# General Sobolev inequalities for $L^\infty$ norm

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 $$u \in C^{\infty}(U)$$?

• 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 $k < n/p$. Commented Mar 31, 2021 at 17:14
• Thanks @WillieWong. Please feel free to turn your comments into an answer so I can accept it. Commented Apr 6, 2021 at 12:26

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.