A two-weight Sobolev inequality for Carnot-Carathéodory spaces

Let X={X1,X2,…,Xm}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X = \{X_1,X_2, \ldots ,X_m\}$$\end{document} be a system of smooth vector fields in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb R}^n}$$\end{document} satisfying the Hörmander’s finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Carathéodory space G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb G$$\end{document} associated to system X1∫BRK(x)dx∫BR|u|tK(x)dx1/t≤CR1∫BR1K(x)dx∫BR|Xu|2K(x)dx1/2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( \frac{1}{\int _{B_R} K(x)\; dx} \int _{B_R} |u|^{t} K(x) \; dx \right) ^{1/t} \le C\, R \left( \frac{1}{\int _{B_R}\frac{1}{K(x)} \; dx} \int _{B_R} \frac{|X u|^2}{K(x)} \; dx \right) ^{1/2}, \end{aligned}$$\end{document}where Xu denotes the horizontal gradient of u with respect to X. We assume that the weight K belongs to Muckenhoupt’s class A2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2$$\end{document} and Gehring’s class Gτ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{\tau }$$\end{document}, where τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is a suitable exponent related to the homogeneous dimension.


Introduction
This paper is devoted to study some basic functional and geometric properties of general families of vector fields that include the Hörmander's type as a special case. Similar to their Euclidean counterparts, such properties play an important role in the analysis of the relevant differential operators (both linear and nonlinear). We are concerned with a two-weight Sobolev type inequality on G, where G denotes the Carnot-Carathèodory space ( , d) (suitably defined -see Sect. 2.1) associated to a system of smooth vector fields X = {X 1 , X 2 , . . . , X m } on R n satisfying the Hörmander's finite rank condition. This fact introduces a kind of degeneracy different from that Euclidean one. Here, is an open (Euclidean) bounded and connected set of R n , n ≥ 2, and d is the metric generated by X .
Let u ∈ Lip (G). We denote by Xu = (X 1 u, . . . , X m u) the horizontal gradient of u with respect to the system X , where X j plays the role of the first order differential operator acting on u given by for j = 1, . . . , m.
Set |Xu| = m j=1 (X j u) 2 1/2 , the length of the horizontal gradient of u. We refer to [5,12] for more details. In our paper we prove a two-weight Sobolev type inequality where the weights K and K −1 form a 2-admissible pair (K −1 , K ), namely 1) K is locally doubling in and K −1 belongs to A 2 (G). 2) Given a compact set V ⊂ there exist t > 2 and C ≥ 1 such that, for every ball B with center in V and 0 < r < 1, it holds Note that inequality (1.1) is the Chanillo-Wheeden condition (see [8]), with exponents t and 2, adapted to the Carnot-Carathèodory geometry (see [18]).
Our main result reads as follows.
where C is the constant in (1.1), 2 < q < t, and B R denotes the ball centered at the origin with radius R > 0. Here, [K −1 ] A 2 and [K ] A 2 stand for A 2 constants of K −1 and K , respectively.
By properties of Muchenoupt's class A p (G), we have that since K ∈ A 2 (G), then K −1 ∈ A 2 (G). Moreover, by [12,Theorem 4.8], the assumption that K belongs to n+2−Q , guarantees that the pair K −1 , K satisfies condition (1.1). Thus, one deduces that K −1 , K is a 2-admissible pair in . We emphasize that the 2-admissible property of K −1 , K will be used in the proof of Theorem 1.1.
The tools used to obtain inequality (1.2) are the classical ones of the Euclidean case. Neverthless, here we deal with a degeneracy into the geometry due to the presence of a differential operator Xu different from the classical gradient ∇u. In particular, this fact causes a change of metric on R n and consequently some of the results valid for Euclidean metric have been enlarged to Carnot-Carathèodory metric.
In the Euclidean setting, Theorem 1.1 generalizes similar result contained in [2], where the authors prove a weighted Sobolev inequality of the same type as (1.2), with the weight K (x) related to the function |u| t and the weight K −1 (x) to the gradient |∇u| 2 .
The result of Theorem 1.1 is a particular case of that contained in [12,Corollary 3.4] with v(x) replaced by K (x) and w(x) replaced by K −1 (x). In [12] the authors show the following more general weighted Sobolev inequality where 1 < p < t < ∞, C > 0 is a constant, and (w, v) is a p-admissible pair in . Herein, we prove inequality (1.2) by using different techniques which rely upon a combination of an estimate for fractional integral of first order with other some properties of A 2 (G) and G τ (G) classes. Moreover, in contrast with the result in [12,Corollary 3.4], we give the explicit value of constant C in our inequality (1.2).
Our paper is organized as follows. In Sect. 2 we give some preliminary results. Actually, in Sect. 2.1 we recall definition and basic properties of Hörmander vector fields, including Carnot-Carathèodory spaces; in Sect. 2.2 we discuss the theory of Muckenhoupt's and Gehring's weights. In Sect. 3 we present the machinery we need to work with the inequality we are interested in. Finally, we prove our main theorem.

Carnot Carathéodory spaces
Let be an open (Euclidean) bounded and connected subset in R n , with n ≥ 2. Let X = {X 1 , . . . , X m } be a system of C ∞ vector fields on R n .
We denote by Lie [X 1 , . . . , X m ] the Lie algebra generated by X 1 , . . . , X m and by their commutators of any order. We say that a field Z belongs to Lie [X 1 , . . . , X m ] if and only if Z is a finite linear combination of terms of this type We define, for any fixed x ∈ R n , the Lie rank as Henceforth, we assume that X satisfies the following Hörmander's finite rank condition in namely there exist a neighborhood 0 of and m ∈ N such that the family of commutators of the vector fields in X up to length m span R n at every point of 0 .
Let C X be the family of absolutely continuous curves γ : We define Carnot-Carathéodory distance d as Note that, owing to Hörmander's finite rank condition (2.1), d is a metric. This fact is not true in general. The Carnot-Carathéodory space G is the pair ( , d) associated The basic properties of these balls have been obtained by Nagel, Stein and Wainger in [25]. In particular, in the following proposition, the authors prove that the metric d is locally Hölder continuous with respect to the Euclidean metric.
Thanks to Proposition 2.1, the topology of Carnot-Carathéodory induced by d on coincides with the Euclidean ones. In the sequel, all the distances will be understood in the sense of the Carnot-Carathéodory metric d. In particular, all the balls will be defined with respect to d.
We denote by | · | the Lebesgue measure in (R n , d) and, by Note that the Lebesgue measure locally satisfies the following doubling condition (see e.g. [25]).

Proposition 2.2 For any compact set E
Let Y 1 , . . . , Y l be the collection of the X j 's and of those commutators which are needed to generate R n . To each Y i it is associated a formal "degree" deg(Y i ) ≥ 1, namely the corresponding order of the commutator. Set I = (i 1 , . . . , i n ), with 1 ≤ i j ≤ l, an n-tuple of integers. We define (see also [25]) the degree of I as For a given compact set E ⊂ R n , we define Q by We define by Just to give an idea, we consider in R 3 the system (see [13]) It is easy to see that l = 4 and Moreover, Q(x) = 3 for all x = 0, whereas for any compact set E containing the origin, Let Y be a metric space and μ a Borel measure in Y . Assume μ finite on bounded sets and satisfying the doubling condition on every open, bounded subset in Y . We say that Q is a homogeneous dimension relative to , if there exists a positive constant C such that for any ball B 0 having center in and radius R 0 < diam, and any ball B centered in x 0 ∈ B 0 and having radius R ≤ R 0 . It is well known that the doubling condition implies the existence of the homogeneous dimension Q. However, Q is not unique and it may change with . Obviously, any Q ≥ Q it is also a homogeneous dimension.
For a bounded open set containing a family of vector fields satisfying the Hörmander's finite rank condition, the homogeneous dimension of the Carnot-Carathéodory space G, defined with the Lebesgue measure, is given by Q = log 2 C d , where C d is the doubling constant.

Some properties of A p and G q classes
In this section, we recall a few properties of Muckenhoupt's and Gehring's classes (see [22,24,27,28]).
We recall that a weight is a positive function in L 1 loc (R n ). We say that a weight w is doubling in if where the constant C is independent by the ball B ⊂ .
We say that w is locally doubling in if for each compact set V ⊂ andR > 0 there exists where the ball B has center in V and radius R <R and 2B is the ball concentric with B and having radius 2-times that of B.
We say that a weight w belongs to the class A p (G) (briefly, w ∈ A p (G)) for some If a weight belongs to a class A p , it is called a Muckenhoupt weight.
A weight w is said to belong to the class G q (G) (briefly, w ∈ G q (G)) for some q ∈ (1, +∞) if If a weight belongs to a class G q , it is called a Gehring weight.
Here, we recall some properties of A p classes with respect to dyadic cubes which we will be used to prove Theorem 1.1.
We use a grid D h of dyadic cubes Q, which are "almost balls", where h is a large negative integer which indexes the edgelengths l(Q) of the smallest cubes Q ∈ D h . In other words, the smallest edgelengths are λ h for an appropriate geometric constant λ > 1 and each cube in the grid has edgelength λ k for some k ≥ h.
In particular, we will make use of a grid of dyadic cubes in the ball B R in the same spirit of [29], where it is proved that there exists a constant λ > 1 such that, for every h ∈ Z, there are points x k j ∈ B R and a family of cubes D h = {Q k j } for j ∈ N and k = h, h + 1, . . . such that We call the family D = h∈Z D h a dyadic cube decomposition of B R and we refer to its sets as dyadic cubes which will be denoted by Q. We observe explicitly that being D a decomposition of B R , then any dyadic cube Q ∈ D is contained in the ball B R .
By [29], making use of (2.3), one can deduce the following lemma.

Lemma 2.3
Let w ∈ A 2 (G) and let Q and Q 0 dyadic cubes in R n such that Q ⊂ Q 0 . If β > 1, then Another important property of A p (G) classes is given by the following proposition (see [20], [30,Chapter 5,p. 195]).

Some preliminary estimates
In order to prove our main theorem, let us prove some preliminary results. The first lemma yields an estimate of the fractional integral of order 1 (see e.g. [5]). In general, the fractional integral of order α ∈ (0, Q) of a locally integrable function g in R n is defined as (2.6) Lemma 2.5 Let g ∈ L 1 loc (G) and assume that g ≥ 0. Then

7)
where c 0 is an absolute constant.
Proof Thanks to a dyadic cube decomposition, we discretize the operator I 1 where the last inequality follows by |Q| = l(Q) Q and, moreover, by B(x, l(Q)) ⊂ 3Q if x ∈ Q. Hence, inequality (2.7) is proved.

Let us consider a Dirichlet problem in this form
where G denotes the canonical sub-Laplacian operator defined as G = m j=1 X 2 j , with {X 1 , ..., X m } the family of smooth vector fields on R n satisfying the Hörmander's finite rank condition.
In [21,Theorem 3.2], the authors proved that, if f ∈ F α (B R ), then there exists a unique solution ϕ ∈ C 2 (B R ) ∩ C 1 (B R ) to problem (2.8), represented by the formula (2.10) Here, x (y) is the fundamental solution of the sub-Laplacian. Thanks to [21, Theorem 2.2], there exists a positive constant c such that (2.11) Consequently, combining (2.10) and (2.11) yields (2.12) The next lemma gives an estimate of the gradient of the solution to problem (2.8) through the fractional integral of order 1. where I 1 ( f ) denotes the fractional integral of order 1 of f .

Proof of main result
The following preliminary lemma will be use in the proof of Theorem 1.1.
Proof For each h ∈ Z, we set Note that, if Q is any dyadic cube such that |u| t−1 K (x) is not identically zero on Q, then Q belongs to only one collection C h . For each h ∈ Z, let us build the collection {Q h j } j of pairwise disjoint maximal dyadic cubes (maximal with respect to inclusion) in C h . If Q ∈ C h , then there exists j ∈ N such that Q ⊂ Q h j . Note also that for each fixed h, the cubes Q h j are disjoint with respect to j. Nevertheless, they may not be disjoint for different values of h.

By (3.3) and (3.4), we deduce
where Owing to (3.5) and (3.7), where (3.9) where the first inequality is a consequence of the fact that h a q t h ≤ h a h q t , the third one holds because, fixed h ∈ Z, Q h j are disjoint in j, the fourth one is due to Fubini's type Theorem. To conclude the proof, we have to evaluate the quantity Then, by (3.9) and (3.11), we obtain with u ∈ C 1 0 (B R ). By Lemma 2.6, we get where c is a positive constant. Thanks to Lemma 2.5, it follows that 13) where c 0 is an absolute constant.
Combining (3.12) and (3.13) yields where C 6 = c c 0 . Note that the last inequality is the consequence of the fact that B R ∩ Q = Q. By (2.5), Coupling inequalities (3.14) and (3.15) tells us that (3.16) By (3.16), the following chain of inequality holds where the first inequality follows by Chanillo-Wheeden condition (1.1), the second one holds since 1/t = 1 − 1/t , the third one is due to Hölder's inequality, for 2 < q < t, and the fifty one comes from the fact that D is a decomposition of B R . Here, constant C 7 = C 6 C K −1

Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.