Embedding of $W^{1,n}(U)\hookrightarrow L^q(U)$

Question

Let $U$ be a $C^1$ bounded region in $\Bbb{R}^n$,we know the Sobolev inequality given as follows:

Asssume $1\le p <n$ for any $u \in W^{1,p}(U)$ we have $u\in L^{p^\star}(U)$ where $p^\star = 1/(\frac{1}{p} - \frac{1}{n})$.moreover the inequality holds.

$$\|u\|_{L^{q^\star}(U)} \le C \|u\|_{W^{1,p}(U)}$$

The question is how to deduced the embedding relationship for $p = n$ that is : $$W^{1,n}(U)\hookrightarrow L^q(U)$$

For any $1\le q <\infty$

Answer 1

Since $L^n(U)\subset L^p(U)$ for $p<n$, $W^{1,n}(U)\subset W^{1,p}(U)\subset L^{p^*}(U)$. Since $p^*\to \infty$ as $p\to n$, the result follows