1 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 \(\mathbb G\), where \(\mathbb G\) denotes the Carnot-Carathèodory space \((\Omega , d)\) (suitably defined - see Sect. 2.1) associated to a system of smooth vector fields \(X = \{X_1,X_2, \ldots ,X_m\}\) on \({{\mathbb R}^n}\) satisfying the Hörmander’s finite rank condition. This fact introduces a kind of degeneracy different from that Euclidean one. Here, \(\Omega \) is an open (Euclidean) bounded and connected set of \({{\mathbb R}^n}\), \(n\ge 2\), and d is the metric generated by X.

Let \(u\in \mathrm {Lip}\,(\mathbb G)\). We denote by \(X u= (X_1 u, \ldots , X_m u)\) 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

$$\begin{aligned} X_j u(x) = \langle X_j(x), \nabla u(x) \rangle \qquad \hbox {for}\;\; j=1, \ldots , m. \end{aligned}$$

Set

$$\begin{aligned} |X u|=\Big ( \sum _{j=1}^{m}(X_j u)^2 \Big )^{1/2}, \end{aligned}$$

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. 1)

    K is locally doubling in \(\Omega \) and \(K^{-1}\) belongs to \(A_2(\mathbb G)\).

  2. 2)

    Given a compact set \(V\subset \Omega \) there exist \(t > 2\) and \(\overline{C}\ge 1\) such that, for every ball B with center in V and \(0<r<1\), it holds

    $$\begin{aligned} r \left( \frac{\int _{rB} K(x)\; dx}{\int _B K(x)\; dx}\right) ^{1/t} \le \overline{C} \left( \frac{\int _{rB} K^{-1}(x)\; dx}{\int _B K^{-1}(x)\; dx} \right) ^{1/2}. \end{aligned}$$
    (1.1)

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.

Theorem 1.1

Let K be in \(A_2(\mathbb G)\cap G_{\tau }(\mathbb G)\) with \(\tau =1+\frac{2(Q-1)}{n+2-Q}\). Let \(t>2\). Then, for every \(u\in \mathcal{{C}}^1_0(B_R)\), there exists a constant \(C\ge 1\) such that

$$\begin{aligned} \left( \frac{1}{\int _{B_R} K(x)\; dx} \int _{B_R} |u|^{t} K(x) \; dx \right) ^{1/t} \le C\, R \left( \frac{1}{\int _{B_R} \frac{1}{K(x)}\; dx} \int _{B_R} \frac{|X u|^2}{K(x)} \; dx \right) ^{1/2} \end{aligned}$$
(1.2)

with

$$\begin{aligned} C= c(Q, n, t, q)\, \overline{C}\, [K^{-1}]_{A_2}^{\frac{1}{2}}\,[K]_{A_2}^{\frac{1}{t'}- \frac{1}{q'}}, \end{aligned}$$

where \(\overline{C}\) 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 \(A_p(\mathbb G)\), we have that since \(K\in A_2(\mathbb G)\), then \(K^{-1}\in A_2(\mathbb G)\). Moreover, by [12, Theorem 4.8], the assumption that K belongs to \(A_2(\mathbb G)\cap G_{\tau }(\mathbb G)\), with \(\tau =1+\frac{2(Q-1)}{n+2-Q}\), guarantees that the pair \(\left( K^{-1}, K\right) \) satisfies condition (1.1). Thus, one deduces that \(\left( K^{-1}, K\right) \) is a 2-admissible pair in \(\Omega \). We emphasize that the 2-admissible property of \(\left( K^{-1}, K\right) \) 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 \(\nabla u\). 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.

Let us emphasize that more general weighted inequalities for Euclidean case have been extensively investigated, and are the subject of a rich literature (see e.g. [1,2,3,4, 6, 8,9,10,11, 14, 15, 26]).

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 \(|\nabla u|^2\).

Problems of this kind, involving weighted Sobolev inequalities for Carnot-Carathèodory space \(\mathbb G\), have been systematically studied in the literature (see e.g. [7, 12, 16, 17, 19]).

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

$$\begin{aligned}&\left( \frac{1}{\int _{B_R} v(x)\; dx} \int _{B_R} |u|^{t} v(x) \; dx \right) ^{1/t} \nonumber \\&\quad \le C\, R \left( \frac{1}{\int _{B_R} w(x)\; dx} \int _{B_R} |X u|^p w(x) \, dx \right) ^{1/p}, \end{aligned}$$
(1.3)

where \(1<p<t<\infty \), \(C>0\) is a constant, and (w, v) is a p-admissible pair in \(\Omega \). 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 \(A_2(\mathbb G)\) and \(G_{\tau }(\mathbb G)\) 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.

2 Preliminary results

2.1 Carnot Carathéodory spaces

Let \(\Omega \) be an open (Euclidean) bounded and connected subset in \({{\mathbb R}^n}\), with \(n\ge 2\). Let \(X=\{X_1, \ldots , X_m\}\) be a system of \(\mathcal {C}^{\infty }\) vector fields on \({{\mathbb R}^n}\).

We denote by \(Lie\, [X_1, \ldots , X_m]\) the Lie algebra generated by \(X_1, \ldots , X_m\) and by their commutators of any order. We say that a field Z belongs to \(Lie \, [X_1, \ldots , X_m]\) if and only if Z is a finite linear combination of terms of this type

$$\begin{aligned}{}[X_{i_1} [X_{i_2}, \ldots , [X_{i_{k-1}}, X_{i_k} ]]] \end{aligned}$$

for \(k \in \mathbb N\), \(1\le i_h \le m\), \(1\le h \le k\).

We define, for any fixed \(x\in {{\mathbb R}^n}\), the Lie rank as

$$\begin{aligned} rank \; Lie\,[X_1, \ldots ,X_m] = \hbox {dim}\; V (x), \end{aligned}$$

where \(V (x) = \{Z(x) : Z\in Lie \, [X_1, \ldots , X_m]\}\) is a subspace of \({{\mathbb R}^n}\). Henceforth, we assume that X satisfies the following Hörmander’s finite rank condition in \(\Omega \)

$$\begin{aligned} rank \, Lie\,[X_1, \ldots ,X_m]=n, \end{aligned}$$
(2.1)

namely there exist a neighborhood \(\Omega _0\) of \(\overline{\Omega }\) and \(m\in \mathbb N\) such that the family of commutators of the vector fields in X up to length m span \({{\mathbb R}^n}\) at every point of \(\Omega _0\).

Let \(\mathcal {C}_X\) be the family of absolutely continuous curves \(\gamma :[a,b] \rightarrow {{\mathbb R}^n}\) such that there exist measurable functions \(c_j:[a,b] \rightarrow \mathbb R\), with \(j=1, \ldots , m\), fulfilling

$$\begin{aligned} \sum _{j=1}^m c_j(t)^2 \le 1 \qquad \mathrm{and} \qquad \gamma \,'(t)= \sum _{j=1}^m c_j(t) X_j(\gamma (t)) \qquad \hbox {for a.e.}\;\;t\in [a,b]. \end{aligned}$$

We define Carnot-Carathéodory distance d as

$$\begin{aligned} d(x,y)=\inf \{T>0: \exists \,\gamma \in \mathcal {C}_X, \gamma (0)=x, \gamma (T)=y \}\qquad \hbox {for}\;\;x,y\in \Omega . \end{aligned}$$
(2.2)

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 \(\mathbb G\) is the pair \((\Omega , d)\) associated to a system of \(\mathcal {C}^{\infty }\) vector fields \(X=\{X_1,\cdots ,X_m\}\) on \({{\mathbb R}^n}\) fulfilling (2.1).

For \(x \in \mathbb R^n\) and \(R > 0\), set \(B(x, R) = \{y \in \mathbb R^n: d(x, y) < R\}\). 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.

Proposition 2.1

([25, Proposition 1.1]) Let \(X_1,\cdots X_m\) be as above. Then, for any compact set \(E\subset \subset \Omega \), there are positive constants \(c_1,c_2\) and \(\lambda \in (0,1]\) such that

$$\begin{aligned} c_1|x-y|\le d(x,y)\le c_2|x-y|^{\lambda } \end{aligned}$$

for every \(x,y\in E\).

Thanks to Proposition 2.1, the topology of Carnot-Carathéodory induced by d on \(\Omega \) 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 \(|\cdot |\) the Lebesgue measure in \(({{\mathbb R}^n}, d)\) and, by \(-\int _{B}f(x)\, dx\), the average of a function f on the ball B, i.e.

$$\begin{aligned} -\int _{B}f(x)\, dx=\frac{1}{|B|}\int _B f(x)\ dx. \end{aligned}$$

Note that the Lebesgue measure locally satisfies the following doubling condition (see e.g. [25]).

Proposition 2.2

For any compact set \(E\subset \subset \Omega \), if \(x_0\in E\), there exist a constant \(C_d \ge 1\), called doubling constant, and \(R_0>0\) such that

$$\begin{aligned} |B(x_0,2R)| \le C_d|B(x_0,R)| \end{aligned}$$

for \(0<R<R_0\).

Let \(Y_1, \ldots , Y_l\) be the collection of the \(X_j\)’s and of those commutators which are needed to generate \(\mathbb R^n\). To each \(Y_i\) it is associated a formal “degree” \(deg(Y_{i} )\ge 1\), namely the corresponding order of the commutator. Set \(I = (i_1, \ldots , i_n)\), with \(1\le i_j \le l\), an n-tuple of integers. We define (see also [25]) the degree of I as

$$\begin{aligned} \tilde{d}(I) =\displaystyle \sum _{j=1}^n deg(Y_{i_j} ). \end{aligned}$$

For a given compact set \(E \subset \mathbb R^n\), we define Q by

$$\begin{aligned} Q = \sup \{\tilde{d}(I) : |a_I (x)| \not = 0, x\in E\}, \end{aligned}$$

the local homogeneous dimension of E with respect to system X, where \(a_I(x) = det (Y_{i_1}, \ldots , Y_{i_n})\).

We define by

$$\begin{aligned} Q(x) = \inf \{\tilde{d}(I) : |a_I (x)| \not = 0\} \end{aligned}$$

the homogeneous dimension at \(x\in {{\mathbb R}^n}\) with respect to X. It is obvious that \(3\le n\le Q(x) \le Q\).

Just to give an idea, we consider in \(\mathbb R^3\) the system (see [13])

$$\begin{aligned} X= \{ X_1, X_2, X_3\}= \Big \{ \frac{\partial }{\partial x_1}, \frac{\partial }{\partial x_2}, x_1\frac{\partial }{\partial x_3} \Big \}. \end{aligned}$$

It is easy to see that \(l = 4\) and

$$\begin{aligned} \{Y_1, Y_2, Y_3, Y_4\} =\{X_1,X_2,X_3, [X_1,X_3]\}. \end{aligned}$$

Moreover, \(Q(x) = 3\) for all \(x \not = 0\), whereas for any compact set E containing the origin, \(Q(0) = Q = 4\).

Let Y be a metric space and \(\mu \) a Borel measure in Y. Assume \(\mu \) finite on bounded sets and satisfying the doubling condition on every open, bounded subset \(\Omega \) in Y. We say that Q is a homogeneous dimension relative to \(\Omega \), if there exists a positive constant C such that

$$\begin{aligned} \frac{\mu (B)}{\mu (B_0)} \ge C\left( \frac{R}{R_0} \right) ^Q \end{aligned}$$

for any ball \(B_0\) having center in \(\Omega \) and radius \(R_0<\mathrm{diam}\), and any ball B centered in \(x_0\in B_0\) and having radius \(R \le R_0\).

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 \(\Omega \). Obviously, any \(Q'\ge Q\) it is also a homogeneous dimension.

For a bounded open set \(\Omega \) containing a family of vector fields satisfying the Hörmander’s finite rank condition, the homogeneous dimension of the Carnot-Carathéodory space \(\mathbb G\), defined with the Lebesgue measure, is given by \(Q=\log _2C_d\), where \(C_d\) is the doubling constant.

2.2 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 \(L^1_{loc}(\mathbb {R}^n)\). We say that a weight w is doubling in \( \Omega \) if

$$\begin{aligned} \int _{2B} w(x)\, dx \le C \int _{B} w(x)\, dx, \end{aligned}$$

where the constant C is independent by the ball \(B\subset \Omega \).

We say that w is locally doubling in \(\Omega \) if for each compact set \(V\subset \Omega \) and \(\bar{R}>0\) there exists \(C_{V,\bar{R}}\) such that

$$\begin{aligned} \int _{2B} w(x)\, dx \le C_{V,\bar{R}} \int _{B} w(x)\, dx, \end{aligned}$$

where the ball B has center in V and radius \(R<\bar{R}\) 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 \(A_p(\mathbb G)\) (briefly, \(w\in A_p(\mathbb G)\)) for some \(p\in (1, +\infty )\) if

$$\begin{aligned}{}[w]_{A_p}=\sup _{B} \left( -\int _{B} w(x)\, dx\right) \left( -\int _{B} w(x)^{1-p'}\, dx\right) ^{p-1} \end{aligned}$$
(2.3)

is finite, where the supremum is taken over all balls \(B\subset \Omega \). Here, \(p'\) denotes the Hölder conjugate of p. The quantity \([w]_{A_p}\) is called the \(A_p\) constant of w.

When \(p=1\), we say that \(w\in A_1(\mathbb G)\) if there exists a constant \(c\ge 1\) such that, for every ball \(B\subset \Omega \),

$$\begin{aligned} \int _{B} w(x)\, dx \le c\;{\displaystyle \mathrm{ess}\inf _{B}}\,w. \end{aligned}$$

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 \(G_q(\mathbb G)\) (briefly, \(w\in G_q(\mathbb G)\)) for some \(q\in (1, +\infty )\) if

$$\begin{aligned}{}[w]_{G_q}=\sup _{B}\frac{\displaystyle \left( -\int _{B}w(x)^q\,dx\right) ^{\frac{1}{q}}}{\displaystyle -\int _ {B}w(x)\,dx} \end{aligned}$$

is finite. The quantity \([w]_{G_q}\) is called the \(G_q\) constant of w.

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 \(\mathcal {D}_h\) of dyadic cubes \(\mathcal {Q} \), which are “almost balls”, where h is a large negative integer which indexes the edgelengths l(Q) of the smallest cubes \(\mathcal {Q} \in \mathcal {D}_h\). In other words, the smallest edgelengths are \(\lambda ^h\) for an appropriate geometric constant \(\lambda >1\) and each cube in the grid has edgelength \(\lambda ^k\) for some \(k \ge h\).

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 \(\lambda >1\) such that, for every \(h\in \mathbb {Z}\), there are points \(x^k_j \in B_R\) and a family of cubes \(\mathcal {D}_h =\{\mathcal {Q} ^k_j\}\) for \(j \in \mathbb N\) and \(k= h, h+1, \ldots \) such that

i):

\( \; B(x^k_j, \lambda ^k) \subset \mathcal {Q} ^k_j \subset B(x^k_j, \lambda ^{k+1})\).

ii):

\( \; \text {For each} \; k= h,h+1, \ldots , \, \text {the family} \, \{\mathcal {Q} ^k_j\} \, \text {is pairwise disjoint in} \, j \, \text {and} \, B_R=\bigcup _j Q^k_j\).

iii):

\( \; \text {If} \; h\le k < l, \,\text {then either}\, \mathcal {Q} ^k_j \cap \mathcal {Q} ^l_j= \emptyset \, \text {or}\; \mathcal {Q} ^k_j \subset \mathcal {Q} ^l_j.\)

We call the family \(\mathcal {D}=\bigcup _{h \in \mathbb {Z}}\mathcal {D}_h\) a dyadic cube decomposition of \(B_R\) and we refer to its sets as dyadic cubes which will be denoted by \(\mathcal {Q} \). We observe explicitly that being \(\mathcal {D}\) a decomposition of \(B_R\), then any dyadic cube \(\mathcal {Q} \in \mathcal {D}\) is contained in the ball \(B_R\).

By [29], making use of (2.3), one can deduce the following lemma.

Lemma 2.3

Let \(w\in A_2(\mathbb G)\) and let \(\mathcal {Q}\) and \(\mathcal {Q}_0\) dyadic cubes in \({{\mathbb R}^n}\) such that \(\mathcal {Q}\subset \mathcal {Q}_0\). If \(\beta >1\), then

$$\begin{aligned} \sum _{\mathcal {Q} \subset \mathcal {Q}_0} \left( \int _\mathcal {Q} w \;dx \right) ^{\beta } \le \left( c(Q, n) [w]_{A_2}\right) ^{\beta -1}\, \left( \int _{\mathcal {Q}_0} w\;dx \right) ^{\beta }. \end{aligned}$$
(2.4)

Another important property of \(A_p(\mathbb G)\) classes is given by the following proposition (see [20, 30, Chapter 5, p. 195]).

Proposition 2.4

If \(w \in A_p(\mathbb G)\), then, for any nonnegative f,

$$\begin{aligned} \left( \frac{1}{|\mathcal {Q} |} \int _{\mathcal {Q}} f(x) \; dx\right) ^p \le \left[ w \right] _{A_p}\; \frac{1}{\int _{\mathcal {Q}}w (x)\; dx} \int _{\mathcal {Q}} |f(x)|^p w(x) \; dx\qquad \forall \mathcal {Q} \subset \mathbb G. \end{aligned}$$
(2.5)

2.3 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 \(\alpha \in (0, Q)\) of a locally integrable function g in \({{\mathbb R}^n}\) is defined as

$$\begin{aligned} I_{\alpha } g (x)=\int _{{{\mathbb R}^n}} \frac{g(y)}{d(x,y)^{Q-\alpha }}\; dy \qquad \hbox { for } x\in {{\mathbb R}^n}. \end{aligned}$$
(2.6)

Lemma 2.5

Let \(g\in L^1_{\text {loc}}(\mathbb G)\) and assume that \(g\ge 0\). Then

$$\begin{aligned} I_{1} g(x)\le c_0 \underset{\mathcal {Q}\in \mathcal {D}}{\sum } \left( |{\mathcal {Q}}|^{\frac{1}{Q}-1} \int _{3\mathcal {Q}} g(y)\; dy \right) \chi _{\mathcal {Q}} (x) \qquad \forall x\in \Omega , \end{aligned}$$
(2.7)

where \(c_0\) is an absolute constant.

Proof

Thanks to a dyadic cube decomposition, we discretize the operator \(I_1\)

$$\begin{aligned} I_{1} g(x)&=\displaystyle \sum _{k\in {Z}}\left( \int _{2^{k-1}<d(x,y)\le 2^k} \frac{g(y)}{d(x,y)^{Q-1}}dy\right) \\&\le c_0\displaystyle \sum _{k\in Z} \displaystyle \sum _{\begin{array}{c} Q\in \mathcal {D} \\ l(\mathcal Q)=2^k \end{array}}\left[ \left( \frac{1}{l(\mathcal {Q})^{Q-1}}\int _{d(x,y)\le l(\mathcal {Q})}g(y)dy \right) \chi _{\mathcal {Q}} (x)\right] \\&\le c_0 \displaystyle \sum _{{\mathcal {Q} \in \mathcal {D}}} \left[ \left( \vert \mathcal {Q} \vert ^{\frac{1-{Q}}{{Q}}} \int _{ 3\mathcal {Q}}g(y)dy \right) \chi _{\mathcal {Q}} (x) \right] , \end{aligned}$$

where the last inequality follows by \(|\mathcal {Q}| =l(\mathcal {Q})^Q\) and, moreover, by \(B(x,l(\mathcal {Q}))\subset 3\mathcal {Q}\) if \(x\in \mathcal {Q}\). Hence, inequality (2.7) is proved. \(\square \)

Let us consider a Dirichlet problem in this form

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _{\mathbb G}\varphi = f(x) &{} \qquad \hbox { in } {B_R} \\ \\ \varphi =0 &{} \qquad \hbox { on } {\partial B_R}, \end{array} \right. \end{aligned}$$
(2.8)

where \(\Delta _{\mathbb G}\) denotes the canonical sub-Laplacian operator defined as \(\Delta _{\mathbb G}=\sum _{j=1}^mX_j^2\), with \(\{X_1,...,X_m\}\) the family of smooth vector fields on \({{\mathbb R}^n}\) satisfying the Hörmander’s finite rank condition.

Let \(\mathcal {F}_\alpha (B_R)\) be the anisotropic Hölder space, with \(\alpha \in (0,1)\), defined by

$$\begin{aligned} \mathcal {F}_\alpha (B_R)=\bigg \{ f: B_R \rightarrow \mathbb R: \sup _{\begin{array}{c} x, y \in B_R \\ x \ne y \end{array}} \frac{f(x)- f(y)}{d(x,y)^\alpha }<\infty \bigg \}, \end{aligned}$$
(2.9)

where d is the Carnot-Carathéodory distance given by (2.2).

In [21, Theorem 3.2], the authors proved that, if \(f\in \mathcal {F}_\alpha (B_R)\), then there exists a unique solution \(\varphi \in \mathcal {C}^2(B_R)\cap \mathcal {C}^1(\overline{B_R})\) to problem (2.8), represented by the formula

$$\begin{aligned} \varphi (x)=\int _{B_R} \Delta _{\mathbb G}\,\varphi \; \Gamma _x(y)\;dy. \end{aligned}$$
(2.10)

Here, \(\Gamma _x(y) \) is the fundamental solution of the sub-Laplacian. Thanks to [21, Theorem 2.2], there exists a positive constant c such that

$$\begin{aligned} \Gamma _x(y)= c \, d(x,y)^{2-Q}. \end{aligned}$$
(2.11)

Consequently, combining (2.10) and (2.11) yields

$$\begin{aligned} \varphi (x) =c \int _{B_R}\frac{f(y)}{d(x,y)^{Q-2}}dy. \end{aligned}$$
(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.

Lemma 2.6

Let \(f\in \mathcal {F}_\alpha (B_R)\) and let \(\varphi \) be the solution to problem (2.8). Then, there exists a positive constant c such that

$$\begin{aligned} \vert X \varphi (x)\vert \le c\, I_1 f (x), \end{aligned}$$
(2.13)

where \(I_1(f)\) denotes the fractional integral of order 1 of f.

Proof

Owing to (2.12), it follows that

$$\begin{aligned} X_j\varphi (x)=c \int _{B_R}\frac{f(y)}{d(x,y)^{Q-1}}X_j(d(x,y))dy. \end{aligned}$$
(2.14)

Thus,

$$\begin{aligned} \vert X \varphi (x)\vert= & {} \Big ( \sum _{j=1}^n \vert X_j\varphi (x) \vert ^2 \Big )^{\frac{1}{2} }\nonumber \\= & {} \Big ( \sum _{j=1}^n \Big \vert c\, \int _{B_R}\frac{f(y)}{d(x,y)^{Q-1}} X_j(d(x,y))dy \Big \vert ^2 \Big )^{\frac{1}{2}}. \end{aligned}$$
(2.15)

Since \(\Big \vert X_j(d(x,y))\Big \vert =1\) (see [23]), by (2.15) and (2.6) one can deduce that

$$\begin{aligned} \vert X \varphi (x)\vert\le & {} \left( \sum _{j=1}^n \Big \vert c \int _{B_R}\frac{f(y)}{d(x,y)^{Q-1}} dy \Big \vert ^2 \right) ^{\frac{1}{2}}\nonumber \\\le & {} n \, c \int _{B_R}\frac{f(y)}{d(x,y)^{Q-1}} dy = c\, I_1 f (x), \end{aligned}$$
(2.16)

where the second inequality is due to the fact that \(a^2+b^2\le (a+b)^2\). \(\square \)

3 Proof of main result

The following preliminary lemma will be use in the proof of Theorem 1.1.

Lemma 3.1

If \(K\in A_2(\mathbb G)\) and \(u \in \mathcal {C}_0^1 (B_R)\), then

$$\begin{aligned} S_1\;= & {} \left[ \sum _{{\mathcal {Q} } \in \mathcal {D}} \left( \int _{\mathcal {Q} } K(x) \;dx \right) ^{\frac{q'}{t'}} \left( \frac{1}{\int _{\mathcal {Q} } K(x) \;dx} \int _{3 \mathcal {Q}} |u|^{t-1}K(x)\;dx \right) ^{q'}\right] ^{\frac{1}{q'}}\!\!\!\!\!\nonumber \\\le & {} C \left( \int _{B_R} |u|^t K(x) \; dx\right) ^{\frac{1}{t'}}\!\!, \end{aligned}$$
(3.1)

where \(2<q<t\) and \(C = c(Q, n,t, q) [K]_{A_2}^{\frac{1}{t'}-\frac{1}{q'}}\).

Proof

For each \(h \in \mathbb {Z}\), we set

$$\begin{aligned} \mathcal {C}^h =\left\{ {\mathcal {Q} } \;\text {dyadic cube}: 2^h < \frac{1}{\int _{\mathcal {Q} } K(x)\;dx}\int _{\mathcal {Q} } |u|^{t-1}K(x)\; dx \le 2^{h+1}\right\} . \end{aligned}$$
(3.2)

Note that, if \({\mathcal {Q} }\) is any dyadic cube such that \(|u|^{t-1}K(x)\) is not identically zero on \({\mathcal {Q}}\), then \({\mathcal {Q}}\) belongs to only one collection \(\mathcal {C}^h\).

For each \(h \in \mathbb {Z}\), let us build the collection \(\{\mathcal {Q}^h_j\}_j\) of pairwise disjoint maximal dyadic cubes (maximal with respect to inclusion) in \(\mathcal {C}^h\). If \(\mathcal {Q} \in \mathcal {C}^h\), then there exists \(j\in \mathbb N\) such that \({\mathcal {Q} }\subset \mathcal {Q}^h_j\). Note also that for each fixed h, the cubes \(\mathcal {Q}^h_j\) are disjoint with respect to j. Nevertheless, they may not be disjoint for different values of h.

By (3.2),

$$\begin{aligned} S_1\le & {} \left( \! \sum _{h\in \mathbb Z} 2^{(h+1)q'}\sum _{{\mathcal {Q} } \in \mathcal {C}^h} \!\!\left( \int _{\mathcal {Q} } K(x) \;dx \right) ^{\frac{q'}{t'}}\! \right) ^{\frac{1}{q'}}\nonumber \\\le & {} \left( \!\sum _{h\in \mathbb Z} 2^{(h+1)q'}\sum _{j\in \mathbb N} \sum _{{\mathcal {Q} } \subset \mathcal {Q}^h_j} \!\!\left( \int _{\mathcal {Q} } K(x) \;dx \right) ^{\frac{q'}{t'}}\! \right) ^{\frac{1}{q'}}. \end{aligned}$$
(3.3)

By Lemma 2.3, since \(\displaystyle \frac{q'}{t'}>1\), we have

$$\begin{aligned} \sum _{{\mathcal {Q} } \subset \mathcal {Q}^h_j} \left( \int _{\mathcal {Q} } K(x)\;dx \right) ^{\frac{q'}{t'}} \le \Big ( c(Q, n) [K]_{A_2}\Big )^{\frac{q'}{t'}-1} \left( \int _{\mathcal {Q}_j^h} K(x) \;dx \right) ^{\frac{q'}{t'}}. \end{aligned}$$
(3.4)

By (3.3) and (3.4), we deduce

$$\begin{aligned} S_1 \le \left( \Big ( c(Q, n) [K]_{A_2}\Big )^{\frac{q'}{t'} -1} \sum _{h\in \mathbb Z} 2^{(h+1)q'} \sum _{j\in \mathbb N} \left( \int _{{\mathcal {Q}}_j^h} K(x) \;dx \right) ^{\frac{q'}{t'}} \right) ^{\frac{1}{q'}}. \end{aligned}$$
(3.5)

Since \(\mathcal {Q}_j^h \in \mathcal {C}^h\), by (3.2)

$$\begin{aligned} \frac{1}{\int _{\mathcal {Q}^h_j } K(x)\;dx}\int _{\mathcal {Q}^h_j } |u|^{t-1}K(x)\; dx> 2^h. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{1}{\int _{\mathcal {Q}^h_j } K(x)\;dx} \int _{\mathcal {Q}_j^h \cap \{x\in B_R:| u|>2^{h-10}\}} |u|^{t-1}K(x) \;dx \ge C_1 \,2^h, \end{aligned}$$
(3.6)

where \(C_1= C_1(Q,n)\) is a constant. Consequently,

$$\begin{aligned} \int _{\mathcal {Q}_j^h} K(x) \;dx \le C_1\, 2^{-h} \int _{\mathcal {Q}_j^h \cap \{x\in B_R:|u|>2^{h-10}\}} |u|^{t-1}K(x) \;dx. \end{aligned}$$
(3.7)

Owing to (3.5) and (3.7),

$$\begin{aligned} S_1 \le C_2\left( \sum _{h\in \mathbb Z} 2^{(h+1)q'} \sum _{j\in \mathbb N} \left( 2^{-h} \int _{\mathcal {Q}_j^h \cap \{x\in B_R: \;|u|>2^{h-10}\}} |u|^{t-1}K(x) \;dx \right) ^{\frac{q'}{t'}} \right) ^{\frac{1}{q'}}, \end{aligned}$$
(3.8)

where \(C_2= C_1(Q, n)\,\left( c(Q, n)[K]_{A_2}\right) ^{\frac{1}{t'} - \frac{1}{q'}}\).

By (3.8), we have

$$\begin{aligned} S_1&\le C_2 \left( \sum _{h\in \mathbb Z} 2^{(h+1)-\frac{h}{t'}} \sum _{j\in \mathbb N} \int _{\mathcal {Q}_j^h \cap \{x\in B_R:\; |u|>2^{h-10}\}} |u|^{t-1}K(x) \;dx \right) ^{\frac{1}{t'}}\nonumber \\&= 2^{\frac{1}{t'}} C_2 \left( \sum _{h\in \mathbb Z} 2^{\frac{h}{t}} \sum _{j\in \mathbb N} \int _{\mathcal {Q}_j^h \cap \{x\in B_R:\; |u|>2^{h-10}\}} |u|^{t-1}K(x) \;dx \right) ^{\frac{1}{t'}}\nonumber \\&\le 2^{\frac{1}{t'}} C_2\left( \sum _{h\in \mathbb Z} 2^h \int _{ \{x\in B_R: \;|u|>2^{h-10}\}} |u|^{t-1}K(x) \;dx \right) ^{\frac{1}{t'}}\nonumber \\&= 2^{\frac{1}{t'}} C_2 \left( \int _{B_R} |u|^{t-1}K(x)\sum _{\{h\in \mathbb Z:\, 2^h < 2^{10}|u|\}} 2^h \; dx\right) ^{\frac{1}{t'}}, \end{aligned}$$
(3.9)

where the first inequality is a consequence of the fact that \(\sum _h a_h^{\frac{q'}{t'}} \le \big [\sum _h a_h \big ]^{\frac{q'}{t'}}\), the third one holds because, fixed \(h\in \mathbb Z\), \(\mathcal {Q}^h_j\) are disjoint in j, the fourth one is due to Fubini’s type Theorem. To conclude the proof, we have to evaluate the quantity

$$\begin{aligned} \sum _{\{h\in \mathbb Z:\, 2^h < 2^{10}|u|\}} 2^h. \end{aligned}$$
(3.10)

Set \(H=\log _2{(2^{10} |u|)}\). Thus, (3.10) yields

$$\begin{aligned} \sum _{h=-\infty }^H 2^h&= \sum _{h=-H}^{+\infty } \left( \frac{1}{2}\right) ^h = \sum _{h=-H}^{+\infty } \left( \frac{1}{2}\right) ^{h +H -H} = \left( \frac{1}{2} \right) ^{-H} \sum _{h=-H}^{\infty } \left( \frac{1}{2}\right) ^{h +H}\nonumber \\&= \left( \frac{1}{2} \right) ^{-H} \sum _{m=0}^{+\infty } \left( \frac{1}{2}\right) ^{m} =\left( \frac{1}{2} \right) ^{-H} 2 = 2^{\log _2{(2^{10} |u|)}} \,2 = 2^{11} |u|. \end{aligned}$$
(3.11)

Then, by (3.9) and (3.11), we obtain

$$\begin{aligned} S_1\le C_3 \left( \int _{B_R} |u|^t K(x)\; dx\right) ^{\frac{1}{t'}}, \end{aligned}$$

with \(C_3 = 2^{\frac{1}{t'}+11} C_1(Q, n)\,\left( c(Q, n)[K]_{A_2}\right) ^{\frac{1}{t'} - \frac{1}{q'}}\) and inequality (3.1) is proved. \(\square \)

Now we are in position to prove our main result.

Proof of Theorem 1.1

By Theorem 3.2 of [21], there exists a solution \(\varphi \) to the following Dirichlet problem for sub-Laplacian

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _{\mathbb G}\varphi = |u|^{t-1} K(x) &{} \qquad \hbox { in } {B_R} \\ \\ \varphi =0 &{} \qquad \hbox { on } {\partial B_R}, \end{array} \right. \end{aligned}$$

with \(u\in {\mathcal C}_0^1(B_R)\). By Lemma 2.6, we get

$$\begin{aligned} |X \varphi (x)|\le c\, I_1 (|u|^{t-1} K(x)) \qquad \forall x \in B_R, \end{aligned}$$
(3.12)

where c is a positive constant.

Thanks to Lemma 2.5, it follows that

$$\begin{aligned} I_{1}(|u|^{t-1} K)(x)\le c_0\,\underset{{\mathcal {Q} } \in \mathcal {D}}{\sum } \left( |{\mathcal {Q} }|^{\frac{1}{n}-1} \int _{3{\mathcal {Q} }} |u(y)|^{t-1} K(y)\; dy\right) \chi _{\mathcal {Q} }(x) \qquad \forall x \in B_R, \end{aligned}$$
(3.13)

where \(c_0\) is an absolute constant.

Combining (3.12) and (3.13) yields

$$\begin{aligned}&\int _{B_R} |u(x)|^t K(x)\; dx \nonumber \\&\quad = \int _{B_R} |u(x)||u(x)|^{t-1} K(x)\; dx = \int _{B_R} |u(x)| \Delta _{\mathbb G} \varphi \; dx\nonumber \\&\quad \le \int _{B_R} |X u|| X \varphi |\; dx \le c \int _{B_R} |X u| I_{1}(|u|^{t-1} K)(x)\; dx \nonumber \\&\quad \le C_6 \int _{B_R} |X u (x)| \underset{{\mathcal {Q} } \in \mathcal {D}}{\sum } \left( |{\mathcal {Q} }|^{\frac{1}{n}-1} \int _{3{\mathcal {Q} }} |u(y)|^{t-1}K(y)\;dy \right) \chi _Q(x) \; dx \nonumber \\&\quad = C_6 \int _{B_R} \underset{{\mathcal {Q} } \in \mathcal {D}}{\sum } |{\mathcal {Q} }|^{\frac{1}{n}-1} \, |X u (x)|\,\chi _{\mathcal {Q} }(x) \left( \int _{3{\mathcal {Q} }} |u(y)|^{t-1}K(y)\;dy \right) \; dx \nonumber \\&\quad = C_6 \underset{{\mathcal {Q} } \in \mathcal {D}}{\sum } |{\mathcal {Q} }|^{\frac{1}{n}-1} \,\int _{B_R\cap {\mathcal {Q} }} |X u (x)| \; dx \left( \int _{3{\mathcal {Q} }} |u(y)|^{t-1}K(y)\;dy \right) \nonumber \\&\quad = C_6 \underset{Q \in \mathcal {D}}{\sum } |{\mathcal {Q} }|^{\frac{1}{n}} \left( \frac{1}{|{\mathcal {Q} }|} \int _{{\mathcal {Q} }} |X u| \;dx \right) \left( \int _{3{\mathcal {Q} }} |u(y)|^{t-1}K(y)\;dy \right) , \end{aligned}$$
(3.14)

where \(C_6 =c \,c_0\). Note that the last inequality is the consequence of the fact that \( B_R \cap {\mathcal {Q} } = {\mathcal {Q} }\).

By (2.5),

$$\begin{aligned} \frac{1}{|{\mathcal {Q} }|} \int _{{\mathcal {Q} }} |X u|\; dx&\le \left[ K^{-1}\right] _{A_{2} }^{\frac{1}{2}} \left( \frac{1}{\int _{\mathcal {Q} } \frac{1}{K(x) }\;dx} \int _{\mathcal {Q} }\frac{|X u|^{2}}{K(x)} \;dx\right) ^{\frac{1}{2}}. \end{aligned}$$
(3.15)

Coupling inequalities (3.14) and (3.15) tells us that

$$\begin{aligned}&\int _{B_R} |u|^t K(x)\; dx N\nonumber \\&\quad \le C_6 \; \left[ K^{-1}\right] _{A_{2} }^{\frac{1}{2}} \;\underset{{\mathcal {Q} } \in \mathcal {D}}{\sum } |{\mathcal {Q} }|^{\frac{1}{n}} \left( \! \frac{1}{\int _{\mathcal {Q} } \frac{1}{K(x) }\;dx} \int _{\mathcal {Q} } \frac{|X u|^2}{K(x)} \;dx\!\right) ^{\frac{1}{2}} \left( \int _{3{\mathcal {Q} }} |u|^{t-1}K(y)\;dy \right) . \end{aligned}$$
(3.16)

By (3.16), the following chain of inequality holds

$$\begin{aligned}&\int _{B_R} |u|^t K(x)\; dx\nonumber \\&\quad \le C_7 |B_R|^{1/n} \frac{ \left( \int _{B_R} K(x) \;dx \right) ^{1/t}}{\left( \int _{B_R} \frac{1}{K(x) }\;dx\right) ^{1/2}} \sum _{{\mathcal {Q} } \in \mathcal {D}} \left( \int _{\mathcal {Q} } K(x) \;dx \right) ^{-1/t}\left( \int _{\mathcal {Q} } \frac{1}{K(x)} \;dx \right) ^{1/2} \nonumber \\&\qquad \times \left( \frac{1}{\int _{\mathcal {Q} } \frac{1}{K(x) }\;dx} \int _{\mathcal {Q} } \frac{|X u|^2}{K(x)} \;dx\right) ^{1/2}\left( \int _{3{\mathcal {Q} }} |u|^{t-1}K(x)\;dx \right) \nonumber \\&\quad = C_7 |B_R|^{1/n} \frac{ \left( \int _{B_R} K(x) \;dx \right) ^{1/t}}{\left( \int _{B_R} \frac{1}{K(x) }\;dx\right) ^{1/2}} \sum _{{\mathcal {Q} } \in \mathcal {D}} \left( \int _{\mathcal {Q} }\frac{1}{K(x)} \;dx \right) ^{1/2}\nonumber \\&\qquad \left( \frac{1}{\int _{\mathcal {Q} } \frac{1}{K(x) }\;dx} \int _{\mathcal {Q} } \frac{|X u|^2}{K(x)} \;dx\right) ^{1/2} \times \left( \int _{\mathcal {Q} } K(x) \;dx \right) ^{1/{t'} -1} \int _{3{\mathcal {Q} }} |u|^{t-1}K(x)\;dx\nonumber \\&\quad \le C_7 |B_R|^{1/n} \frac{ \left( \int _{B_R} K(x) \;dx \right) ^{1/t}}{\left( \int _{B_R} \frac{1}{K(x) }\;dx\right) ^{1/2}} \!\!\left[ \sum _{{\mathcal {Q} } \in \mathcal {D}} \left( \int _{\mathcal {Q} } \frac{1}{K(x)} \;dx \right) ^{q/2} \!\!\right. \nonumber \\&\left. \qquad \left( \frac{1}{\int _{\mathcal {Q} } \frac{1}{K(x) }\;dx} \int _{\mathcal {Q} } \frac{|X u|^2}{K(x)} \;dx\right) ^{q/2}\! \right] ^{1/q} \nonumber \\&\qquad \times \left[ \sum _{{\mathcal {Q} } \in \mathcal {D}} \left( \int _{\mathcal {Q} } K(x) \;dx \right) ^{q'/{t'}} \left( \frac{1}{\int _{\mathcal {Q} } K(x) \;dx} \int _{3{\mathcal {Q} }} |u|^{t-1}K(x)\;dx \right) ^{q'}\right] ^{1/{q'}}\nonumber \\&\quad = C_7 |B_R|^{1/n} \frac{ \left( \int _{B_R} K(x) \;dx \right) ^{1/t}}{\left( \int _{B_R} \frac{1}{K(x) }\;dx\right) ^{1/2}} \!\!\left[ \sum _{{\mathcal {Q} } \in \mathcal {D}} \!\!\left( \int _{\mathcal {Q} } \frac{|X u|^2}{K(x)} \;dx\right) ^{q/2}\! \right] ^{1/q}\nonumber \\&\qquad \Bigg [ \sum _{{\mathcal {Q} } \in \mathcal {D}} \left( \int _{\mathcal {Q} } K(x) \;dx \right) ^{q'/{t'}} \left( \frac{1}{\int _{\mathcal {Q} } K(x) \;dx} \int _{3{\mathcal {Q} }} |u|^{t-1}K(x)\;dx \right) ^{q'}\Bigg ]^{1/{q'}}\nonumber \\&\qquad \le C_7 |B_R|^{1/n} \frac{ \left( \int _{B_R} K(x) \;dx \right) ^{1/t}}{\left( \int _{B_R} \frac{1}{K(x) }\;dx\right) ^{1/2}} \!\!\left( \int _{B_R } \frac{|X u|^2}{K(x)} \;dx\right) ^{1/2}\! \Bigg [ \sum _{{\mathcal {Q} } \in \mathcal {D}} \left( \int _{\mathcal {Q} } K(x) \;dx \right) ^{q'/{t'}}\nonumber \\&\qquad \left( \frac{1}{\int _{\mathcal {Q} } K(x) \;dx} \int _{3{\mathcal {Q} }} |u|^{t-1}K(x)\;dx \right) ^{q'}\Bigg ]^{1/{q'}}\nonumber \\&\quad = C_7 |B_R|^{1/n} \frac{ \left( \int _{B_R} K(x) \;dx \right) ^{1/t}}{\left( \int _{B_R} \frac{1}{K(x) }\;dx\right) ^{1/2}} \; \left( \int _{B_R } \frac{|X u|^2}{K(x)} \;dx\right) ^{1/2}\! \; S_1, \end{aligned}$$
(3.17)

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 \(\mathcal {D}\) is a decomposition of \(B_R\). Here, constant \(C_7= C_6\; \overline{C} \; \left[ K^{-1}\right] _{A_{2} }^{\frac{1}{2}} \). The quantity \(S_1\) is introduced in Lemma 3.1 above.

Combining (3.17) and (3.1) shows that

$$\begin{aligned} \left( \int _{B_R} |u|^t K(x)\; dx \right) ^{1/t} \le C_8 |B_R|^{1/n} \frac{\displaystyle \left( \int _{B_R} K(x) \; dx \right) ^{1/t}}{\displaystyle \left( \int _{B_R} \frac{1}{K(x)} \; dx \right) ^{1/2}} \left( \int _{B_R} \frac{|X u|^2}{K(x)}\; dx \right) ^{1/2}, \end{aligned}$$
(3.18)

where \(C_8= c(Q, n, t, q) \,\overline{C} \;[K^{-1}]_{A_2}^{\frac{1}{2}} [K]_{A_2}^{\frac{1}{t'} - \frac{1}{q'}}\). Then, inequality (1.2) follows. \(\square \)