Consider the smooth vector fields $X=(X\_1,X\_2,...,X\_m)$ defined in a open bounded set $\Omega\in R^n$. And the non-isotropic dimension is $Q$ (see <https://arxiv.org/pdf/1502.06332.pdf> page 398)
In the paper above (page 398) the author gave a embedding that if $f\in H\_{X,0}^1$,
$$\|f\|\_{L^{p^{\*}}\left(\Omega^{\prime}\right)} \leqslant C\left\|X f\right\|\_{L^{p}\left(\Omega^{\prime}\right)}$$
$$\frac{1}{p^{\*}}=\frac{1}{p}-\frac{1}{Q}, \quad 1 \leqslant p<Q$$
Now what we concern about is that whether this embedding can be improved to be a compact embedding? We are in trouble in doing the proof. Could anybody give some reference or information about this topic?
All advices are greatful. Thank you! 



subelliptic Sobolev compact embedding theorem
The compactness for exponents below the embedding exponent is standard and true in a great generality. In your case compactness follows from results in Section 8 of [1]. Sobolev embeddings for vector fields are discussed in Section 11 of [1]. See also Theorem 4 in [2] for a very general compactness criteria. Both papers are available on my website. 


In many cases one can prove the compactness using the following general result (Theorem 4 in [2]).



> 
> **Theorem.** Let $X$ be a set equipped with a finite measure $\mu$. Assume that a linear normed space $W$ of measurable functions on $X$
>  has the following two peoperties.
> 
> 
> 1. There is $q>1$ such that the embedding $W\subset L^q(X,\mu)$ is bounded.
> 2. Every bounded sequence in $W$ contains a subsequence that converges almost everywhere.
> 
> 
> Then the embedding $W\subset L^s(X,\mu)$ is compact for every $1\leq
>  s<q$.
> 
> 
> 


In the case of Sobolev spaces associated with vector fields in order to prove compactness below the embedding exponent it suffices to verify a Poincare inequality on every ball $B\subset\Omega$ such that $2B\subset\Omega$, see Theorem 8.1 in [1].


**[1]** **P. Hajlasz, P. Koskela,** Sobolev met Poincare, *Memoirs Amer. Math. Soc.* 688 (2000), 1--101. 


**[2]** **P. Hajlasz, P. Koskela,** Isoperimetric inequalities and imbedding theorems in irregular domains, *J. London Math. Soc.* 58 (1998), 425-450. 


