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I want to know if certain conditions hold after applying the Sobolev Extension Theorem:

Assume $U$ is a bounded open subset of $\mathbb{R}^{n}$ and $\partial U$ is $C^{1}$. Suppose $1 \leq p < n$.

If we fix $1 \leq q < p^{*}$, where $p^{*}$ is the Sobolev conjugate of $p$, then since we also have that $U$ is bounded it follows from Nirenberg-Gagliardo-Sobolev Inequaity that $W^{1,p}(U) \subset L^{q}(U)$ and $||u||_{L^{q}(U)} \leq C||u||_{W^{1,p}(U)}$.

If we consider the linear extension operator $P: W^{1,p}(U) \rightarrow W^{1,p}(\mathbb{R}^{n})$, then does the above assumptions still hold? In other words is the following true?

$W^{1,p}(\mathbb{R}^{n}) \subset L^{q}(\mathbb{R}^{n})$ and the inequality $||Pu||_{L^{q}(\mathbb{R}^{n})} \leq C||Pu||_{W^{1,p}(\mathbb{R}^{n})}$.

Thanks for any assistance.

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Take the your case $1 \leq q < p^{*}$ where $u \in W^{1,p}(U)$. Let $Pu := v$ where $P$ is an extension operator $P: W^{1,p}(U) \rightarrow W^{1,p}(\mathbb{R}^{n})$. Consider $v \in W^{1,p}(\mathbb{R}^{n})$ where $\text{spt}(v) \subset V$, it follows then that $v \in W^{1,p}(V)$.

By Gagliardo-Nirenberg-Sobolev we have:

$||v||_{L^{p^{*}}(\mathbb{R}^{n})} = (\int_{\mathbb{R}^{n}}|v|^{p^{*}})^{\frac{1}{p^{*}}} \leq C||Dv||_{L^{p}(\mathbb{R}^{n})}$

Using Holder's Inequality we have the following:

$\int_{\mathbb{R}^{n}}|v|^{q}dx =$ $\int_{V}|v|^{q}dx \leq (\int_{V}(|v|^{q})^{\frac{p^{*}}{q}}dx)^{\frac{q}{p^{*}}}$ $(\int_{V} 1^{\frac{\frac{p^{*}}{q}}{\frac{p^{*}}{q}-1}}dx)^{\frac{\frac{p^{*}}{q}-1}{\frac{p^{*}}{q}}}$ $\leq C(V)^{q}||Dv||_{L^{p}(V)}^{q}$ where $C(V)$ is a constant depending on the bounded set $V$ (Since it is bounded it therefore has finite measure).

It then immediately follows that $||v||_{L^{q}(V)} \leq C(V)||Dv||_{L^{p}(V)} \leq C||v||_{W^{1,p}(V)}$ which is the result you were seeking.

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There is no such thing as the extension operator. The validity of norm inequality for $P$ may depend on what extension operator is used. For example, one can extend Sobolev functions on the unit ball to Sobolev functions on $\mathbb R^n$ so that the extension vanishes for $|x|>2$. In such a case, the embedding results for bounded domains still apply.

More concretely, you asked whether the continuous embedding $W^{1,p}(\mathbb R^n)\subset L^q(\mathbb R^n)$ holds. The answer is positive for $p\le q\le p^*$, negative for $1\le q<p$.

Reasons: the embedding holds for $q=p$ by the definition of $W^{1,p}$, and for $q=p^*$ by the GNS inequality. For intermediate $q$ it holds by interpolation (log-convexity of $L^p$ norm).

If $1\le q<p$, consider $f(x)=(1+|x|)^{-n/q}$. A direct computation shows that $f\in W^{1,p}(\mathbb R^n)$, but $f\notin L^q(\mathbb R^n)$.

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