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The Higher Order Riesz Transform and BMO Type Space Associated with Schrödinger Operators on Stratified Lie Groups
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).
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  and Li  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 . 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 . 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 . 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. ). 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 , 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 , we deduce that there exists a constant such that, for any ,
It follows from  that , , are skew adjoint; that is, . Let be the sub-Laplacian on . This operator (which is hypoelliptic by Hörmander’s theorem in ) 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  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 . The dual space of is the type space investigated by Lin and Liu in . 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
The dual space of is the type space (cf. ). 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 .
Similar to Remark 1 in , 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  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  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 , 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  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 . 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 ).
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  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 , we have
By , 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  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 .
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.
Proof. Let be the solution of in the ball . By Lemma 3.2 in , 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
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.
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 , 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 .
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 , 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