High Order Fefferman-Phong Type Inequalities in Carnot Groups and Regularity for Degenerate Elliptic Operators plus a Potential

and Applied Analysis 3 Remark 5. The main difference between Theorem 4 and [6, Theorem 3.1] is clear; that is, the right-hand side term ‖Xu‖ L p (B) in [6] is replaced by ‖Xu‖ L p (B) here. Of course, the class involving V is not the same. We observe an important relation between the Stummel class here and the reverse Hölder class: if V ∈ B q ∩ L 1 (G), q > Q/2, then V ∈ S p (G), 1 < p ≤ 2q/Q. From it and Theorem 3, the following result follows. Theorem 6. Under the same assumptions on a ij as in Theorem 3, ifV ∈ B q ∩L 1 (G), q > Q/2, then for 1 < p ≤ 2q/Q and u ∈ C 0 (B), the estimate (11) holds. Remark 7. When q = Q/2 and V ∈ B Q/2 ⋂L 1 (G), by the important property of the B q class (see [21]), there exists ε > 0 such that V ∈ B Q/2+ε . Therefore, estimate (11) holds for 1 < p ≤ 1 + 2ε/Q and u ∈ C 0 (B). The paper is organized as follows. In Section 2 we recall some basic facts about Carnot groups and function spaces. In Section 3 we first give the proof of Theorem 4. Then combining with the known result in [12, Theorem 2] and proving an estimate with the potential V, we finish the proof of Theorem 3. The proof of Theorem 6 is given in Section 4. In Section 5, we restate Theorems 3 and 4 for the Euclidean case and elliptic operators without proofs. Dependence of Constants. Throughout this paper, the letter c denotes a positive constant which may vary from line to line. 2. Preliminaries 2.1. Background on Carnot Groups. We collect some facts about Carnot groups that will be needed in the sequel and refer the readers to [22–25] for further details. Definition 8 (Carnot group). A Carnot groupG = (R; ∘) is a simply connected nilpotent Lie group such that its Lie algebra g admits a stratification g = V 1 ⊕ V 2 ⊕ ⋅ ⋅ ⋅ ⊕ V r = ⊕ r j=1 V j , (14) where [V 1 , V j ] = V j+1 , j = 1, . . . , r − 1, and [V 1 , V r ] = {0}. Here r is called the step of G. For k = 1, . . . , r, letX 1,k , . . . , X mk ,k be a basis ofV k consisting of commutators of length k, where m k is the dimension of V k . The horizontal vector fields are ones in the first layer V 1 and for convenience, we set m 1 = m and denote X i,1 = X i , i = 1, . . . , m. Clearly, vector fields X 1 , . . . , X m satisfy Hormander’s condition [26]. Let {δ λ } λ>0 be a family of nonisotropic dilations on G defined by δ λ : G 󳨀→ G, δ λ (ξ) = (λξ 1 , λ 2 ξ 2 , . . . , λ r ξ r ) , (15) for any λ > 0 and ξ = (ξ 1 , . . . , ξ r ) = (x 1,1 , . . . , x m1 ,1 , . . . , x 1,r , . . . , x mr ,r ) ∈ G.The integerQ = ∑r k=1 km k is said to be the homogeneous dimension of G. In general, we assume Q ≥ 4. We call that a vector field X i,k ∈ g is left invariant if for any smooth function f one has X x i,k (f (y ∘ x)) = (X i,k f) (y ∘ x) , for any x ∈ G, (16) and X i,k is sth homogeneous if for any smooth function f, it follows that X i,k (f (δ λ (x))) = λ s (X i,k f) (δ λ (x)) , for any x ∈ G. (17) As in [23], the homogeneous norm of ξ ∈ G is defined by


Introduction and the Main Results
The classical   estimates for nondivergence elliptic operators with potentials of the form have been extensively investigated and many results have been proved; see [1][2][3][4][5] and so forth.In particular, when (  ) × = , the identity matrix, and  belongs to the reverse Hölder class   (/2 ≤  < ∞), Shen [2] established   (1 <  ≤ ) boundedness for the Schrödinger operator −Δ +  and showed that the range of  is optimal.It is noted that  ∈   ( > 1) means that  ∈   loc (R  ),  ≥ 0, and there exists a positive constant  such that the reverse Hölder inequality holds for every ball  in R  .More recently, when  ∈   (/2 ≤  < ∞), a priori   (R  ) (1 <  ≤ ) estimate for L in (1) with  coefficients has been deduced by Bramanti et al. [1] by using the representation formula for  in terms of L, which generalized the result in [2].The aim of this paper is to establish high order Fefferman-Phong type inequalities in Carnot groups and prove   regularity of degenerate elliptic operators plus a potential.
Sometimes we will call   () the Stummel modulus of .
is the special case of the function class in [7, page 56] with  = 2 and G = R  ( ≥ 5).Also, note that the function where  is the Carnot-Carathéodory distance (see Section 2).
Nondivergence degenerate elliptic operators similar to (3) including the form − ∑  =1  2  +  have been studied by some authors; see [8][9][10][11] and so forth.The local   estimate for operator (3) with the vanishing potential  = 0 on the homogeneous group has been verified by Bramanti and Brandolini [12].For the study of related operators, we refer to [13,14] and references therein.We will prove regularity for the operator  in (3) on G if   satisfy (4)-( 5) and   ∈   (G); see Theorem 3 below.Our methods are different from the Euclidean case by Bramanti et al. [1], where estimates of integral operators and their commutators were used as a main tool.
Since Fefferman [15] proved the well-known imbedding inequality with  belonging to the classical Morrey class  ,−2 , 1 <  ≤ /2, it has been extended to many more general settings and applied to infer regularity for partial differential operators; see [6,[16][17][18][19]  We mention that the homogeneous dimension  of G, the horizontal gradient , the second order horizontal gradient  2 , the horizontal Sobolev spaces  2, (G) and  2,  (G), the polynomial   (), and the reverse Hölder class   in our setting will be described in Section 2. Now we are in a position to state main results.Theorem 3.Under the assumptions ( 4)- (5), if   ∈   (G), 1 <  < ∞, then there exists a positive constant  = (, , , , G) such that, for any  ∈  ∞ 0 (G), it follows that where  in  depends only on the  moduli   of the coefficients   .Furthermore, (11) holds for  ∈

2,𝑝 𝑉 (G).
It is noted that the   estimates of the operators similar to (3) with discontinuous leading coefficients and bounded lower terms were obtained by Bramanti and Brandolini [12,20].Here the potential  in Theorem 3 may be unbounded on G.
The key for the proof of Theorem 3 is the following high order Fefferman-Phong type inequality.
, then there exists a first order polynomial   () such that, for any  ∈  ∞ (), one has where the positive constant  is independent of  and .Moreover, for any  ∈  ∞ 0 (), one has where  > 0 is independent of  and .
The above  2  is a set of      for all ,  = 1, . . ., .We will define  2  precisely in Section 2.
Remark 5.The main difference between Theorem 4 and [6, Theorem 3.1] is clear; that is, the right-hand side term ‖‖   () in [6] is replaced by ‖ 2 ‖   () here.Of course, the class involving  is not the same.
The paper is organized as follows.In Section 2 we recall some basic facts about Carnot groups and function spaces.In Section 3 we first give the proof of Theorem 4. Then combining with the known result in [12,Theorem 2] and proving an estimate with the potential , we finish the proof of Theorem 3. The proof of Theorem 6 is given in Section 4. In Section 5, we restate Theorems 3 and 4 for the Euclidean case and elliptic operators without proofs.
Dependence of Constants.Throughout this paper, the letter  denotes a positive constant which may vary from line to line.

Preliminaries
2.1.Background on Carnot Groups.We collect some facts about Carnot groups that will be needed in the sequel and refer the readers to [22][23][24][25] for further details.Definition 8 (Carnot group).A Carnot group G = (R  ; ∘) is a simply connected nilpotent Lie group such that its Lie algebra g admits a stratification where Here  is called the step of G.
The homogeneous degree of monomial   is the sum =1  , and the homogeneous degree of () is max{() |   ̸ = 0}.From [28], the left invariant vector fields  1 , . . .,   can induce the corresponding Carnot-Carathéodory distance : for any  > 0, let () be the set of absolutely continuous curves  : [0, 1] → G such that for a.e. ∈ [0, 1], By [29], it is known that for  large enough the set () is nonempty.We define the Carnot-Carathéodory distance by It is well known that the distance  is equivalent to the pseudo distance  G (see [28]).In this paper, we will mainly use the Carnot-Carathéodory distance  to study regularity of (3).
Associated with the distance, we define the metric ball of center  and radius  in G by The Lebesgue measure in R  is the Haar measure on G ([25, page 619]).Due to (15), one has where |  ()| is the measure of   () and   is a positive constant.

Function
with the norm where Analogously to [1], the space Definition 10 (Reverse Hölder class).(1) A nonnegative locally   integrable function () on G is said to belong to the reverse Hölder class   (1 <  < ∞), if there exists a positive constant  such that for any metric ball  in G.

Proofs of Theorems 3 and 4
We first prove Theorem 4 and then prove Theorem 3.
Proof of Theorem 3. We consult the way in [1, pages 342-343] and apply our previous results.By the basic theorem on the partition of unity (e.g., see [32, we obtain (11).

Proof of Theorem 6
Several preliminary conclusions are necessary.( The result is proved.
Proof of Theorem 6.By Lemma 16 and Theorem 3, we immediately obtain Theorem 6.

Results to the Euclidean Case and Elliptic Operators
Here for convenience of readers, we restate Theorems 3 and 4 corresponding to the Euclidean case but omit their proofs because the proofs are analogous to Theorems 3 and 4. It will be assumed for the leading coefficients   in (1) that

Remark 17 .
In order to assure the convergence of the series∑ ∞ =0 (2 − ) 2−/in the proof of Lemma 16, we require the assumption  ≤ 2/, which leads to the range of  in Theorem 6 smaller than [1, Theorem 1].