Sobolev Embedding Theorem, when $n = p$

Question

I am studying Sobolev spaces by using the book by Evans.

I am wondering about the Embedding Theorem for $p = n$. It is said that it is considered in chapter 5.8.1, where I only find the Poincare inequalities. For now I don't understand how they are related to the case $p = N$.

Does anyone know a reference or a proof?

Edit: I am speaking about the Sobolev Embedding Theorems. Generally I hope that for $p = n$ we have something like

$$W^{1,p}(\mathbb{R}^N) \to L^p(\mathbb{R}^N)$$

What kinds of embedding theorem do you expect? In particular, which $L^q$ should $W^{1,n}$ be embedded into? — Arctic Char, May 16 '23 at 15:56
Actually, this has already been asked (and answered) before. See e.g. this question. — cs89, May 16 '23 at 16:10
You are completely right, I have overseen this question. Thanks! — Metalhead, May 16 '23 at 16:16

score 0 · Accepted Answer · answered May 16 '23 at 16:07

A great reference for such results is the following book. In particular, case (3) of Theorem 2.31 page 69 should correspond to what you are looking for. It says

If $p = N$, then for every $p \leq q < +\infty$, we have $W^{1,N}(\mathbb{R}^N) \hookrightarrow L^q(\mathbb{R}^N)$.

Note that it is stated with $N = mp$ and $W^{m,N}$ but I take it from your question that you are interested in the case $m=1$.

Demengel, Françoise; Demengel, Gilbert, Functional spaces for the theory of elliptic partial differential equations. Transl. from the French by Reinie Erné, Universitext. Berlin: Springer (ISBN 978-1-4471-2806-9/pbk; 978-1-4471-2807-6/ebook). xviii, 465 p. (2012). ZBL1239.46001.

Sobolev Embedding Theorem, when $n = p$

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