I am reading Luca Capogna's article [An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations][1]. In this article, the authors proved the following theorem

> (Theorem 2.3) Let $U\subset \mathbb{R}^n$ be a bounded open set and denote by $Q$ the homogeneous dimension relative to $U$. Let $1<p<Q$. Then there exist $C>0$ and $R_{0}>0$ such that for any $x\in U$, $B_{R}=B(x,R)$ ($B_{R}$ is the subunit ball) with $R\leq R_{0}$, we have
$$ \left(\frac{1}{|B_{R}|}\int_{B_{R}}|u|^{sp}dx \right)^{\frac{1}{sp}}\leq CR\left(\frac{1}{|B_{R}|}\int_{B_{R}}|D_{L}u|^{p}dx\right)^{\frac{1}{p}}$$
for any $u\in S_{0}^{1,p}(B_{R})$, Here, $1\leq s\leq \frac{Q}{Q-p}$.

The author says that a standard partition of the unity argument implies
$$ S_{c}^{1,p}(U)\hookrightarrow L^{q}(U)$$
for any $U\subset\subset \mathbb{R}^n$.
I don't know how to use the partition of unity to obtain this claim. Can someone show it in detail?. Furthermore, can we get something like 
$$ \left(\int_{U}|u|^{q}dx\right)^{\frac{1}{q}}\leq C\left(\int_{U}|D_{L}u|^{p}dx \right)^{\frac{1}{p}},$$
for $U\subset\subset \mathbb{R}^n$ instead the subunit ball $B_{R}$?


  [1]: http://www.math.purdue.edu/~danielli/commpde.pdf