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Abstract and Applied Analysis
Volume 2014, Article ID 274859, 8 pages
http://dx.doi.org/10.1155/2014/274859
Research Article

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

Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710129, China

Received 4 June 2014; Accepted 13 October 2014; Published 10 November 2014

Academic Editor: Sung G. Kim

Copyright © 2014 Pengcheng Niu and Kelei Zhang. 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

Let be the basis of space of horizontal vector fields in a Carnot group . We prove high order Fefferman-Phong type inequalities in . As applications, we derive a priori estimates for the nondivergence degenerate elliptic operators with coefficients and a potential belonging to an appropriate Stummel type class introduced in this paper. Some of our results are also new even for the usual Euclidean space.

1. 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 [15] and so forth. In particular, when , the identity matrix, and belongs to the reverse Hölder class , Shen [2] established boundedness for the Schrödinger operator and showed that the range of is optimal. It is noted that means that , , and there exists a positive constant such that the reverse Hölder inequality holds for every ball in . More recently, when , a priori estimate for in (1) with coefficients has been deduced by Bramanti et al. [1] by using the representation formula for in terms of , 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.

Let be horizontal vector fields in a Carnot group , (see Section 2.1). In this paper we consider the nondivergence degenerate elliptic operator of the kind where the leading coefficient satisfies for , and there exists a constant such that, for any and , furthermore, we assume which shows that, for , Here . denotes a metric ball of radius and center associated with the Carnot-Carathéodory distance (see Section 2) by

As to the potential , inspired by Di Fazio and Zamboni [6, Definition 2.4], we introduce the following Stummel type class .

Definition 1 (Stummel type class). Let , , . One says that , if for every , is finite and where is the Carnot-Carathéodory distance; see Section 2.

Sometimes we will call the Stummel modulus of .

Remark 2. We note that and is the special case of the function class in [7, page 56] with and . Also, note that the function on belongs to the classes for , where is the Carnot-Carathéodory distance (see Section 2).

Nondivergence degenerate elliptic operators similar to (3) including the form have been studied by some authors; see [811] and so forth. The local estimate for operator (3) with the vanishing potential 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 if satisfy (4)-(5) and ; 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 , , it has been extended to many more general settings and applied to infer regularity for partial differential operators; see [6, 1619] and so forth. One of the main jobs of this paper is to establish a high order Fefferman-Phong type inequality in Carnot groups (see Theorem 4), which is motivated by Di Fazio and Zamboni [6, Theorem 3.1]. So far as we know, there is not any result in literature on high order Fefferman-Phong inequalities. Using this inequality and proving several estimates with the potential, a priori estimate for is obtained.

We mention that the homogeneous dimension of , the horizontal gradient , the second order horizontal gradient , the horizontal Sobolev spaces and , 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 , then there exists a positive constant such that, for any , it follows that where in depends only on the moduli of the coefficients . Furthermore, (11) holds for .

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 .

The key for the proof of Theorem 3 is the following high order Fefferman-Phong type inequality.

Theorem 4. Let be any metric ball in . If , then there exists a first order polynomial such that, for any , one has where the positive constant is independent of and . Moreover, for any , one has where is independent of and .

The above is a set of for all . We will define 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 here. Of course, the class involving is not the same.

We observe an important relation between the Stummel class here and the reverse Hölder class: if , , then , . From it and Theorem 3, the following result follows.

Theorem 6. Under the same assumptions on as in Theorem 3, if , , then for and , the estimate (11) holds.

Remark 7. When and , by the important property of the class (see [21]), there exists such that . Therefore, estimate (11) holds for and .

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.

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 [2225] for further details.

Definition 8 (Carnot group). A Carnot group is a simply connected nilpotent Lie group such that its Lie algebra admits a stratification where , , and . Here is called the step of .

For , let be a basis of consisting of commutators of length , where is the dimension of . The horizontal vector fields are ones in the first layer and for convenience, we set and denote , . Clearly, vector fields satisfy Hormander’s condition [26].

Let be a family of nonisotropic dilations on defined by for any and . The integer is said to be the homogeneous dimension of . In general, we assume . We call that a vector field is left invariant if for any smooth function one has and is th homogeneous if for any smooth function , it follows that

As in [23], the homogeneous norm of is defined by It is natural to define the pseudo distance by the homogeneous norm where is the inverse of . A polynomial on [24] is a function which can be expressed in exponential coordinates (see, e.g., [22, Section ] and [27]) as where are multi-indices and The homogeneous degree of monomial is the sum and the homogeneous degree of is .

From [28], the left invariant vector fields can induce the corresponding Carnot-Carathéodory distance : for any , let be the set of absolutely continuous curves such that for a.e. , 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 (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 by The Lebesgue measure in is the Haar measure on ([25, page 619]). Due to (15), one has where is the measure of and is a positive constant.

2.2. Function Spaces

Denote , , , , and .

Definition 9 (Horizontal Sobolev space). For any and a domain , one defines the Horizontal Sobolev spaces by with the norm where , .

Analogously to [1], the space is the closure of in the norm

Definition 10 (Reverse Hölder class). (1) A nonnegative locally integrable function on is said to belong to the reverse Hölder class , if there exists a positive constant such that for any metric ball in .
(2) Let a.e. and ; one says if there exists a positive constant such that

It is easy to see that , .

3. Proofs of Theorems 3 and 4

We first prove Theorem 4 and then prove Theorem 3.

3.1. Proof of Theorem 4

The following lemma is due to Lu and Wheeden [30, 31]. It will play a key role in our proof.

Lemma 11. Let be a metric ball in . If , then there exists a first order polynomial such that, for a.e. , where the positive constant is independent of , , and . Moreover, if , then for a.e. , where is independent of , , and .

Proof of Theorem 4. By (31), Fubini’s Theorem, and Hölder’s inequality, we have Now a computation yields Therefore, It implies (12).
By using (32) and repeating the argument above for (12), we immediately obtain (13).

3.2. Proof of Theorem 3

Let us recall estimates for the operator by Bramanti and Brandolini [12, Theorem 2].

Lemma 12. Under the assumptions (4) and (5), for every , there exist positive constants and , where denotes the moduli of coefficients , such that, for any and sprt u   ( any metric ball of radius ),

Based on it and Theorem 4, we have the following estimates for in (3).

Lemma 13. Under the assumptions (4) and (5), for every and , there exist positive constants and such that, for any and sprt   ,

Proof. By Theorem 4, Applying Lemma 12, it follows that Choosing such that , we derive (37).

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, page 66]), there exists a partition of unity of nonnegative functions in such that with in Lemma 13 and a family of metric balls satisfying the finite overlapping property. We have from Lemma 13 that Combining with the interpolation inequality (see [12, Proposition 2]) we obtain (11).

4. Proof of Theorem 6

Several preliminary conclusions are necessary.

Lemma 14. If  , , then there exists a constant such that, for any and ,

Proof. By Hölder’s inequality and (29), it yields

Remark 15. If we take and , then Lemma 14 gives the version in [2, Lemma 1.2].

Lemma 16. If , , then , .

Proof. For any , it follows that By (42), it yields Also, we have Therefore combining (45) and (46) gets The result is proved.

Proof of Theorem 6. By Lemma 16 and Theorem 3, we immediately obtain Theorem 6.

Remark 17. In order to assure the convergence of the series in the proof of Lemma 16, we require the assumption , which leads to the range of in Theorem 6 smaller than [1, Theorem 1].

5. 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

for all and there exists a positive constant such that, for any and ,

; that is, for , where .

A function for means that, for each , is finite and

Theorem 18. Under assumptions and , if , , then there exists a positive constant such that, for any , one has where depends only on the moduli of the coefficients .

Remark 19. If or , the theory of (1) with discontinuous leading coefficients was intensively studied and the result was proved in [3336] and so forth. Bramanti et al. [1] obtained a prior estimate for (1) with and . Here and .

Theorem 20. Let be any ball in . If  , then there exists a first order polynomial in such that, for any , one has where the positive constant is independent of and . Moreover, for any , one has where is independent of and .

Conflict of Interests

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

Acknowledgments

The first author was supported by the National Natural Science Foundation of China (Grant no. 11271299) and Natural Science Foundation Research Project of Shaanxi Province (Grant no. 2012JM1014). The authors are very grateful to the anonymous referee who read carefully the manuscript and offered valuable suggestions.

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