For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the *celebrated Sobolev inequalities*:

If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying $\frac{1}{q}=\frac{1}{p}-\frac{k}{n}$,

If $k > n/p$ then $u$ lies in a particular Hölder space.

It is also known that this *doesn't work* for the borderline case $k=n/p$ (which is related to the Sobolev conjugate $p^*\to \infty$ as $p\to n$), although one would expect/hope for $u\in L^\infty(U)$.

**An exception as Denis mentions: it works for $(k,p)=(n,1)$ via the fundamental theorem of calculus.

**As a counterexample, the function $u(x)=\log\log(1+\frac{1}{|x|})$ with domain $U=int(\mathbb{D}^n)$ lies in $W^{1,n}$ but not in $L^\infty$ (for $n>1$). Instead, apparently the result we end up with is that the functions lie in *"the space of functions with bounded mean oscillation".*

1) **Is there an intuitive / deeper reason as to what goes wrong?**

2) **Is there some sort of geometric reason why the Sobolev embedding theorem has to fail for $k=n/p$?** I've been told that this critical case seems to arise often in geometry/topology.