I am reading Luca Capogna's article [An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations](http://www.math.purdue.edu/~danielli/commpde.pdf). 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
$$ S\_{0}^{1,p}(U)\hookrightarrow L^{q}(U)$$
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 deduce the following fact ? 
$$ \left(\int\_{U}|u|^{q}dx\right)^{\frac{1}{q}}\leq C\left(\int\_{U}|D\_{L}u|^{p}dx \right)^{\frac{1}{p}},$$
for $U\subset\subset \mathbb{R}^n$ instead the subunit ball $B\_{R}$?


**My approach:** since $\overline{U}$ is a compact set, then there exist $n$ subunit ball $B\_{i}(x\_{i},r\_{i}) (i=1,\ldots,n)$ which cover $\overline{U}$ (We can assume that each $r\_{i}\leq R\_{0}$). Then there exists a partition of unity of $B\_{i}(x\_{i},r\_{i}) (i=1,\ldots,n)$ satisfy 


(1)$0\leq \phi\_{i}\leq 1, \text{supp}\phi\_{i}\subset B\_{i}(x\_{i},r\_{i}) $ and $\phi\_{i}\in C\_{0}^{\infty}(\mathbb{R}^n)$. 


(2) $$ \sum\_{i=1}^{n}\phi\_{i}=1 \qquad \forall x\in U $$


Then for a function $f\in S\_{0}^{1,p}(U)$, we have 
$$ f=\sum\_{i=1}^{n}f\phi\_{i} $$
\begin{align\*}
 \|f\|\_{L^{q}(U)}&=\|\sum\_{i=1}^{n}\phi\_{i}f\|\_{L^{q}(U)}\\
&\leq \sum\_{i=1}^{n}\|\phi\_{i}f\|\_{L^{q}(U)}\\
&=\sum\_{i=1}^{n}\|\phi\_{i}f\|\_{L^{q}(B\_{i}(x\_{i},r\_{i}))}\\
&\leq \sum\_{i=1}^{n} \|D\_{L}(\phi\_{i}f)\|\_{L^{p}(B\_{i}(x\_{i},r\_{i}))}
\end{align\*}
I don't know if 
**$$\sum\_{i=1}^{n} \|D\_{L}(\phi\_{i}f)\|\_{L^{p}(B\_{i}(x\_{i},r\_{i}))}\leq C\|D\_{L}f\|\_{L^p(U)}$$** holds or not.
Then I stuck here and don't know how to continue, Can some one help me? thank you very much!



how to use the sobolev inequality to obtain the embedding theorem
*Disclaimer: Not an expert in analysis/PDE, happen to know tangential results while studying Whitney-type embeddings.*



> 
> $S\_{0}^{1,p}(U)\hookrightarrow L^{q}(U)$ for any $U\subset\subset \mathbb{R}^n$. I don't know how to use the partition of unity to obtain this claim.
> 
> 
> 


There is a more general proof for Thm 2.3 in [1], shown in details by the same authors in a later paper [2].


Theorem 2.3 in [1] is the same as Theorem 1.2 in [2] with different notations. And the partition of unity should follow the same lines in the proof of Theorem 1.3 of [2]. The proof in details you asked for is in Sec 4 of [2]. 


In [2] they proved the theorem as a special case of Theorem 1.1 of [2], and in the process of proving Theorem 1.1 of [2], they also proved (Theorem 1.3 in [2]) the sub-elliptic inequality
$$\left|\left\{ x\in B\_{R}:\left|u(x)\right|>\lambda\right\} \right|^{\frac{Q-1}{Q}}\leq C\_{3}\frac{1}{\lambda}E|B\_{R}|^{\frac{-1}{Q}}\int\_{B\_{R}}|D\_{X}u|dx$$
for some constant $C\_3$. So the roadmap is $Thm1.3\rightarrow Thm1.4\text{(isoperimetric)}\rightarrow Thm1.1\rightarrow Thm1.2\text{(special case,=Thm2.3 in [1])}$. The partition of unity occurs in the proof of Thm 1.3
$$I\_{\alpha}^{1}f(x)=\int\_{B(x,\epsilon)}|f(\xi)|\frac{d(x,\xi)^{\alpha}}{|B(x,d(x,\xi))|}d\xi$$
$$I\_{\alpha}^{2}f(x)=\int\_{B(x,\epsilon)^{C}\cap B\_{R}}|f(\xi)|\frac{d(x,\xi)^{\alpha}}{|B(x,d(x,\xi))|}d\xi$$
and then in the proof of Thm2.1 of [2], the author argued that both $I\_\alpha$ can be bounded from above.
This is formally a partition of unity, but its spirit is more like Calderon-Zygmund type argument.
$$I\_{\alpha}^{1}f(x)\leq C\_{4}Mf(x)\epsilon^{\alpha}$$
where $M$ is the Hardy-Littlewood maximal operator over $B(x,r),r>0$,
$$I\_{\alpha}^{2}f(x)\leq C\_{5}R^{Q}|B\_{R}|^{-1}\epsilon^{\alpha-Q}\left\Vert f\right\Vert \_{L^{1}(B\_{R})}$$
with $C\_4,C\_5$ being constants. Let $I\_\alpha=I\_\alpha^1+I\_\alpha^2$, then let $\alpha=1$ we reach Thm 1.3 and hence Thm 1.4 hence Thm 1.1. As a corollary we get Thm 1.2(Your Thm2.3).The details in proving the bound of $I\_\alpha$'s are extremely obscure(for me), so please do read [2] for details.


**Reference**


[1]Capogna, Luca, Donatella Danielli, and Nicola Garofalo. "An embedding theorem and the Harnack inequality for nonlinear subelliptic equations." Communications in Partial Differential Equations 18.9-10 (1993): 1765-1794.


[2]Capogna, Luca, Donatella Danielli, and Nicola Garofalo. "The geometric Sobolev embedding for vector fields and the isoperimetric inequality." Comm. Anal. Geom 2.2 (1994): 203-215.


