^{1}

^{2}

^{1}

^{2}

Assume that _{L}

In recent years, some problems related to Schrödinger operators on the Euclidean space

Firstly, we recall some basic facts of stratified Lie groups (cf. [

The number

Let

The Carnot-Caratheodory metric

It follows from [

A nonnegative locally

Moreover, a locally bounded nonnegative function

Furthermore, it is easy to see that

Let

It follows from [

Next, we recall the definition of

A function

Assume

The dual space of

Let

It is clear that

Let

Similar to Remark 1 in [

Our main results are given as follows.

Suppose

Suppose

It shoud be noted that because the left invariant vector fields in

This paper is organized as follows. In Section

Throughout this paper, we will use

In this section, we collect some known results about auxiliary function

There exist constants

There exists

If

There exist

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

Let

Let

There exists a positive constant

Equations (

By (

Moreover, we need some other basic facts of fundamental solutions for sub-Laplacian on the stratified Lie group

In the first place, we use the standard notations

A measurable function

A differential operator

For sub-Laplacian

By [

Let

For each

If

The operator

Moreover, we also need other estimates for the kernel

Assume

Let

For

By (

This finishes the proof of Lemma

Suppose

Note that

For

Firstly, by (

Now, we turn to estimating

Then,

A similar argument implies that

The proof is completed.

Suppose

The above lemmas hold true due to Theorem 4.1 and Theorem

To prove Theorem

Suppose

Let

Suppose

Therefore, we prove that

Since

The proof is completed.

Similar to the proof of Theorem 1.7 in [

Suppose

Due to Lemma

Let

Thus

Suppose

Since

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

Assume

for any

Let

Following from [

Therefore,

If

Therefore,

Let

Then

Thus,

Assume that

By the equivalence of two quasi-norm in the finite dimension quasi-normed linear space, we conclude that, for any polynomial

For any

Assume that

By [

The gradient

By the equivalence of two quasi-norm in the finite dimension quasi-normed linear space, we also conclude that, for any polynomial

For any

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

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.