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In Evans' book on PDEs, section 5.6.3 states the general Sobolev inequalities. Specifically, let $U$ be a bounded open subset of $\mathbb{R}^n$ with $C^1$ boundary. Assuming $u \in W^{k,p}(U)$, and $k<\frac{n}{p}$, the proof states that: Since $D^{\alpha}u \in L^p(U)$ for all $|\alpha| \leq k$ (using multi-index notation here), the Gagliardo-Nirenberg-Sobolev inequality implies:

$$ ||D^{\beta} u ||_{L^{p^*}(U)} \leq C ||u||_{W^{k,p}(U)} \quad \text{if } |\beta| \leq k-1, $$ where $p^* = \frac{np}{n - p}$ is the Sobolev conjugate of $p$ if $1 \leq p < n$.

What does the equation above look like in case of $L^{\infty}$ for the norm on the left, i.e. when $p^* = \infty$? Using the definition of the Sobolev conjugate, this would lead to $p=n$, but then how would you satisfy $k<\frac{n}{p}$ in that case? And how would the norm on the right look like?

Also, is it possible to extend this inequality for $u \in C^{\infty}(U)$?

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    $\begingroup$ The $L^\infty$ case doesn't work due to Logarithmic divergences. If you go a few pages more in Evan's book you see a discussion of the $k > n/p$ case where you get Morrey's inequality. You can get "scaling sharp" versions like the Gagliardo-Nirenberg-Sobolev interpoation inequalities, but they all require interpolating between one norm with derivative $k > n/p$ and one with $k < n/p$. $\endgroup$ Mar 31, 2021 at 17:14
  • $\begingroup$ Thanks @WillieWong. Please feel free to turn your comments into an answer so I can accept it. $\endgroup$ Apr 6, 2021 at 12:26

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(Converted from comment)

The $L^\infty$ case doesn't work due to "logarithmic divergences".

If you go a few more pages into Evans' book, you see a discussion of the $k > n/p$ case which often goes by the name of "Morrey's inequality". You can get versions that are "scaling sharp" which looks a bit like the Gagliardo-Nirenberg-Sobolev interpolation inequalities, but they all require interpolating between one norm with derivative $k > n/p$ and one with $k < n/p$.

If you are willing to look at modifications of Sobolev spaces, then sometimes you can hit the $k = n/p$ endpoint; this requires modifying the definition of Sobolev spaces in a "log" sense. For example, Besov spaces can be regarded as one such modification, and in the Besov hierarchy you can in fact write down "Sobolev inequalities" that use up exactly $n/p$ derivatives.

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