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

Assume that G is a stratified Lie group andQ is the homogeneous dimension of G. Let −Δ be the sub-Laplacian on G andW ̸ ≡ 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∇, LW, and L (−Δ) on the space BMO L (G).


Introduction
In recent years, some problems related to Schrödinger operators on the Euclidean space R  with nonnegative potentials have been investigated by a number of scholars (cf.[1][2][3][4][5][6][7][8][9][10][11][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][14][15][16][17][18][19][20][21][22][23][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  1 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.
(1) Assume that  is a Lie group with underlying manifold R  for some positive integer . inherits dilations from g: if  ∈  and  > 0, we write where 1 ≤  1 ≤ ⋅⋅⋅ ≤   .The map  →    is an automorphism of .The left (or right) Haar measure on  is simply  1 ⋅ ⋅ ⋅   , which is the Lebesgue measure on g.For any measurable set  ⊆ , denote by || the measure of .
Let  = { 1 , . . .,   } be a basis for  1 (viewed as leftinvariant 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  > 0 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  = () > 1 such that, for any , ℎ ∈ , It follows from [28] that   ,  = 1, 2, . . ., , are skew adjoint; that is,  *  = −  .Let Δ = − ∑  =1  2  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 R  .The gradient operator ∇ is denoted by ∇ = ( 1 , . . .,   ).Definition 1.A nonnegative locally   integrable function  on  is said to belong to the reverse Hölder class   (1 <  < ∞) if there exists  > 0 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 1 <  < ∞.
Let  = −Δ+ be a Schrödinger operator on the stratified Lie group , where  ̸ ≡ 0 is a nonnegative potential belonging to the reverse Hölder class   for some  ≥ /2.Denote by R  = ∇ 2  −1 the higher order Riesz transform.Accordingly, denote by R =  −1 ∇ 2 its dual operator.
It follows from [13] that the integral operators  −1 and (−Δ) −1 are bounded on   () for 1 ≤  ≤  and R  is bounded on   () for 1 <  ≤ .Lin et al. introduced the Hardy type space  1  () related to the Schrödinger operator  on the Heisenberg group  in [14].The dual space of  1  () is the  type space   () investigated by Lin and Liu in [15]. 1  () and   () were also introduced as applications of results in [11,22].
The norm of such a function is defined by Assume  ∈   for  > /2.The auxiliary function (, ) is defined by It follows from Lemma 9 in Section 2 that 0 < (, ) < ∞ for any  ∈ .
The dual space of  1  () is the  type space   () (cf.[22]).Let  be a locally integrable function on  and  = (, ) be a ball.Set Our main results are given as follows.
It shoud be noted that because the left invariant vector fields in  1 ⊆ 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 (, ).Section 3 gives some estimates of kernel for some operators in this paper.Section 4 gives the proof of the boundedness of  −1 ,  −1 (−Δ) on the space   ().In Section 5, we establish the   boundedness of  −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  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 1/ ≤ / ≤  and  ≤   , respectively.

Estimates for the Kernels
In this section we will investigate some necessary estimates about the kernel of the operators in the paper.
We only need to show that (32) holds true, because ( 31) and ( 33) can be proved similarly.By ( 26) and ( 28), Firstly, for  1 , 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 D, E, and D  for the spaces of  ∞ functions with compact support,  ∞ functions, and distributions on D.
A measurable function  on  will be called homogeneous of degree  ( ∈ C) if  ∘   =    for all  > 0. Likewise, a distribution  ∈ D  will be called homogeneous of degree  if ⟨,  ∘   ⟩ =  −− ⟨, ⟩ for all  ∈ D and  > 0. 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  ∈ D and  > 0. Since  is stratified,  ∈ g is homogeneous of degree  if and only if  ∈   .In particular, sub-Laplacian Δ 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  − .
(i) For each  ∈ N there exists   > 0 such that (ii) If  ∈   , then for each  ∈ N there exists   > 0 such that where the above estimate can be deduced by Lemma 5.1 in [13].
Lemma 15.Suppose  ∈   for some  ≥ /2.Let   be the conjugate index of .
The above lemmas hold true due to Theorem 4.1 and Theorem A in [13], respectively.
The proof is completed.

Example
In this section, we give some examples for the potentials which satisfy the assumption in Theorem 6.
Case 1. Assume  = R  .At this time, the homogeneous norm | ⋅ | on  is defined as , where  is a positive constant.
Following from [33], we know that if () is a polynomial and  > 0, then where  1 and  2 are positive constants.Thus, where  is a positive constant.By the above argument, we conclude that  ∈  ∞ (R  ).