Let $\Omega\subset\mathbb{R}^n$ be a open and bounded set with the Carnot-Caretheodary distance and assume that $X=(X_1,X_2,\ldots,X_n)$ is a smooth H"ormander vector field with $rank\,(Lie(X))=n$. Let $V^{1,p}(\Omega)$ be the space of measurable functions $u$ such that $$ \|u\|_{V^{1,p}(\Omega)}:=\|u\|_{L^p(\Omega)}+\|Xu\|_{L^p(\Omega)}<\infty, $$ where $Xu=(X_1 u,X_2 u,\ldots,X_n u)$.
In the Euclidean case, the classical Sobolev space $W^{1,p}(\Omega)$ is the space of measurable functions $u$ such that $$ \|u\|_{W^{1,p}(\Omega)}:=\|u\|_{L^p(\Omega)}+\|\nabla u\|_{L^p(\Omega)}<\infty, $$ where $\nabla u=(\frac{\partial u}{\partial x_1},\ldots,\frac{\partial u}{\partial x_n})$. We know that the embedding $W^{1,p}(\Omega)\to L^q(\Omega)$ is compact for $1<q<p^*=\frac{np}{n-p}$ for $1<p<n$.
Can somebody please inform if an analogous compact embedding result holds for the space $V^{1,p}(\Omega)\to L^q(\Omega)$ for some $q$? If so, what is the range of $q$?
Thanks.