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 ...2; 0 a nonnegative potential belonging to certain reverse Hölder class [subscript] B s [/subscript] for s ...5; 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 [superscript] L - 1 [/superscript] [superscript] ∇ 2 [/superscript] , [superscript] L - 1 [/superscript] W , and [superscript] L - 1 [/superscript] ( - Δ ) on the space BMO[subscript]L[/subscript] (G).

(ProQuest: ... denotes non-US-ASCII text omitted.)

Yu Liu 1 and Jianfeng Dong 2

Academic Editor:Yoshihiro Sawano

1, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
2, Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 1 October 2013; Accepted 7 November 2013

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In recent years, some problems related to Schrödinger operators on the Euclidean space ^{... 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 B M O , B L O , 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. Dziubanski and Zienkiewicz proved that Riesz transforms related to Schrödinger operators are bounded from Hardy spaces associated with Schrödinger operators into ^{L 1} in [1]. Endpoint boundedness of Riesz transforms related to Schrödinger operators had been investigated in [11, 25]. Dong and Liu established the B M O 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 B M O estimates for the higher order Riesz transform in [27]. The aim of this paper is to obtain the B M O 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 ... admits a vector space decomposition ... = _{V 1} [ecedil]5; ... [ecedil]5; _{V m} such that [ _{V 1} , _{V k} ] = _{V k + 1} for 1 ...4; k < m and [ _{V 1} , _{V m} ] = 0 . If G is stratified, its Lie algebra admits a family of dilations, namely, [figure omitted; refer to PDF] Assume that G is a Lie group with underlying manifold ^{... n} for some positive integer n . G inherits dilations from ... : if x ∈ G and r > 0 , we write [figure omitted; refer to PDF] where 1 ...4; _{d 1} ...4; ... ...4; _{d n} . The map x [arrow right] _{δ r} x is an automorphism of G . The left (or right) Haar measure on G is simply d _{x 1} ... d _{x n} , which is the Lebesgue measure on ... . 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: [figure omitted; refer to PDF] for some polynomials _{p 1} , ... , _{p n} in _{x 1} , ... , _{x n} , _{y 1} , ... , _{y n} .

The number Q = ^{∑ j = 1 m} ... j ( dim _{V 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 r} x | = r | x | for all x ∈ G , r > 0 , | ^{x - 1} | = | x | for all x ∈ G , and | x | > 0 if x ...0; 0 . The homogeneous norm induces a quasi-metric d which is defined by d ( x , y ) : = | ^{x - 1} y | . In particular, d ( e , x ) = | x | and d ( x , y ) = d ( e , ^{x - 1} y ) . The ball of radius r centered at x is written by [figure omitted; refer to PDF] The measure of B ( x , r ) is [figure omitted; refer to PDF] where b is a constant. In particular set V ( r ) = | B ( x , r ) | for r > 0 and x ∈ G .

Let X = { _{X 1} , ... , _{X l} } be a basis for _{V 1} (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 [figure omitted; refer to PDF] Let us define [figure omitted; refer to PDF]

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 , [figure omitted; refer to PDF]

It follows from [28] that _{X j} , j = 1,2 , ... , l , are skew adjoint; that is, ^{X j *} = - _{X j} . Let Δ = - ^{∑ i = 1 l} ... ^{X i 2} 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 ^{... n} . The gradient operator ∇ is denoted by ∇ = ( _{X 1} , ... , _{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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 ...2; 0 is a nonnegative potential belonging to the reverse Hölder class _{B s} for some s ...5; Q / 2 . Denote by ^{... L} = ^{∇ 2 L - 1} the higher order Riesz transform. Accordingly, denote by ^{... ~ L} = ^{L - 1 ∇ 2} its dual operator.

It follows from [13] that the integral operators W ^{L - 1} and ( - Δ ) ^{L - 1} are bounded on ^{L p} ( G ) for 1 ...4; p ...4; s and ^{... L} is bounded on ^{L p} ( G ) for 1 < p ...4; s . Lin et al. introduced the Hardy type space ^{H L 1} ( G ) related to the Schrödinger operator L on the Heisenberg group G in [14]. The dual space of ^{H L 1} ( G ) is the B M O type space B M _{O L} ( G ) investigated by Lin and Liu in [15]. ^{H L 1} ( G ) and B M _{O L} ( G ) were also introduced as applications of results in [11, 22].

Next, we recall the definition of ^{H L 1} ( G ) and B M _{O L} ( G ) . Since W ...5; 0 and W ∈ ^{L loc s} ( G ) , the Schrödinger operator L = - Δ + W generates a ( _{C 0} ) semigroup _{{ ^{T t L} } t > 0} = _{{ ^{e - t L} } t > 0} . The maximal function with respect to the semigroup _{{ ^{T t L} } t > 0} is given by [figure omitted; refer to PDF] The Hardy space ^{H L 1} ( G ) associated with the Schrödinger operator L is defined as follows in terms of the maximal function mentioned above.

Definition 2.

A function f ∈ ^{L 1} ( G ) is said to be in ^{H L 1} ( G ) if the semigroup maximal function ^{M L} f belongs to ^{L 1} ( G ) . The norm of such a function is defined by [figure omitted; refer to PDF]

Assume W ∈ _{B s} for s > Q / 2 . The auxiliary function m ( x , W ) is defined by [figure omitted; refer to PDF] It follows from Lemma 9 in Section 2 that 0 < m ( x , W ) < ∞ for any x ∈ G .

The dual space of ^{H L 1} ( G ) is the B M O type space B M _{O L} ( G ) (cf. [22]). Let f be a locally integrable function on G and B = B ( x , r ) be a ball. Set [figure omitted; refer to PDF]

Definition 3.

Let f be a locally integrable function on G . One says f ∈ B M _{O L} ( G ) if [figure omitted; refer to PDF]

It is clear that ^{L ∞} ( G ) ⊂ B M _{O L} ( G ) ⊂ B M O ( G ) and || f _{|| B M O} ...4; 2 || f _{|| B M O L} . Some remarks are given as follows.

Remark 4.

Let 1 ...4; p < ∞ . If f ∈ B M _{O L} ( G ) , then there exists a positive constant C : [figure omitted; refer to PDF] 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 ∈ B M _{O 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 ...5; ρ ( x ) such that [figure omitted; refer to PDF]

Our main results are given as follows.

Theorem 5.

Suppose W ∈ _{B s} for some s ...5; Q / 2 . Then the operators ^{L - 1} W and ^{L - 1} ( - Δ ) are bounded on the space B M _{O L} ( G ) .

Theorem 6.

Suppose W ∈ _{B s} for some s ...5; Q , | ∇ W | ∈ _{B s 1} for some _{s 1} ...5; Q / 2 , and ρ ( x ) [<, ~] _{ρ 1} ( x ) = 1 / m ( x , | ∇ W | ) and _{ρ 1} ( x ) ...4; M for some positive constant M . Then operator ^{L - 1 ∇ 2} is bounded on the space B M _{O L} ( G ) .

It shoud be noted that because the left invariant vector fields in _{V 1} ⊆ ... 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 - 1} W , ^{L - 1} ( - Δ ) on the space B M _{O L} ( G ) . In Section 5, we establish the B M _{O 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 [variant prime]} such that 1 / C ...4; A / B ...4; C and A ...4; ^{C [variant prime]} 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 ...2; 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 [figure omitted; refer to PDF]

Lemma 8.

There exist constants C , _{k 0} > 0 such that [figure omitted; refer to PDF] In particular, m ( x , W ) ~ m ( y , W ) if d ( x , y ) ...4; C / m ( x , W ) .

Lemma 9.

There exists C > 0 such that, for 0 < r < R < ∞ , [figure omitted; refer to PDF]

Lemma 10.

If r = 1 / m ( x , W ) , then [figure omitted; refer to PDF] Moreover, [figure omitted; refer to PDF]

Lemma 11.

There exist C > 0 and _{l 1} > 0 such that [figure omitted; refer to PDF] Moreover, if W ∈ _{B Q} , then there exists C > 0 such that [figure omitted; refer to PDF]

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 - 1} x ) be the heat kernel of the semigroup ^{e - t Δ} , t ...5; 0 , associated with - Δ . Via Theorem 4.2 of [31], the following estimates hold true; that is, there exist positive constants C and ^{C [variant prime]} such that [figure omitted; refer to PDF] where e is the unit element of G . Moreover, for any 1 ...4; i , j , k ...4; m and x , y ∈ G , by using (3.5) in [13] we obtain [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] Then for λ ...5; 0 , [figure omitted; refer to PDF]

Let _{Γ 0} ( x , y , λ ) be the fundamental solution of the operator - Δ + λ for λ ∈ [ 0 , ∞ ) . In particular, we denote by _{Γ 0} ( x , y , 0 ) = _{Γ 0} ( x , y ) . Then we have the following.

Proposition 12.

There exists a positive constant _{C k} such that [figure omitted; refer to PDF] for x ...0; 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), [figure omitted; refer to PDF] Firstly, for _{I 1} , we have [figure omitted; refer to PDF] In addition, for any positive integer k , [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] Secondly, we have [figure omitted; refer to PDF] 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 ...9F; , ... , and ^{...9F; [variant prime]} for the spaces of ^{C ∞} functions with compact support, ^{C ∞} functions, and distributions on ...9F; .

A measurable function f on G will be called homogeneous of degree λ ( λ ∈ ... ) if f [composite function] _{δ r} = ^{r λ} f for all r > 0 . Likewise, a distribution τ ∈ ^{...9F; [variant prime]} will be called homogeneous of degree λ if Y9; τ , u [composite function] _{δ r} YA; = ^{r - Q - λ} Y9; τ , u YA; for all u ∈ ...9F; 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 [composite function] _{δ r} ) = ^{r λ} ( D u ) [composite function] _{δ r} for all u ∈ ...9F; and r > 0 . Since G is stratified, X ∈ ... 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 D K is a kernel of type α - λ .

For sub-Laplacian Δ , _{X i X j Γ 0} is homogeneous of - Q due to the fact that _{Γ 0} is homogeneous of 2 - Q , where _{X i} , _{X j} ∈ _{V 1} . In addition, by using Proposition 1.7 in [28], we have [figure omitted; refer to PDF]

By [28], the left-invariant fields X ∈ _{V 1} are formally skew-adjoint; that is, [figure omitted; refer to PDF] Moreover, X interacts with convolution in the following way: [figure omitted; refer to PDF]

Let Γ ( x , y , λ ) be the fundamental solution of L + λ for λ ∈ [ 0 , ∞ ) . Then, Theorem 3.6 in [13] implies that [figure omitted; refer to PDF] 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 ∈ ... there exists _{C k} > 0 such that [figure omitted; refer to PDF]

(ii) If W ∈ _{B Q} , then for each k ∈ ... there exists _{C k} > 0 such that [figure omitted; refer to PDF] where the above estimate can be deduced by Lemma 5.1 in [13].

The operator ^{∇ 2 ( - Δ + W ) - 1} is defined by [figure omitted; refer to PDF] where the kernel K ( x , y ) = ^{∇ x 2} Γ ( x , y ) . Also, its adjoint operator ^{( - Δ + W ) - 1 ∇ 2} is defined by [figure omitted; refer to PDF] where the kernel K ~ ( x , y ) = ^{∇ y 2} Γ ( x , y ) . Since Γ ( x , y ) = Γ ( y , x ) , then ^{∇ y 2} Γ ( x , y ) = ^{∇ y 2} Γ ( 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 _{x 0} , _{y 0} ∈ G and R = ( 1 / 2 ) d ( _{x 0} , _{y 0} ) . Then for any k ∈ ... there exists _{C k} > 0 such that [figure omitted; refer to PDF] where _{k 0} is the constant appearing in Lemma 8 and _{l 1} is the constant appearing in Lemma 11.

Proof.

Let u be the solution of - Δ + W = 0 in the ball B ( _{x 0} , 2 R ) . By Lemma 3.2 in [13], we choose η ∈ ^{C 0 ∞} ( B ( _{x 0} , 2 R ) ) such that 0 ...4; η ...4; 1 , η = 1 on B ( _{x 0} , R / ^{C 2} ) , | ∇ η | ...4; ^{C *} / R , and | ^{∇ 2} η | ...4; ^{C *} / ^{R 2} , where C ...5; 1 and ^{C *} are fixed constants, which are independent of _{x 0} and R .

For x ∈ B ( _{x 0} , R / 2 ^{C 2} ) , [figure omitted; refer to PDF] By (31) and (32) we have [figure omitted; refer to PDF]

By (9), the Calderón-Zygmund estimates, and Lemma 11, [figure omitted; refer to PDF] Let u ( x ) = Γ ( x , _{y 0} ) and R = ( 1 / 2 ) d ( _{x 0} , _{y 0} ) . Then u ( x ) = Γ ( x , _{y 0} ) is a solution of - Δ + W = 0 in B ( _{x 0} , R ) . By the above inequality and Lemma 8, we immediately have [figure omitted; refer to PDF]

This finishes the proof of Lemma 13.

Lemma 14.

Suppose W ∈ _{B s} for some s > Q and | ∇ W | ∈ _{B s 1} for some _{s 1} > Q / 2 . Then [figure omitted; refer to PDF] where _{δ 1} = 2 - Q / _{s 1} and δ = 2 - Q / s , if d ( x , y ) ...4; A / m ( x , | ∇ W | ) for some positive constant A .

Proof.

Note that ( - Δ + W ) Γ ( x , y ) = 0 . Let _{Γ 0} ( x , y ) be the fundamental solution of - Δ . Then, we have [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] By (41), we have [figure omitted; refer to PDF] Set R = ( 1 / 8 ) d ( x , y ) . Thus, we have [figure omitted; refer to PDF]

For _{I 1} , [figure omitted; refer to PDF]

Firstly, by (43) and (30), it holds that [figure omitted; refer to PDF] Secondly, via Lemmas 9 and 11, we similarly have [figure omitted; refer to PDF] where _{δ 1} = 2 - ( Q / _{s 1} ) .

Now, we turn to estimating _{I 13} . By Lemma 7, (43), and (30), we have that [figure omitted; refer to PDF] if we choose k large enough.

Then, [figure omitted; refer to PDF]

A similar argument implies that [figure omitted; refer to PDF]

The proof is completed.

Lemma 15.

Suppose W ∈ _{B s} for some s ...5; Q / 2 . Let ^{s [variant prime]} be the conjugate index of s .

(1) ^{L - 1} ( - Δ ) and ^{L - 1} W are bounded on the space ^{L p} ( G ) , where ^{s [variant prime]} ...4; p ...4; ∞ .

(2) ^{L - 1 ∇ 2} is bounded on the space ^{L p} ( G ) for ^{s [variant prime]} ...4; p < ∞ .

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

4. The Boundedness of ^{L - 1} W and ^{L - 1} ( - Δ ) on _{B M O 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 ∈ B M _{O L} ( G ) and B = B ( _{x 0} , r ) . Firstly, we suppose r ...5; ρ ( _{x 0} ) . Set ^{B *} = B ( _{x 0} , 2 r ) . Then, we have [figure omitted; refer to PDF] where _{χ A} denotes the characteristic function of the set A . Since ^{L - 1} W is bounded on ^{L 2} ( G ) , by Remark 4, we have [figure omitted; refer to PDF]

Let x ∈ B . By Lemma 8, 1 / m ( x , W ) ...4; C r . Set _{T j} = B ( _{x 0} , ^{2 j} r ) . By using (43) and Lemma 11, we obtain [figure omitted; refer to PDF] where we choose k large enough. Thus [figure omitted; refer to PDF] The above argument also shows that ^{L - 1} W is well defined on B M _{O L} ( G ) without the ambiguity of an additive constant.

Suppose r < ρ ( _{x 0} ) . Set ^{B [music natural]} = B ( _{x 0} , 2 ρ ( _{x 0} ) ) . Then, we can write [figure omitted; refer to PDF] Via Lemma 8, m ( x , W ) ~ m ( _{x 0} , W ) for any x ∈ B . Similar to (66), we have [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] Set _{T j} = B ( x , ^{2 j} ρ ( _{x 0} ) ) ( j ∈ ^{... -} ∪ { 0,1 } ) . By Hölder inequality and (9), we have, for any x ∈ B , [figure omitted; refer to PDF] where δ = 2 - ( Q / s ) . Then, we have [figure omitted; refer to PDF]

Therefore, we prove that ( ^{L - 1} W ) f ∈ B M _{O L} ( G ) and _{|| ( ^{L - 1} W ) f || B M O L} ...4; C _{|| f || B M O L} , where C is an absolute constant independent of f .

Since ^{L - 1} W + ^{L - 1} ( - Δ ) = I d , as an immediate consequence, ^{L - 1} ( - Δ ) is a bounded operator on B M _{O L} ( G ) .

The proof is completed.

5. The Boundedness of ^{L - 1 ∇ 2} on B M _{O 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 ∈ B M _{O L} ( G ) and B = B ( _{x 0} , r ) . Firstly, we suppose r ...5; ρ ( _{x 0} ) . Set ^{B *} = B ( _{x 0} , 2 r ) . Then, we decompose f as [figure omitted; refer to PDF]

Due to Lemma 15, we conclude that ^{R ~ L} is bounded on ^{L 2} ( G ) . By Remark 4, we have [figure omitted; refer to PDF]

Let x ∈ B . By Lemma 8, 1 / m ( x , W ) ...4; C r . Set _{T j} = B ( _{x 0} , ^{2 j} r ) . Then, by Hölder inequality and Lemma 13, we have [figure omitted; refer to PDF] where we choose k sufficiently large.

Thus [figure omitted; refer to PDF] The above argument also shows that ^{R ~ L} are well defined on B M _{O L} ( G ) without the ambiguity of an additive constant.

Suppose r < ρ ( _{x 0} ) . Set ^{B [music natural]} = B ( _{x 0} , 2 ρ ( _{x 0} ) ) . Then, we can write f as follows: [figure omitted; refer to PDF] Note that m ( x , W ) ~ m ( _{x 0} , W ) for any x ∈ B . Similar to (66), we have [figure omitted; refer to PDF] To complete the proof of the theorem, by Remark 4, we only need to prove that there exists a constant _{C B} such that [figure omitted; refer to PDF] The left side of (78) is bounded by [figure omitted; refer to PDF] 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 - k} ρ ( _{x 0} ) ) , k = 0,1 , ... . Note that m ( x , W ) ~ m ( _{x 0} , W ) . It is clear that | f ( _{B x , 0} ) | ...4; C || f _{|| B M O L} , [figure omitted; refer to PDF] (cf. [28, Page 148]); therefore, [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] By Lemma 14, we get [figure omitted; refer to PDF]

Since y ∈ ^{B [music natural]} = B ( _{x 0} , 2 ρ ( _{x 0} ) ) , m ( _{x 0} , W ) ~ m ( y , W ) . It is easy to see that [figure omitted; refer to PDF] By the same argument and noting that M ...4; m ( _{x 0} , | ∇ W | ) ...4; C m ( _{x 0} , W ) ...4; 1 / r , that is, r ...4; ρ ( _{x 0} ) ...4; C _{ρ 1} ( _{x 0} ) ...4; M , [figure omitted; refer to PDF] Because | ∇ W | ∈ _{B s 1} for _{s 1} ...5; Q / 2 , then _{δ 1} ...5; 0 . Thus, the last series converges. Using the fractional integral and the condition ρ ( _{x 0} ) ...4; C _{ρ 1} ( _{x 0} ) ...4; M , we have [figure omitted; refer to PDF] where ( 1 / p ) + ( 1 / ^{p [variant prime]} ) = 1 and 1 / p = ( 1 / _{s 1} ) - ( 1 / Q ) . Thus [figure omitted; refer to PDF] It remains to show [figure omitted; refer to PDF] Let _{B k} = B ( _{x 0} , ^{2 1 - k} ρ ( _{x 0} ) ) , k = 0,1 , ... , _{k 0} , where _{k 0} satisfies ^{2 - _{k 0} - 1} ρ ( _{x 0} ) ...4; r < ^{2 - _{k 0}} ρ ( _{x 0} ) . Note that R ~ _{f 1} = R ~ ( _{f 1} - f ( _{B k 0} ) ) . Set [figure omitted; refer to PDF] Since R ~ are bounded on ^{L 2} ( G ) , by Remark 4, we have [figure omitted; refer to PDF] Since ^{∇ 2} _{Γ 0} is homogeneous of degree - Q , then by (39), [figure omitted; refer to PDF] Then, for x ∈ B , we have [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] For the third term, we have [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] 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 = ^{... n} . At this time, the homogeneous norm | · | on G is defined as [figure omitted; refer to PDF]

for any x ∈ ^{... n} .

Example 1.

Let W ( x ) = A + | x | = A + ^{( x 1 2 + ... + x n 2 ) 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 ∞} ( ^{... n} ) . For 1 < p < ∞ , it is easy to see _{B ∞} ( ^{... n} ) ⊆ _{B p} ( ^{... n} ) .

Therefore, W ( x ) = A + | x | = A + ^{( x 1 2 + ... + x n 2 ) 1 / 2} ∈ _{B ∞} ( ^{... n} ) .

If _{W 1} ( x ) ...4; _{C 1 W 2} ( x ) , then [figure omitted; refer to PDF] where _{C 1} and _{C 2} are positive constants. Thus, [figure omitted; refer to PDF]

Therefore, | ∇ W ( x ) | = 1 . Clearly, | ∇ W ( x ) | ∈ _{B ∞} ( ^{... n} ) . So W ( x ) ...5; A | ∇ W ( x ) | . Therefore, ρ ( x ) ...4; _{B 1 ρ 1} ( x ) and _{ρ 1} ( x ) = _{M 1} , where _{M 1} and _{B 1} are positive constants. Then the potential W ( x ) = A + | x | = A + ^{( x 1 2 + ... + x n 2 ) 1 / 2} satisfies the assumption of Theorem 6.

Example 2.

Let W ( x ) = A + ^{| x | 3} = A + ^{( x 1 2 + ... + x n 2 ) 3 / 2} , where A is a positive constant. By the above argument, we conclude that W ∈ _{B ∞} ( ^{... n} ) .

Then [figure omitted; refer to PDF]

Thus, | ∇ W ( x ) | = 3 ^{| x | 2} = 3 ( ^{x 1 2} + ... + ^{x n 2} ) . Clearly, | ∇ W ( x ) | ∈ _{B ∞} ( ^{... n} ) . From [5], we know that m ( x , | ∇ W | ) ~ 1 + | x | . Furthermore, ρ ( x ) ...4; _{B 2 ρ 1} ( x ) and _{ρ 1} ( x ) ...4; _{M 2} , where _{B 2} and _{M 2} are positive constants. Then the potential W ( x ) = A + | x | = A + ^{( x 1 2 + ... + x n 2 ) 3 / 2} satisfies the assumption of Theorem 6.

Case 2.

Assume that G = ^{... n} . ^{... 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 ^{... n} is a Lie group with the underlying manifold ^{... 2 n} × ... and the multiplication [figure omitted; refer to PDF] A basis for the Lie algebra of left-invariant vector fields on ^{... n} is given by [figure omitted; refer to PDF] The gradient _{∇ ^{... n}} is defined by [figure omitted; refer to PDF] Q = 2 n + 2 is the homogeneous dimension of ^{... n} . We define a homogeneous norm function on ^{... n} by [figure omitted; refer to PDF]

By the equivalence of two quasi-norm in the finite dimension quasi-normed linear space, we conclude that, for any polynomial P ( g ) on ^{... n} and α > 0 , W ( g ) = ^{| P ( g ) | α} belongs to _{B ∞} . For 1 < p < ∞ , we also have _{B ∞} ( ^{... n} ) ⊆ _{B p} ( ^{... n} ) .

Example 3.

For any g = ( _{x 1} , ... , _{x 2 n} , t ) ∈ ^{... n} , let W ( g ) = A + ^{( x 1 2 + ... + x 2 n 2 ) 1 / 2} , where A is a positive constant. Then W ( g ) ∈ _{B ∞} ( ^{... n} ) . Then [figure omitted; refer to PDF] and | ∇ W ( g ) | = 1. Clearly, | ∇ W ( g ) | ∈ _{B ∞} ( ^{... n} ) . So W ( g ) ...5; A | ∇ W ( g ) | . Therefore, ρ ( x ) ...4; _{B 3 ρ 1} ( x ) and _{ρ 1} ( x ) = _{M 3} , where _{M 3} and _{B 3} are positive constants. Then the potential W ( g ) = A + ^{( x 1 2 + ... + x 2 n 2 ) 1 / 2} satisfies the assumption of Theorem 6.

Case 3.

Assume that G is a group of Heisenberg type with Lie algebra ... = _{V 1} [ecedil]5; _{V 2} . Denote by { _{X 1} , ... , _{X l} } and { _{Y 1} , ... , _{Y k} } two orthonormal basis of _{V 1} and _{V 2} , respectively. The group of Heisenberg type G is a Lie group of step two with the underlying manifold ^{... l} × ^{... k} and the multiplication [figure omitted; refer to PDF] where x , ^{x [variant prime]} ∈ ^{... l} , y , ^{y [variant prime]} ∈ ^{... k} , and S ( x , ^{x [variant prime]} ) is a skew-symmetric bilinear function from ^{... l} × ^{... k} to ^{... k} with integer coefficients when expressed in the standard bases of ^{... l} and ^{... k} .

By [35], we know that there exist real constants _{c i j m} with i , j = 1 , ... , l , m = 1 , ... , k , such that [figure omitted; refer to PDF]

The gradient _{∇ G} is defined by [figure omitted; refer to PDF] Q = l + 2 k is the homogeneous dimension of G . We define a homogeneous norm function on G by [figure omitted; refer to PDF]

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 + ^{( x 1 2 + ... + x l 2 ) 1 / 2} , where A is a positive constant. Then W ( g ) ∈ _{B ∞} ( G ) . Then [figure omitted; refer to PDF] and | ∇ W ( g ) | =1. Clearly, | ∇ W ( g ) | ∈ _{B ∞} ( G ) . So W ( g ) ...5; A | ∇ W ( g ) | . Therefore, ρ ( x ) ...4; _{B 4 ρ 1} ( x ) and _{ρ 1} ( x ) = _{M 4} , where _{M 4} and _{B 4} are positive constants. Then the potential W ( g ) = A + ^{( x 1 2 + ... + x l 2 ) 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.

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