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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. "An isoperimetric inequality and the geometric Sobolev embedding for vector fields." Math. Res. Lett 1.2 (1994): 263-268.

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.

Disclaimer: Not an expert in analysis/PDE, happen to know tangential results while studying Whitney-type embeddings.

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. "An isoperimetric inequality and the geometric Sobolev embedding for vector fields." Math. Res. Lett 1.2 (1994): 263-268.

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. "An isoperimetric inequality and the geometric Sobolev embedding for vector fields." Math. Res. Lett 1.2 (1994): 263-268.

Disclaimer: Not an expert in analysis/PDE, happen to know tangential results while studying Whitney-type embeddings.

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. "An isoperimetric inequality and the geometric Sobolev embedding for vector fields." Math. Res. Lett 1.2 (1994): 263-268.

Disclaimer: Not an expert in analysis/PDE, happen to know tangential results while studying Whitney-type embeddings.

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. "An isoperimetric inequality and the geometric Sobolev embedding for vector fields." Math. Res. Lett 1.2 (1994): 263-268.