 10. If 𝑟=1/𝑚(𝑥,𝑊), then 1𝑟^𝑄−2_{𝐵(𝑥,𝑟)}𝑊𝑦𝑑𝑦=1. (21) Moreover, 1𝑟^𝑄−2_{𝐵(𝑥,𝑟)}𝑊𝑦1𝑑𝑦∼1iff𝑟∼𝑚().𝑥,𝑊 (22)

Lemma 11. There exist 𝐶>0 and 𝑙₁>0 such that _{𝐵(𝑥,𝑟)}𝑊𝑦𝑑𝑥,𝑦^𝑄−2≤𝐶𝑑𝑦𝑟^𝑄−2_{𝐵(𝑥,𝑟)}𝑊𝑦𝑑𝑦≤𝐶(1+𝑟𝑚(𝑥,𝑊))^𝑙1. (23) Moreover, if 𝑊∈𝐵_𝑄, then there exists 𝐶>0 such that _{𝐵(𝑥,𝑟)}𝑊𝑦𝑑𝑦𝑑𝑥,𝑦^𝑄−11≤𝐶𝑟^𝑄−1_{𝐵(𝑥,𝑟)}𝑊𝑦𝑑𝑦. (24)

3. Estimates for the Kernels

In this section we will investigate some necessary estimates about the kernel of the operators in the paper.

Let 𝑝_𝑡(𝑥,𝑦)=ℎ_𝑡(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 ∇²ℎ_𝑡(𝑥)≤𝐶𝑡⁻¹𝑉√𝑡⁻¹exp−𝐶^′𝑑(𝑒,𝑥)𝑡,∇∀𝑥∈𝐺,∀𝑡>0,₃ℎ_𝑡(𝑥)≤𝐶𝑡^−3/2𝑉√𝑡⁻¹exp−𝐶^′𝑑(𝑒,𝑥)𝑡,∀𝑥∈𝐺,∀𝑡>0, (25) 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 𝑋_𝑖^(𝑥)𝑋_𝑗^(𝑥)𝑋_𝑘^(𝑦)𝑝_𝑡𝑥,𝑦≤𝐶𝑡^−3/2𝑉⁻¹√𝑡exp−𝐶^′𝑑²𝑥,𝑦𝑡. (26) Let 𝐻_𝑡𝑓(𝑥)≐_𝐺𝑓𝑦ℎ_𝑡𝑦⁻¹𝑥𝑑𝑦,∀𝑡>0. (27) Then for 𝜆≥0, (𝜆−Δ)⁻¹𝑓(𝑥)=^∞₀𝑒^−𝜆𝑡𝐻_𝑡𝑓(𝑥)𝑑𝑡. (28)

Let Γ₀(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 constant 𝐶_𝑘 such that Γ₀≤𝐶𝑥,𝑦,𝜆^𝑘1+𝜆^1/2𝑑𝑥,𝑦^𝑘1𝑑^𝑄−2,∇𝑥,𝑦_𝑥Γ₀≤𝐶𝑥,𝑦,𝜆^𝑘1+𝜆^1/2𝑑𝑥,𝑦^𝑘1𝑑^𝑄−1,∇𝑥,𝑦_2𝑥Γ₀≤𝐶𝑥,𝑦,𝜆^𝑘1+𝜆^1/2𝑑𝑥,𝑦^𝑘1𝑑^𝑄,∇𝑥,𝑦_2𝑥∇_𝑦Γ₀≤𝐶𝑥,𝑦,𝜆^𝑘1+𝜆^1/2𝑑𝑥,𝑦^𝑘1𝑑^𝑄+1,∇𝑥,𝑦_3𝑥Γ₀≤𝐶𝑥,𝑦,𝜆^𝑘1+𝜆^1/2𝑑𝑥,𝑦^𝑘1𝑑^𝑄+2𝑥,𝑦 (29)(30)(31)(32)(33) 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), 𝑋²_𝑖𝑋^𝑦_𝑗Γ₀=𝑥,𝑦,𝜆_∞0𝑒^−𝜆𝑡𝑋²_𝑖𝑋^𝑦_𝑗𝑝_𝑡≤𝑥,𝑦𝑑𝑡_{𝑑20(𝑥,𝑦)}𝑒^−𝜆𝑡𝑋²_𝑖𝑋^𝑦_𝑗𝑝_𝑡+𝑥,𝑦𝑑𝑡_{∞𝑑2𝑥,𝑦}𝑒^−𝜆𝑡𝑋²_𝑖𝑋^𝑦_𝑗𝑝_𝑡𝑥,𝑦𝑑𝑡≐𝐼₁+𝐼₂. (34) Firstly, for 𝐼₁, we have 𝐼₁≤𝐶^𝑑2₀^(𝑥,𝑦)𝑒^−𝜆𝑡𝑡^−3/2𝑉⁻¹√𝑡exp−𝐶^′𝑑²𝑥,𝑦𝑡𝑑𝑡≤𝐶^𝑑2₀^(𝑥,𝑦)𝑒^−𝜆𝑡𝑡^−3/21√𝑡^𝑄exp−𝐶^′𝑑²𝑥,𝑦𝑡𝑑𝑡≤𝐶^𝑑2₀^(𝑥,𝑦)𝑒^−𝜆𝑡𝑡^−3/21√𝑡^𝑄√𝑡𝑑𝑥,𝑦^𝑄+2≤𝐶𝑑𝑡𝑑^𝑄+1.𝑥,𝑦 (35) In addition, for any positive integer 𝑘, 𝐼₁≤𝐶^𝑑2₀^(𝑥,𝑦)𝑒^−𝜆𝑡𝑡^−3/2𝑉⁻¹√𝑡exp−𝐶^′𝑑²𝑥,𝑦𝑡𝑑𝑡≤𝐶^𝑑2₀^(𝑥,𝑦)𝑒^−𝜆𝑡𝑡^−3/21√𝑡^𝑄exp−𝐶^′𝑑²𝑥,𝑦𝑡𝑑𝑡≤𝐶^𝑑2₀^(𝑥,𝑦)𝑒^−𝜆𝑡𝑡^−3/21√𝑡^𝑄√𝑡𝑑𝑥,𝑦^𝑄+𝑘+1𝑑𝑡≤𝐶^𝑑2₀^(𝑥,𝑦)𝑒^−𝜆𝑡𝑡^(𝑘/2)−11𝑑𝑡𝑑^𝑄+𝑘+1𝑥,𝑦=𝐶^𝜆𝑑2₀^(𝑥,𝑦)𝑒^−𝑡𝑡^(𝑘/2)−11𝑑𝑡𝜆^1/2𝑑𝑥,𝑦^𝑘𝑑^𝑄+1≤𝐶𝑥,𝑦_𝑘𝑑^𝑄+1𝑥,𝑦𝜆^1/2𝑑𝑥,𝑦^𝑘. (36) Therefore, 𝐼₁≤𝐶_𝑘1+𝜆^1/2𝑑𝑥,𝑦^𝑘1𝑑^𝑄+1.𝑥,𝑦 (37) Secondly, we have 𝐼₂≤𝐶^∞_{𝑑2𝑥,𝑦}𝑒^−𝜆𝑡𝑡^−3/2𝑉⁻¹√𝑡exp−𝐶^′𝑑²𝑥,𝑦𝑡𝑑𝑡≤𝐶^∞_{𝑑2𝑥,𝑦}𝑒^−𝜆𝑡𝑡^−3/2𝑑𝑡𝑉⁻¹𝑑𝑥,𝑦≤𝐶𝐶_𝑘^∞_{𝑑2𝑥,𝑦}𝑡^−3/211+𝜆^1/2𝑡^1/2𝑘𝑑𝑡𝑉⁻¹𝑑𝑥,𝑦≤𝐶𝐶_𝑘^∞_{𝑑2𝑥,𝑦}𝑡^−3/21𝑑𝑡1+𝜆^1/2𝑑𝑥,𝑦^𝑘𝑉⁻¹𝑑𝑥,𝑦≤𝐶𝐶_𝑘11+𝜆^1/2𝑑𝑥,𝑦^𝑘1𝑑^𝑄+1.𝑥,𝑦 (38) Therefore, (32) holds true.

Moreover, we need some other basic facts of fundamental solutions for sub-Laplacian on the stratified Lie group 𝐺 (see [32]).

In the first place, we use the standard notations 𝒟, ℰ, and 𝒟^′ for the spaces of 𝐶^∞ functions with compact support, 𝐶^∞ functions, and distributions on 𝒟.

A measurable function 𝑓 on 𝐺 will be called homogeneous of degree 𝜆(𝜆∈ℂ) if 𝑓∘𝛿_𝑟=𝑟^𝜆𝑓 for all 𝑟>0. Likewise, a distribution 𝜏∈𝒟^′ will be called homogeneous of degree 𝜆 if ⟨𝜏,𝑢∘𝛿_𝑟⟩=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 operator 𝐷 will be called homogeneous of degree 𝜆 if 𝐷(𝑢∘𝛿_𝑟)=𝑟^𝜆(𝐷𝑢)∘𝛿_𝑟 for all 𝑢∈𝒟 and 𝑟>0. Since 𝐺 is stratified, 𝑋∈𝔤 is homogeneous of degree 𝑗 if and only if 𝑋∈𝑉_𝑗. 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-Laplacian Δ, 𝑋_𝑖𝑋_𝑗Γ₀ is homogeneous of −𝑄 due to the fact that Γ₀ is homogeneous of 2−𝑄, where 𝑋_𝑖,𝑋_𝑗∈𝑉₁. In addition, by using Proposition  1.7 in [28], we have Γ₀𝑥𝑦−Γ₀(𝑥𝑦)≤𝐶|𝑥|^𝑄+1𝑦≤|𝑥|whenever2. (39)

By [28], the left-invariant fields 𝑋∈𝑉₁ are formally skew-adjoint; that is, 𝑋𝑓𝑦_𝐺𝑔𝑦𝑑𝑦=−_𝐺𝑓𝑦𝑋𝑔𝑦𝑑𝑦. (40) Moreover, 𝑋 interacts with convolution in the following way: 𝑋𝑓∗𝑔=𝑓∗𝑋𝑔. (41)

Let Γ(𝑥,𝑦,𝜆) be the fundamental solution of 𝐿+𝜆 for 𝜆∈[0,∞). Then, Theorem  3.6 in [13] implies that Γ≤1𝑥,𝑦,𝜆1+𝜆^1/2𝑑𝑥,𝑦^𝑘×𝐶_𝑘1+𝑚(𝑥,𝑊)𝑑𝑥,𝑦^𝑘1𝑑^𝑄−2.𝑥,𝑦 (42) 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 each 𝑘∈ℕ there exists 𝐶_𝑘>0 such that Γ≤𝐶𝑥,𝑦^𝑘1+𝑚(𝑥,𝑊)𝑑𝑥,𝑦^𝑘1𝑑^𝑄−2.𝑥,𝑦 (43)(ii)If 𝑊∈𝐵_𝑄, then for each 𝑘∈ℕ there exists 𝐶_𝑘>0 such that ≤𝐶∇Γ𝑥,𝑦^𝑘1+𝑚(𝑥,𝑊)𝑑𝑥,𝑦^𝑘1𝑑^𝑄−1,𝑥,𝑦 (44) where the above estimate can be deduced by Lemma  5.1 in [13].

The operator ∇²(−Δ+𝑊)⁻¹ is defined by ∇²(−Δ+𝑊)⁻¹𝑓(𝑥)=_𝐺𝐾𝑥,𝑦𝑓𝑦𝑑𝑦, (45) where the kernel 𝐾(𝑥,𝑦)=∇²_𝑥Γ(𝑥,𝑦). Also, its adjoint operator (−Δ+𝑊)⁻¹∇² is defined by (−Δ+𝑊)⁻¹∇²𝑓(𝑥)=_𝐺𝐾𝑥,𝑦𝑓𝑦𝑑𝑦, (46) where the kernel 𝐾(𝑥,𝑦)=∇²_𝑦Γ(𝑥,𝑦). Since Γ(𝑥,𝑦)=Γ(𝑦,𝑥), then ∇²_𝑦Γ(𝑥,𝑦)=∇²_𝑦Γ(𝑦,𝑥).

Moreover, we also need other estimates for the kernel 𝐾(𝑥,𝑦) and 𝐾(𝑥,𝑦) in order to prove the main results.

Lemma 13. Assume 𝑊∈𝐵_𝑠 for 𝑠>𝑄. Let 𝑥₀,𝑦₀∈𝐺 and 𝑅=(1/2)𝑑(𝑥₀,𝑦₀). Then for any 𝑘∈ℕ there exists 𝐶_𝑘>0 such that ∇²Γ𝑥,𝑦_{0𝐿𝑠(𝐵(𝑥0,𝑅))}≤𝐶_𝑘𝑉(𝑅)^1/𝑠1+𝑅𝑚𝑥₀,𝑊^{(𝑘/(𝑘0+1))−𝑙1}1𝑉(𝑅), (47) where 𝑘₀ is the constant appearing in Lemma 8 and 𝑙₁ is the constant appearing in Lemma 11.

Proof. Let 𝑢 be the solution of −Δ+𝑊=0 in the ball 𝐵(𝑥₀,2𝑅). By Lemma  3.2 in [13], we choose 𝜂∈𝐶^∞₀(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.
For 𝑥∈𝐵(𝑥₀,𝑅/2𝐶²), 𝑢(𝑥)𝜂(𝑥)=_𝐺Γ₀()(𝑧)=𝑥,𝑧,𝜆)(−Δ+𝜆𝑢𝜂𝑑𝑧_𝐺Γ₀()=𝑥,𝑧,𝜆−𝑊𝑢𝜂−2∇𝑢⋅∇𝜂−𝑢Δ𝜂𝑑𝑧_𝐺Γ₀(𝑥,𝑧,𝜆)−𝑊𝑢𝜂+𝑢Δ𝜂𝑑𝑧+2_𝐺∇_𝑧Γ₀(𝑥,𝑧,𝜆)⋅∇𝜂𝑢𝑑𝑧. (48) By (31) and (32) we have ∇²_𝑥𝑢(𝑥=_𝐺)∇²_𝑥Γ₀(𝑥,𝑧,𝜆)𝑊𝑢𝜂𝑑𝑧+_𝐺∇²_𝑥Γ₀(𝑥,𝑧,𝜆)𝑢Δ𝜂𝑑𝑧+2_𝐺∇²_𝑥∇_𝑧Γ₀()≤𝑥,𝑧,𝜆⋅∇𝜂𝑢𝑑𝑧_𝐺∇²_𝑥Γ₀(𝑥,𝑧,𝜆)𝑊𝑢𝜂𝑑𝑧+_𝐺∇²_𝑥Γ₀(𝑥,𝑧,𝜆)𝑢Δ𝜂𝑑𝑧+2_𝐺∇²_𝑥∇_𝑧Γ₀(𝑢|≤𝑥,𝑧,𝜆)∇𝜂|𝑑𝑧_{𝐵(𝑥0,2𝑅)}∇²_𝑥Γ₀(𝑥,𝑧,𝜆)𝑊𝑢𝜂𝑑𝑧+2_{𝐵(𝑥0,2𝑅)}∇²_𝑥∇_𝑧Γ₀(𝑢|≤𝑥,𝑧,𝜆)∇𝜂|𝑑𝑧_{𝐵(𝑥0,2𝑅)}∇²_𝑥Γ₀()+𝐶𝑥,𝑧,𝜆𝑊𝑢𝜂𝑑𝑧𝑅²𝑉(𝑅)_{𝐵(𝑥0,2𝑅)}|𝑢(𝑧≤)|𝑑𝑧_{𝐵(𝑥0,2𝑅)}∇²_𝑥Γ₀()𝑊𝑢|+𝐶𝑥,𝑧,𝜆|𝑑𝑧𝑅²𝑉(𝑅)_{𝐵(𝑥0,2𝑅)}|𝑢(𝑧)|𝑑𝑧. (49)
By (9), the Calderón-Zygmund estimates, and Lemma 11, ∇²𝑢_{𝐿𝑠(𝐵(𝑥0,𝑅))}≤𝐶𝑊𝑢𝜂_𝑠+𝐶𝑉(𝑅)^1/𝑠𝑅⁻²sup_{𝐵𝑥0,2𝑅}|𝑢|≤𝐶𝑉(𝑅)^1/𝑠×1𝑉(𝑅)_{𝐵(𝑥0,𝑅)}𝑊^𝑠(𝑧)𝑑𝑧^1/𝑠+𝑅⁻²sup_{𝐵𝑥0,2𝑅}|𝑢|≤𝐶𝑉(𝑅)^1/𝑠1𝑉(𝑅)_{𝐵(𝑥0,𝑅)}𝑊(𝑧)𝑑𝑧+𝑅⁻²sup_{𝐵𝑥0,2𝑅}|𝑢|≤𝐶𝑉(𝑅)^1/𝑠𝑅⁻²𝑅²𝑉(𝑅)_{𝐵(𝑥0,𝑅)}𝑊(𝑧)𝑑𝑧+1sup_{𝐵𝑥0,2𝑅}|𝑢|≤𝐶𝑉(𝑅)^1/𝑠𝑅⁻²1+𝑅𝑚𝑥₀,𝑊^𝑙1⋅sup_{𝐵𝑥0,2𝑅}|𝑢|. (50) 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 ∇²Γ(𝑥,𝑦₀)_{𝐿𝑠(𝐵(𝑥0,𝑅))}≤𝐶𝑉(𝑅)^1/𝑠𝑅⁻²1+𝑅𝑚𝑥₀,𝑊^𝑙1⋅sup_{𝐵𝑥0,2𝑟}|𝑢|≤𝐶_𝑘𝑉(𝑅)^1/𝑠1+𝑅𝑚𝑥₀,𝑊^{(𝑘/(𝑘0+1))−𝑙1}1𝑉(𝑅). (51)
This finishes the proof of Lemma 13.

Lemma 14. Suppose 𝑊∈𝐵_𝑠 for some 𝑠>𝑄 and |∇𝑊|∈𝐵_𝑠1 for some 𝑠₁>𝑄/2. Then ∇²Γ𝑥,𝑦−∇²Γ₀≤𝐶𝑥,𝑦𝑑^𝑄−2𝑥,𝑦^{𝐵(𝑥,𝑑(𝑥,𝑦))}|∇𝑊(𝑧)|𝑑𝑧𝑑^𝑄−1()+𝐶𝑥,𝑧𝑑^𝑄−1𝑑𝑚|𝑥,𝑦𝑥,𝑦𝑦,∇𝑊|^𝛿1+𝐶𝑑^𝑄𝑑𝑚𝑥,𝑦𝑥,𝑦𝑦,𝑊^𝛿, (52) 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 −ΔΓ𝑥,𝑦−Γ₀𝑥,𝑦=−𝑊(𝑥)Γ𝑥,𝑦. (53) It follows that Γ𝑥,𝑦−Γ₀𝑥,𝑦=(−Δ)⁻¹−𝑊(𝑥)Γ𝑥,𝑦. (54) By (41), we have ∇Γ𝑥,𝑦−∇Γ₀𝑥,𝑦=−_𝐺∇Γ₀(𝑥,𝑧)𝑊(𝑧)Γ𝑧,𝑦𝑑𝑧=−_𝐺Γ₀(𝑥,𝑧)∇𝑊(𝑧)Γ𝑧,𝑦𝑑𝑧=−_𝐺Γ₀(𝑥,𝑧)∇𝑊(𝑧)Γ𝑧,𝑦+𝑊(𝑧)∇Γ𝑧,𝑦𝑑𝑧. (55) Set 𝑅=(1/8)𝑑(𝑥,𝑦). Thus, we have ∇²Γ𝑥,𝑦−∇²Γ₀𝑥,𝑦≤_𝐺∇Γ₀((𝑧Γ+𝑥,𝑧)|∇𝑊)|𝑧,𝑦𝑑𝑧_𝐺∇Γ₀(𝑥,𝑧)|𝑊(𝑧)|∇Γ𝑧,𝑦𝑑𝑧≐𝐼₁+𝐼₂. (56)
For 𝐼₁, 𝐼₁≤_{𝑑(𝑥,𝑧)<𝑅}+_{𝑑(𝑦,𝑧)<𝑅}+_{𝑑(𝑥,𝑧)≥𝑅,𝑑(𝑦,𝑧)≥𝑅}=𝐼₁₁+𝐼₁₂+𝐼₁₃. (57)
Firstly, by (43) and (30), it holds that 𝐼₁₁=_{𝑑(𝑥,𝑧)<𝑅}∇Γ₀((𝑧Γ≤𝐶𝑥,𝑧)|∇𝑊)|𝑧,𝑦𝑑𝑧𝑑^𝑄−2𝑥,𝑦^{𝐵(𝑥,𝑅)}|∇𝑊(𝑧)|𝑑^𝑄−1(𝑥,𝑧)𝑑𝑧. (58) Secondly, via Lemmas 9 and 11, we similarly have 𝐼₁₂=_{𝑑(𝑦,𝑧)<𝑅}∇Γ₀(⋅|(𝑧⋅Γ≤𝐶𝑥,𝑧)∇𝑊)|𝑧,𝑦𝑑𝑧𝑑^𝑄−1𝑥,𝑦^{𝐵(𝑦,𝑅)}|∇𝑊(𝑧)|𝑑^𝑄−2≤𝐶𝑦,𝑧𝑑𝑧𝑑^𝑄−11𝑥,𝑦𝑑^𝑄−2𝑥,𝑦^{𝐵(𝑦,𝑅)}|(𝑧≤𝐶∇𝑊)|𝑑𝑧𝑑^𝑄−1𝑑𝑚(|𝑥,𝑦𝑥,𝑦𝑥,∇𝑊|)^𝛿1, (59) where 𝛿₁=2−(𝑄/𝑠₁).
Now, we turn to estimating 𝐼₁₃. By Lemma 7, (43), and (30), we have that 𝐼₁₃=_{𝑑(𝑥,𝑧)≥𝑅,𝑑(𝑦,𝑧)≥𝑅}∇Γ₀(⋅|(𝑧⋅Γ≤𝐶𝑥,𝑧)∇𝑊)|𝑧,𝑦𝑑𝑧𝑑^𝑄−1×𝑥,𝑦_{∞𝑗=12𝑗−1𝑅≤𝑑(𝑦,𝑧)<2𝑗𝑅}|(𝑧×𝑑∇𝑊)|_𝑄−2𝑦,𝑧1+𝑑𝑦,𝑧𝑚𝑦,𝑊^𝑘−1≤𝐶𝑑𝑧𝑑^𝑄−1×𝑥,𝑦_∞𝑗=1𝐶^𝑗2^𝑘𝑗12^𝑗𝑐𝑅^{𝑄−2𝑑(𝑦,𝑧)≤𝑅}|(𝑧≤𝐶∇𝑊)|𝑑𝑧𝑑^𝑄−1×𝑥,𝑦_∞𝑗=1𝐶2^{𝑘+𝑛−2𝑗}1𝑅^𝑄−2_{𝑑(𝑦,𝑧)≤𝑅}|(𝑧≤𝐶∇𝑊)|𝑑𝑧𝑑^𝑄−1𝑑𝑚||𝑥,𝑦𝑥,𝑦𝑦,∇𝑊^𝛿1 (60) if we choose 𝑘 large enough.
Then, 𝐼₁≤𝐶𝑑^𝑄−2𝑥,𝑦^{𝐵(𝑥,𝑅)}|∇𝑊(𝑧)|𝑑^𝑄−1()+𝐶𝑥,𝑧𝑑𝑧𝑑^𝑄−1𝑑𝑚||𝑥,𝑦𝑥,𝑦𝑦,∇𝑊^𝛿1. (61)
A similar argument implies that 𝐼₂≤𝐶𝑑^𝑄𝑑𝑥,𝑦𝑥,𝑦𝑚𝑦,𝑊^𝛿. (62)
The proof is completed.

Lemma 15. Suppose 𝑊∈𝐵_𝑠 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 𝐿^𝑝(𝐺), where 𝑠^′≤𝑝≤∞.(2)𝐿⁻¹∇² is bounded on the space 𝐿^𝑝(𝐺) for 𝑠^′≤𝑝<∞.

The above lemmas hold true due to Theorem 4.1 and Theorem A in [13], respectively.

4. The Boundedness of 𝐿⁻¹𝑊 and 𝐿⁻¹(−Δ) on 𝐵𝑀𝑂_𝐿(𝐺)

Proof of Theorem 5. To prove Theorem 5, we adopt the method used in the proof of Theorem  1.6 in [27].
Suppose 𝑓∈𝐵𝑀𝑂_𝐿(𝐺) and 𝐵=𝐵(𝑥₀,𝑟). Firstly, we suppose 𝑟≥𝜌(𝑥₀). Set 𝐵^∗=𝐵(𝑥₀,2𝑟). Then, we have 𝑓=𝑓𝜒_𝐵∗+𝑓𝜒_(𝐵∗)^𝑐=𝑓₁+𝑓₂, (63) where 𝜒_𝐴 denotes the characteristic function of the set 𝐴. Since 𝐿⁻¹𝑊 is bounded on 𝐿²(𝐺), by Remark 4, we have 1|𝐵|_𝐵𝐿⁻¹𝑊𝑓₁(𝑥≤1)𝑑𝑥|𝐵|_𝐵𝐿⁻¹𝑊𝑓₁(𝑥)²𝑑𝑥^1/2≤𝐶|𝐵|_𝐵^∗𝑓(𝑥)²𝑑𝑥^1/2≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (64)
Let 𝑥∈𝐵. By Lemma 8, 1/𝑚(𝑥,𝑊)≤𝐶𝑟. Set 𝑇_𝑗=𝐵(𝑥₀,2^𝑗𝑟). By using (43) and Lemma 11, we obtain 𝐿⁻¹𝑊𝑓₂(𝑥≤_{(𝐵∗)𝑐})Γ𝑊𝑦𝑓𝑦𝑥𝑥,𝑦𝑑𝑦≤𝑚₀,𝑊^∞−𝑘_𝑗=0𝐶_𝑘2^𝑗𝑟^{𝑄−2+𝑘}×_{𝑇𝑗+1⧵𝑇𝑗}𝑊𝑦𝑓𝑦𝑥𝑑𝑦≤𝑚₀,𝑊^−𝑘×^∞_𝑗=0𝐶_𝑘2^𝑗𝑟^𝑘−21𝑇_𝑗+1^𝑇_𝑗+1𝑊𝑦^𝑠𝑑𝑦^1/𝑠𝑓_{𝐵𝑀𝑂𝐿}≤𝐶_𝑘^∞_𝑗=02^{−𝑗(𝑘−𝑙1)}×1𝑟𝑚𝑥₀,𝑊^𝑘−𝑙1𝑓_{𝐵𝑀𝑂𝐿}≤𝐶𝑓_{𝐵𝑀𝑂𝐿}, (65) where we choose 𝑘 large enough. Thus 1|𝐵|_𝐵𝐿⁻¹𝑊𝑓₂(𝑥)𝑑𝑥≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (66) The above argument also shows that 𝐿⁻¹𝑊 is well defined on 𝐵𝑀𝑂_𝐿(𝐺) without the ambiguity of an additive constant.
Suppose 𝑟<𝜌(𝑥₀). Set 𝐵^♮=𝐵(𝑥₀,2𝜌(𝑥₀)). Then, we can write 𝑓=𝑓𝜒_𝐵♮+𝑓𝜒_(𝐵♮)^𝑐=𝑓₁+𝑓₂. (67) Via Lemma 8, 𝑚(𝑥,𝑊)∼𝑚(𝑥₀,𝑊) for any 𝑥∈𝐵. Similar to (66), we have 1|𝐵|_𝐵𝐿⁻¹𝑊𝑓₂(𝑥)𝑑𝑥≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (68) Note that 𝑓()(𝑓(≤𝐶𝐵,𝑊−𝑓2𝐵,𝑊)≤𝐶2𝐵,𝑊)|2𝐵|_2𝐵𝑓𝑦𝑓𝐵𝑑𝑦,_♮,𝑊≤𝑓_{𝐵𝑀𝑂𝐿}. (69) Set 𝑇_𝑗=𝐵(𝑥,2^𝑗𝜌(𝑥₀))(𝑗∈ℤ⁻∪{0,1}). By Hölder inequality and (9), we have, for any 𝑥∈𝐵, 𝐿⁻¹𝑊𝑓₁(𝑥≤_𝐵♮)Γ𝑊𝑦𝑓𝑦≤𝑥,𝑦𝑑𝑦_1𝑗=−∞𝐶_𝑘2^𝑗𝜌𝑥₀^𝑄−2𝑇_𝑗^⧵𝑇_𝑗+1𝑊𝑦𝑓𝑦𝑑𝑦≤𝐶¹_𝑗=−∞12^𝑗𝜌𝑥₀⁻²×1𝑇_𝑗+1^𝑇_𝑗+1𝑊𝑦^𝑠𝑑𝑦^1/𝑠𝑗+1𝑓_{𝐵𝑀𝑂𝐿}≤𝐶¹_𝑗=−∞2^𝑗𝛿𝑗+1𝑓_{𝐵𝑀𝑂𝐿}≤𝐶𝑓_{𝐵𝑀𝑂𝐿}, (70) where 𝛿=2−(𝑄/𝑠). Then, we have 1|𝐵|_𝐵𝐿⁻¹𝑊𝑓₁(𝑥)−𝑓₁𝐵^♮,𝑊𝑑𝑥≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (71)
Therefore, we prove that (𝐿⁻¹𝑊)𝑓∈𝐵𝑀𝑂_𝐿(𝐺) and ‖(𝐿⁻¹𝑊)𝑓‖_{𝐵𝑀𝑂𝐿}≤𝐶‖𝑓‖_{𝐵𝑀𝑂𝐿}, where 𝐶 is an absolute constant independent of 𝑓.
Since 𝐿⁻¹𝑊+𝐿⁻¹(−Δ)=𝐼𝑑, as an immediate consequence, 𝐿⁻¹(−Δ) is a bounded operator on 𝐵𝑀𝑂_𝐿(𝐺).
The proof is completed.

5. The Boundedness of 𝐿⁻¹𝛁² on 𝐵𝑀𝑂_𝐿(𝐺)

Proof of Theorem 6. Similar to the proof of Theorem  1.7 in [27], we show that Theorem 6 holds true.
Suppose 𝑓∈𝐵𝑀𝑂_𝐿(𝐺) and 𝐵=𝐵(𝑥₀,𝑟). Firstly, we suppose 𝑟≥𝜌(𝑥₀). Set 𝐵^∗=𝐵(𝑥₀,2𝑟). Then, we decompose 𝑓 as 𝑓=𝑓𝜒_𝐵∗+𝑓𝜒_(𝐵∗)^𝑐=𝑓₁+𝑓₂. (72)
Due to Lemma 15, we conclude that 𝑅^𝐿 is bounded on 𝐿²(𝐺). By Remark 4, we have 1|𝐵|_𝐵𝑅^𝐿𝑓₁(𝑥1)𝑑𝑥≤|𝐵|_𝐵𝑅^𝐿𝑓₁(𝑥)²𝑑𝑥^1/2≤𝐶|𝐵|_𝐵^∗𝑓(𝑥)²𝑑𝑥^1/2≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (73)
Let 𝑥∈𝐵. By Lemma 8, 1/𝑚(𝑥,𝑊)≤𝐶𝑟. Set 𝑇_𝑗=𝐵(𝑥₀,2^𝑗𝑟). Then, by Hölder inequality and Lemma 13, we have 𝑅^𝐿𝑓₂(𝑥≤_{(𝐵∗)𝑐})∇²Γ𝑓𝑦=𝑦,𝑥𝑑𝑦_{∞𝑗=1𝑇𝑗+1⧵𝑇𝑗}∇²Γ𝑓𝑦≤𝑦,𝑥𝑑𝑦_{∞𝑗=1𝑇𝑗+1⧵𝑇𝑗}∇²Γ𝑦,𝑥^𝑠𝑑𝑦^1/𝑠_{𝑇𝑗+1⧵𝑇𝑗}𝑓𝑦^𝑠′𝑑𝑦^1/𝑠′≤^∞_𝑗=1𝐶1+2^{𝑗(𝑘/(𝑘0+1))−𝑙1}1𝑇_𝑗^𝑇_𝑗𝑓𝑦^𝑠′𝑑𝑦^1/𝑠′≤𝐶𝑓_{𝐵𝑀𝑂𝐿}, (74) where we choose 𝑘 sufficiently large.
Thus 1|𝐵|_𝐵𝑅^𝐿𝑓₂(𝑥)𝑑𝑥≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (75) The above argument also shows that 𝑅^𝐿 are well defined on 𝐵𝑀𝑂_𝐿(𝐺) without the ambiguity of an additive constant.
Suppose 𝑟<𝜌(𝑥₀). Set 𝐵^♮=𝐵(𝑥₀,2𝜌(𝑥₀)). Then, we can write 𝑓 as follows: 𝑓=𝑓𝜒_𝐵♮+𝑓𝜒_(𝐵♮)^𝑐=𝑓₁+𝑓₂. (76) Note that 𝑚(𝑥,𝑊)∼𝑚(𝑥₀,𝑊) for any 𝑥∈𝐵. Similar to (66), we have 1|𝐵|_𝐵𝑅^𝐿𝑓₂(𝑥)𝑑𝑥≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (77) To complete the proof of the theorem, by Remark 4, we only need to prove that there exists a constant 𝐶_𝐵 such that 1|𝐵|_𝐵𝑅^𝐿𝑓₁(𝑥)−𝐶_𝐵𝑑𝑥≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (78) The left side of (78) is bounded by 1|𝐵|_𝐵𝑅^𝐿𝑓₁(𝑥)−𝑅𝑓₁(𝑥1)𝑑𝑥+|𝐵|_𝐵𝑅𝑓₁(𝑥)−𝐶_𝐵𝑑𝑥, (79) where 𝑅=(−Δ)⁻¹∇² is the dual operators of classical higher order Riesz transform 𝑅=∇²(−Δ)⁻¹. Let 𝑥∈𝐵 and 𝐵_𝑥,𝑘=𝐵(𝑥,2^2−𝑘𝜌(𝑥₀)),𝑘=0,1,…. Note that 𝑚(𝑥,𝑊)∼𝑚(𝑥₀,𝑊). It is clear that |𝑓(𝐵_𝑥,0)|≤𝐶‖𝑓‖_{𝐵𝑀𝑂𝐿}, 𝑓𝐵_𝑥,𝑘−𝑓𝐵_{𝑥,𝑘−1}≤𝐶𝑓_𝐵𝑀𝑂 (80) (cf. [28, Page 148]); therefore, 𝑓𝐵_𝑥,𝑘≤𝐶(𝑘+1)𝑓_{𝐵𝑀𝑂𝐿}. (81) It follows that _{𝐵𝑥,𝑘}𝑓(𝑥)𝑑𝑥≤𝐶(𝑘+1)𝐵_𝑥,𝑘𝑓_{𝐵𝑀𝑂𝐿}. (82) By Lemma 14, we get 𝑅^𝐿𝑓₁(𝑥)−𝑅𝑓₁(𝑥≤_𝐵♮)∇²_𝑦Γ𝑦,𝑥−∇²_𝑦Γ₀𝑓𝑦≤𝑦,𝑥𝑑𝑦_{∞𝑘=0𝐵𝑥,𝑘⧵𝐵𝑥,𝑘+1}∇²_𝑦Γ𝑦,𝑥−∇²_𝑦Γ₀𝑦,𝑥𝑓𝑦𝑑𝑦≤𝐶^∞_{𝑘=0𝐵𝑥,𝑘⧵𝐵𝑥,𝑘+1}1𝑑^𝑄−2𝑥,𝑦^{𝐵(𝑥,𝑑(𝑥,𝑦))}|∇𝑊(𝑧)|𝑑𝑧𝑑^𝑄−1+1𝑥,𝑦𝑑^𝑄−1𝑑𝑚||𝑥,𝑦𝑥,𝑦𝑦,∇𝑊^𝛿1+𝐶𝑑^𝑄𝑑𝑚𝑥,𝑦𝑥,𝑦𝑦,𝑊^𝛿×𝑓𝑦𝑑𝑦=𝐼₁+𝐼₂+𝐼₃. (83)
Since 𝑦∈𝐵^♮=𝐵(𝑥₀,2𝜌(𝑥₀)), 𝑚(𝑥₀,𝑊)∼𝑚(𝑦,𝑊). It is easy to see that 𝐼₃≤𝐶^∞_𝑘=02^−𝑘𝛿12^−𝑘𝜌𝑥₀^{𝑄𝐵𝑥,𝑘}𝑓𝑦𝑑𝑦≤𝐶^∞_𝑘=0(𝑘+1)2^−𝑘𝛿𝑓_{𝐵𝑀𝑂𝐿}≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (84) By the same argument and noting that 𝑀≤𝑚(𝑥₀,|∇𝑊|)≤𝐶𝑚(𝑥₀,𝑊)≤1/𝑟, that is, 𝑟≤𝜌(𝑥₀)≤𝐶𝜌₁(𝑥₀)≤𝑀, 𝐼₂≤𝐶^∞_𝑘=02^{−𝑘(1+𝛿1)}12^−𝑘𝜌𝑥₀^{𝑄𝐵𝑥,𝑘}𝑓𝑦𝑑𝑦≤𝐶^∞_𝑘=0(𝑘+1)2^{−𝑘(1+𝛿1)}𝑓_{𝐵𝑀𝑂𝐿}≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (85) Because |∇𝑊|∈𝐵_𝑠1 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 𝐼₁≤𝐶^∞_𝑘=02^−𝑘𝜌𝑥₀²2^−𝑘𝜌𝑥₀^{𝑄𝐵𝑥,𝑘−1}|∇𝑊|^𝑠1𝑑𝑦^1/𝑠1×_{𝐵𝑥,𝑘−1}𝑓𝑦^𝑝′𝑑𝑦^1/𝑝′≤𝐶𝜌𝑥₀^∞_𝑘=0(𝑘+1)2^{−𝑘(1+𝛿1)}𝑓_{𝐵𝑀𝑂𝐿}≤𝐶𝑓_{𝐵𝑀𝑂𝐿}, (86) where (1/𝑝)+(1/𝑝^′)=1 and 1/𝑝=(1/𝑠₁)−(1/𝑄). Thus 1|𝐵|_𝐵𝑅^𝐿𝑓₁(𝑥)−𝑅𝑓₁(𝑥)𝑑𝑥≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (87) It remains to show 1|𝐵|_𝐵𝑅𝑓₁(𝑥)−𝐶_𝐵𝑑𝑥≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (88) Let 𝐵_𝑘=𝐵(𝑥₀,2^1−𝑘𝜌(𝑥₀)),𝑘=0,1,…,𝑘₀, where 𝑘₀ satisfies 2^−𝑘0−1𝜌(𝑥₀)≤𝑟<2^−𝑘0𝜌(𝑥₀). Note that 𝑅𝑓₁=𝑅(𝑓₁−𝑓(𝐵_𝑘0)). Set 𝑓₁−𝑓𝐵_𝑘0=𝑓₁−𝑓𝐵_𝑘0𝜒_𝐵𝑘0+𝑓₁−𝑓𝐵_𝑘0𝜒_{𝐵0⧵𝐵𝑘0}−𝑓𝐵_𝑘0𝜒_(𝐵0)^𝑐=𝑓_1,1+𝑓_1,2+𝑓_1,3. (89) Since 𝑅 are bounded on 𝐿²(𝐺), by Remark 4, we have 1|𝐵|_𝐵𝑅𝑓_1,1(𝑥1)𝑑𝑥≤|𝐵|_𝐵𝑅𝑓_1,1(𝑥)²𝑑𝑥^1/2≤𝐶𝐵_𝑘0^𝐵_𝑘0𝑓₁−𝑓𝐵_𝑘0²𝑑𝑥^1/2≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (90) Since ∇²Γ₀ is homogeneous of degree −𝑄, then by (39), ∇²_𝑦Γ₀𝑦,𝑥−∇²_𝑦Γ₀𝑦,𝑥₀≤𝐶𝑑𝑥,𝑥₀𝑑𝑦,𝑥₀^𝑄+1for𝑑𝑥,𝑥₀≤𝑑𝑦,𝑥₀2. (91) Then, for 𝑥∈𝐵, we have 𝑅𝑓_1,2(𝑥)−𝑅𝑓_1,2𝑥₀≤^𝑘0−1_{𝑘=0𝐵𝑘⧵𝐵𝑘+1}∇²_𝑦Γ₀𝑦,𝑥−∇²_𝑦Γ₀𝑦,𝑥₀×𝑓₁𝑦−𝑓𝐵_𝑘0𝑑𝑦≤𝐶^𝑘0−1_𝑘=02^𝑘−𝑘0𝐵_𝑘𝐵𝑘𝑓𝑦−𝑓𝐵_𝑘+𝑓𝐵_𝑘−𝑓𝐵_𝑘0𝑑𝑦≤𝐶^𝑘0−1_𝑘=0𝑘₀−𝑘+12^𝑘−𝑘0𝑓_𝐵𝑀𝑂≤𝐶𝑓_𝐵𝑀𝑂. (92) Thus 1|𝐵|_𝐵𝑅𝑓_1,2(𝑥)−𝑅𝑓_1,2𝑥₀𝑑𝑥≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (93) For the third term, we have 𝑅𝑓_1,3(𝑥)−𝑅𝑓_1,3𝑥₀≤𝑓𝐵_{𝑘0(𝐵0)𝑐}∇²Γ₀𝑦,𝑥−∇²Γ₀𝑦,𝑥₀𝑘𝑑𝑦≤𝐶₀+12^−𝑘0𝑓_{𝐵𝑀𝑂𝐿}≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (94) Then 1|𝐵|_𝐵𝑅𝑓_1,3(𝑥)−𝑅𝑓_1,3𝑥₀𝑑𝑥≤𝐶𝑓_{𝐵𝑀𝑂𝐿}. (95) The proof of Theorem 6 is completed.

6. Example

In this section, we give some examples for the potentials which satisfy the assumption in Theorem 6.

Case 1. Assume 𝐺=ℝ^𝑛. At this time, the homogeneous norm |⋅| on 𝐺 is defined as |𝑥|=𝑥²₁+⋯+𝑥²_𝑛^1/2 (96)
for any 𝑥∈ℝ^𝑛.

Example 1. Let 𝑊(𝑥)=𝐴+|𝑥|=𝐴+(𝑥²₁+⋯+𝑥²_𝑛)^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_∞(ℝ^𝑛). For 1<𝑝<∞, it is easy to see 𝐵_∞(ℝ^𝑛)⊆𝐵_𝑝(ℝ^𝑛).
Therefore, 𝑊(𝑥)=𝐴+|𝑥|=𝐴+(𝑥²₁+⋯+𝑥²_𝑛)^1/2∈𝐵_∞(ℝ^𝑛).
If 𝑊₁(𝑥)≤𝐶₁𝑊₂(𝑥), then 1𝑚𝑥,𝑊₁≐sup_𝑟>01𝑟∶𝑟^𝑛−2_{𝐵(𝑥,𝑟)}𝑊₁𝑦≥𝐶𝑑𝑦≤1₂𝑚𝑥,𝑊₂≐sup_𝑟>01𝑟∶𝑟^𝑛−2_{𝐵(𝑥,𝑟)}𝑊₂𝑦𝑑𝑦≤1, (97) where 𝐶₁ and 𝐶₂ are positive constants. Thus, (𝑥)=𝑥∇𝑊¹𝑥²₁+⋯+𝑥²_𝑛^1/2𝑥,…,^𝑛𝑥²₁+⋯+𝑥²_𝑛^1/2. (98)
Therefore, |∇𝑊(𝑥)|=1. Clearly, |∇𝑊(𝑥)|∈𝐵_∞(ℝ^𝑛). 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+(𝑥²₁+⋯+𝑥²_𝑛)^1/2 satisfies the assumption of Theorem 6.

Example 2. Let 𝑊(𝑥)=𝐴+|𝑥|³=𝐴+(𝑥²₁+⋯+𝑥²_𝑛)^3/2, where 𝐴 is a positive constant. By the above argument, we conclude that 𝑊∈𝐵_∞(ℝ^𝑛).
Then (𝑥)=3𝑥∇𝑊₂₁+⋯+𝑥²_𝑛^1/2𝑥₁,…,3𝑥²₁+⋯+𝑥²_𝑛^1/2𝑥_𝑛. (99)
Thus, |∇𝑊(𝑥)|=3|𝑥|²=3(𝑥²₁+⋯+𝑥²_𝑛). Clearly, |∇𝑊(𝑥)|∈𝐵_∞(ℝ^𝑛). 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+(𝑥²₁+⋯+𝑥²_𝑛)^3/2 satisfies the assumption of Theorem 6.

Case 2. Assume that𝐺=ℍ^𝑛. ℍ^𝑛 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 ℍ^𝑛 is a Lie group with the underlying manifold ℝ^2𝑛×ℝ and the multiplication (𝑥,𝑡)𝑦,𝑠=𝑥+𝑦,𝑡+𝑠+2^𝑛_𝑗=1𝑥_𝑛+𝑗𝑦_𝑗−𝑥_𝑗𝑦_𝑛+𝑗. (100) A basis for the Lie algebra of left-invariant vector fields on ℍ^𝑛 is given by 𝑋_𝑗=𝜕𝜕𝑥_𝑗+2𝑥_𝑛+𝑗𝜕𝜕𝑡,𝑋_𝑛+𝑗=𝜕𝜕𝑥_𝑛+𝑗−2𝑥_𝑗𝜕,𝜕𝑡𝑗=1,…,𝑛,𝑋_2𝑛+1=𝜕.𝜕𝑡 (101) The gradient ∇_ℍ^𝑛 is defined by ∇_ℍ^𝑛=𝑋₁,…,𝑋_2𝑛. (102)𝑄=2𝑛+2 is the homogeneous dimension of ℍ^𝑛. We define a homogeneous norm function on ℍ^𝑛 by 𝑔=(|𝑥|²+𝑡|^21/4,𝑔=(𝑥,𝑡)∈ℍ^𝑛. (103)
By the equivalence of two quasi-norm in the finite dimension quasi-normed linear space, we conclude that, for any polynomial 𝑃(𝑔) on ℍ^𝑛 and 𝛼>0, 𝑊(𝑔)=|𝑃(𝑔)|^𝛼 belongs to 𝐵_∞. For 1<𝑝<∞, we also have 𝐵_∞(ℍ^𝑛)⊆𝐵_𝑝(ℍ^𝑛).

Example 3. For any 𝑔=(𝑥₁,…,𝑥_2𝑛,𝑡)∈ℍ^𝑛, let 𝑊(𝑔)=𝐴+(𝑥²₁+⋯+𝑥²_2𝑛)^1/2, where 𝐴 is a positive constant. Then 𝑊(𝑔)∈𝐵_∞(ℍ^𝑛). Then 𝑔=𝑥∇𝑊¹𝑥²₁+⋯+𝑥²_2𝑛^1/2𝑥,…,^2𝑛𝑥²₁+⋯+𝑥²_2𝑛^1/2 (104) and |∇𝑊(𝑔)| = 1. Clearly, |∇𝑊(𝑔)|∈𝐵_∞(ℍ^𝑛). 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+(𝑥²₁+⋯+𝑥²_2𝑛)^1/2 satisfies the assumption of Theorem 6.

Case 3. Assume that 𝐺 is a group of Heisenberg type with Lie algebra 𝔤=𝑉₁⊕𝑉₂. Denote by {𝑋₁,…,𝑋_𝑙} and {𝑌₁,…,𝑌_𝑘} 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 ℝ^𝑙×ℝ^𝑘 and the multiplication 𝑥,𝑦𝑥^′,𝑦^′=𝑥+𝑥^′,𝑦+𝑦^′+𝑆𝑥,𝑥^′, (105) where 𝑥,𝑥^′∈ℝ^𝑙,  𝑦,𝑦^′∈ℝ^𝑘, and 𝑆(𝑥,𝑥^′) is a skew-symmetric bilinear function from ℝ^𝑙×ℝ^𝑘 to ℝ^𝑘 with integer coefficients when expressed in the standard bases of ℝ^𝑙 and ℝ^𝑘.
By [35], we know that there exist real constants 𝑐_𝑖𝑗𝑚 with 𝑖,𝑗=1,…,𝑙,  𝑚=1,…,𝑘, such that 𝑋_𝑖=𝜕𝜕𝑥_𝑖+^𝑙𝑘_𝑚=1𝑗=1𝑐_𝑖𝑗𝑚𝑥_𝑚𝜕𝜕𝑦_𝑗,𝑖=1,…,𝑙. (106)
The gradient ∇_𝐺 is defined by ∇_𝐺=𝑋₁,…,𝑋_𝑙. (107)𝑄=𝑙+2𝑘 is the homogeneous dimension of 𝐺. We define a homogeneous norm function on 𝐺 by 𝑔=(|𝑥|⁴+𝑦|^21/4,𝑔=𝑥,𝑦∈𝐺. (108)
By the equivalence of two quasi-norm in the finite dimension quasi-normed linear space, we also conclude that, for any polynomial 𝑃(𝑔) on 𝐺 and 𝛼>0, 𝑊(𝑔)=|𝑃(𝑔)|^𝛼 belongs to 𝐵_∞. For 1<𝑝<∞, we also have 𝐵_∞(𝐺)⊆𝐵_𝑝(𝐺).

Example 4. For any 𝑔=(𝑥,𝑦)∈𝐺, let 𝑊(𝑔)=𝐴+(𝑥²₁+⋯+𝑥²_𝑙)^1/2, where 𝐴 is a positive constant. Then 𝑊(𝑔)∈𝐵_∞(𝐺). Then 𝑔=𝑥∇𝑊¹𝑥²₁+⋯+𝑥²_𝑙^1/2𝑥,…,^𝑙𝑥²₁+⋯+𝑥²_𝑙^1/2 (109) 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+(𝑥²₁+⋯+𝑥²_𝑙)^1/2 satisfies the assumption of Theorem 6.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to Professor Jie Xiao and Professor Hongquan Li for their helpful advice on this paper. This paper is supported by Research Fund for the Doctoral Program of Higher Education of China under Grant no. 20113108120001, the Shanghai Leading Academic Discipline Project (J50101), the National Natural Science Foundation of China under grant no. 10901018, and the Fundamental Research Funds for the Central Universities and Program for New Century Excellent Talents in University.