Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 483951, 13 pages

http://dx.doi.org/10.1155/2013/483951

Research Article

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

Yu Liu¹ and Jianfeng Dong²

¹School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

²Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 1 October 2013; Accepted 7 November 2013

Academic Editor: Yoshihiro Sawano

Copyright © 2013 Yu Liu and Jianfeng Dong. 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.

Abstract.

Assume that is a stratified Lie group and is the homogeneous dimension of . Let be the sub-Laplacian on and a nonnegative potential belonging to certain reverse Hölder class for . Let be a Schrödinger operator on the stratified Lie group . In this paper, we prove the boundedness of some integral operators related to , such as , , and () on the space BMOL(G).

1. Introduction

In recent years, some problems related to Schrödinger operators on the Euclidean space 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 , , 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 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 in [1]. Endpoint boundedness of Riesz transforms related to Schrödinger operators had been investigated in [11, 25]. Dong and Liu established the 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 estimates for the higher order Riesz transform in [27]. The aim of this paper is to obtain the 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 is called stratified if it is nilpotent, connected, and simple connected, and its Lie algebra admits a vector space decomposition such that for and . If is stratified, its Lie algebra admits a family of dilations, namely,

Assume that is a Lie group with underlying manifold for some positive integer . inherits dilations from : if and , we write

where . The map is an automorphism of . The left (or right) Haar measure on is simply , which is the Lebesgue measure on . For any measurable set , denote by the measure of . The inverse of any is simply . The group law has the following form:

for some polynomials in .

The number is called the homogeneous dimension of . We fix a homogeneous norm function on , which is smooth away from , where is the unit element of . Thus, for all for all , and if . The homogeneous norm induces a quasi-metric which is defined by . In particular, and . The ball of radius centered at is written by

The measure of is

where is a constant. In particular set for and .

Let be a basis for (viewed as left-invariant vector fields on ). Following [29], one can define a left invariant metric associated with which is called the Carnot-Caratheodory metric: let , and for every define

Let us define

The Carnot-Caratheodory metric is equivalent to the quasi-metric . From the results of Nagel et al. in [29], we deduce that there exists a constant such that, for any ,

It follows from [28] that , , are skew adjoint; that is, . Let be the sub-Laplacian on . This operator (which is hypoelliptic by Hörmander’s theorem in [30]) plays the same fundamental role on as the ordinary does on . The gradient operator is denoted by .

Definition 1. A nonnegative locally integrable function on is said to belong to the reverse Hölder class if there exists such that the reverse Hölder inequality

holds for every ball in .

Moreover, a locally bounded nonnegative function if there exists a positive constant such that

holds for every ball in .

Furthermore, it is easy to see that for any .

Let be a Schrödinger operator on the stratified Lie group , where is a nonnegative potential belonging to the reverse Hölder class for some . Denote by the higher order Riesz transform. Accordingly, denote by its dual operator.

It follows from [13] that the integral operators and are bounded on for and is bounded on for . Lin et al. introduced the Hardy type space related to the Schrödinger operator on the Heisenberg group in [14]. The dual space of is the type space investigated by Lin and Liu in [15]. and were also introduced as applications of results in [11, 22].

Next, we recall the definition of and . Since and , the Schrödinger operator generates a () semigroup . The maximal function with respect to the semigroup is given by

The Hardy space associated with the Schrödinger operator is defined as follows in terms of the maximal function mentioned above.

Definition 2. A function is said to be in if the semigroup maximal function belongs to . The norm of such a function is defined by

Assume for . The auxiliary function is defined by

It follows from Lemma 9 in Section 2 that for any .

The dual space of is the type space (cf. [22]). Let be a locally integrable function on and be a ball. Set

Definition 3. Let be a locally integrable function on . One says if

It is clear that and . Some remarks are given as follows.

Remark 4. Let . If , then there exists a positive constant :

The above inequality can be easily deduced by Lemma  3.1 in [11].

Similar to Remark  1 in [15], we conclude that a function if and only if there exist some suitable constants and depending on and satisfying whenever such that

Our main results are given as follows.

Theorem 5. Suppose for some . Then the operators and are bounded on the space .

Theorem 6. Suppose for some , for some , and and for some positive constant . Then operator is bounded on the space .

It shoud be noted that because the left invariant vector fields in 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 . Section 3 gives some estimates of kernel for some operators in this paper. Section 4 gives the proof of the boundedness of on the space . In Section 5, we establish the boundedness of . Finally, we give some examples for the potentials which satisfy the assumptions in Theorem 6 in different settings.

Throughout this paper, we will use to denote the positive constant, which is not necessarily the same at each occurrence and may depend on the dimension , and the constant in (9). By and , we mean that there exist some constants such that and , respectively.

2. Some Lemmas about the Auxiliary Function

In this section, we collect some known results about auxiliary function . We refer to [13] for the details. Throughout this section, unless otherwise indicated, we always assume that for some .

Lemma 7. is a doubling measure; that is, there exists a constant such that

Lemma 8. There exist constants such that

In particular, if .

Lemma 9. There exists such that, for ,

Lemma 10. If , then

Moreover,

Lemma 11. There exist and such that

Moreover, if , then there exists such that

3. Estimates for the Kernels

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

Let be the heat kernel of the semigroup ,  , associated with . Via Theorem 4.2 of [31], the following estimates hold true; that is, there exist positive constants and such that

where is the unit element of . Moreover, for any and , by using (3.5) in [13] we obtain

Let

Then for ,

Let be the fundamental solution of the operator for . In particular, we denote by . Then we have the following.

Proposition 12. There exists a positive constant such that

for .

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),

Firstly, for , we have

In addition, for any positive integer ,

Therefore,

Secondly, we have

Therefore, (32) holds true.

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

In the first place, we use the standard notations , , and for the spaces of functions with compact support, functions, and distributions on .

A measurable function on will be called homogeneous of degree if for all . Likewise, a distribution will be called homogeneous of degree if for all and . A distribution which is away from and homogeneous of degree will be called a kernel of type .

A differential operator will be called homogeneous of degree if for all and . Since is stratified, is homogeneous of degree if and only if . In particular, sub-Laplacian is homogeneous of degree 2. It follows from [32] that if is a kernel of type and is homogeneous of degree , then is a kernel of type .

For sub-Laplacian , is homogeneous of due to the fact that is homogeneous of , where . In addition, by using Proposition  1.7 in [28], we have

By [28], the left-invariant fields are formally skew-adjoint; that is,

Moreover, interacts with convolution in the following way:

Let be the fundamental solution of for . Then, Theorem  3.6 in [13] implies that

In particular, is the fundamental solution of Schrödinger operator , which satisfies the following.

(i)For each there exists such that

(ii)If , then for each there exists such that

where the above estimate can be deduced by Lemma  5.1 in [13].

The operator is defined by

where the kernel . Also, its adjoint operator is defined by

where the kernel . Since , then .

Moreover, we also need other estimates for the kernel and in order to prove the main results.

Lemma 13. Assume for . Let and . Then for any there exists such that

where is the constant appearing in Lemma 8 and is the constant appearing in Lemma 11.

Proof. Let be the solution of in the ball . By Lemma  3.2 in [13], we choose such that on , , and , where and are fixed constants, which are independent of and .

For ,

By (31) and (32) we have

By (9), the Calderón-Zygmund estimates, and Lemma 11,

Let and . Then is a solution of in . By the above inequality and Lemma 8, we immediately have

This finishes the proof of Lemma 13.

Lemma 14. Suppose for some and for some . Then

where and , if for some positive constant .

Proof. Note that . Let be the fundamental solution of . Then, we have

It follows that

By (41), we have

Set . Thus, we have

For ,

Firstly, by (43) and (30), it holds that

Secondly, via Lemmas 9 and 11, we similarly have

where .

Now, we turn to estimating . By Lemma 7, (43), and (30), we have that

if we choose large enough.

Then,

A similar argument implies that

The proof is completed.

Lemma 15. Suppose for some . Let be the conjugate index of .

(1) and are bounded on the space , where .(2) is bounded on the space for .

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

4. The Boundedness of and on

Proof of Theorem 5. To prove Theorem 5, we adopt the method used in the proof of Theorem  1.6 in [27].

Suppose and . Firstly, we suppose . Set . Then, we have

where denotes the characteristic function of the set . Since is bounded on , by Remark 4, we have

Let . By Lemma 8, . Set . By using (43) and Lemma 11, we obtain

where we choose large enough. Thus

The above argument also shows that is well defined on without the ambiguity of an additive constant.

Suppose . Set . Then, we can write

Via Lemma 8, for any . Similar to (66), we have

Note that

Set . By Hölder inequality and (9), we have, for any ,

where . Then, we have

Therefore, we prove that and , where is an absolute constant independent of .

Since , as an immediate consequence, is a bounded operator on .

The proof is completed.

5. The Boundedness of on

Proof of Theorem 6. Similar to the proof of Theorem  1.7 in [27], we show that Theorem 6 holds true.

Suppose and . Firstly, we suppose . Set . Then, we decompose as

Due to Lemma 15, we conclude that is bounded on . By Remark 4, we have

Let . By Lemma 8, . Set . Then, by Hölder inequality and Lemma 13, we have

where we choose sufficiently large.

Thus

The above argument also shows that are well defined on without the ambiguity of an additive constant.

Suppose . Set . Then, we can write as follows:

Note that for any . Similar to (66), we have

To complete the proof of the theorem, by Remark 4, we only need to prove that there exists a constant such that

The left side of (78) is bounded by

where is the dual operators of classical higher order Riesz transform . Let and . Note that . It is clear that ,

(cf. [28, Page 148]); therefore,

It follows that

By Lemma 14, we get

Since , . It is easy to see that

By the same argument and noting that , that is, ,

Because for , then . Thus, the last series converges. Using the fractional integral and the condition , we have

where and . Thus

It remains to show

Let , where satisfies . Note that . Set

Since are bounded on , by Remark 4, we have

Since is homogeneous of degree , then by (39),

Then, for , we have

Thus

For the third term, we have

Then

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 . At this time, the homogeneous norm on is defined as

for any .

Example 1. Let , where is a positive constant.

Following from [33], we know that if is a polynomial and , then belongs to . For , it is easy to see .

Therefore, .

If , then

where and are positive constants. Thus,

Therefore, . Clearly, . So . Therefore, and , where and are positive constants. Then the potential satisfies the assumption of Theorem 6.

Example 2. Let , where is a positive constant. By the above argument, we conclude that .

Then

Thus, . Clearly, . From [5], we know that . Furthermore, and , where and are positive constants. Then the potential satisfies the assumption of Theorem 6.

Case 2. Assume that. 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 is a Lie group with the underlying manifold and the multiplication

A basis for the Lie algebra of left-invariant vector fields on is given by

The gradient is defined by

is the homogeneous dimension of . We define a homogeneous norm function on by

By the equivalence of two quasi-norm in the finite dimension quasi-normed linear space, we conclude that, for any polynomial on and , belongs to . For , we also have .

Example 3. For any , let , where is a positive constant. Then . Then

and = 1. Clearly, . So . Therefore, and , where and are positive constants. Then the potential satisfies the assumption of Theorem 6.

Case 3. Assume that is a group of Heisenberg type with Lie algebra . Denote by and two orthonormal basis of and , respectively. The group of Heisenberg type is a Lie group of step two with the underlying manifold and the multiplication

where ,  , and is a skew-symmetric bilinear function from to with integer coefficients when expressed in the standard bases of and .

By [35], we know that there exist real constants with ,  , such that

The gradient is defined by

is the homogeneous dimension of . We define a homogeneous norm function on by

By the equivalence of two quasi-norm in the finite dimension quasi-normed linear space, we also conclude that, for any polynomial on and , belongs to . For , we also have .

Example 4. For any , let , where is a positive constant. Then . Then

and =1. Clearly, . So . Therefore, and , where and are positive constants. Then the potential 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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