how to use the sobolev inequality to obtain the embedding theorem

Question

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!

Answer 1

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.