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I am reading Luca Capogna's article An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations. 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*} \left(\int_{U}|f|^{q}dx \right)^{\frac{1}{q}}&=\left(\int_{U}|\sum_{i=1}^{n}f\phi_{i}|^{q}dx \right)^{\frac{1}{q}} \end{align*} Then I stuck here and don't know how to continue, it seems far away to the right side. Can some one help me? thank you very much!

I am reading Luca Capogna's article An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations. 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!

I am reading Luca Capogna's article An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations. 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 $B_{R}$?

I am reading Luca Capogna's article An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations. 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*} \left(\int_{U}|f|^{q}dx \right)^{\frac{1}{q}}&=\left(\int_{U}|\sum_{i=1}^{n}f\phi_{i}|^{q}dx \right)^{\frac{1}{q}} \end{align*} Then I stuck here and don't know how to continue, it seems far away to the right side. Can some one help me? thank you very much!

I am reading Luca Capogna's article <An Embedding theorem and the harnack inequalitiy for nonlinear subelliptic equations>. In this article, the authors proved the following theorem

(Theorem 2.3) Let $U\subset \mathbb{R}^n$ be a bounded open set and denot by $Q$ the homogeneous dimession 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).wiht $R\leq R_{0}$, 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 said that using a standard partition of the unity argument can imply $$ S_{c}^{1,p}(U)\hookrightarrow L^{q}(U)$$ for any $U\subset\subset \mathbb{R}^n$ I don't know how to use the partition of unity to obtain that result. Can someone show it in detail?. Further more, Can we get some thing like $$ \left(\int_{U}|u|^{q}dx\right)^{\frac{1}{q}}\leq C\left(\int_{U}|D_{L}u|^{p}dx \right)^{\frac{1}{p}}$$ ? that use $U\subset\subset \mathbb{R}^n$ instead the subunit ball $B_{R}$  ?

I am reading Luca Capogna's article An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations. 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 $B_{R}$?