TITLE: Open problems in Sobolev spaces
QUESTION [29 upvotes]: What are the open problems in the theory of Sobolev spaces?

I would like to see problems that are yes or no only. Also I would like to see problems with the statements that are short and easy to understand for someone who has a basic knowledge in the theory, say at the level of the book by Evans and Gariepy.
The problems do not have to be a well know ones. Just the problems you think are interesting. 

Please, list each problem as a separate answer.

That will allow people to leave comments related exclusively to this particular problem.
I have been working with Sobolev spaces for most of my adult live and I have some of my favorite problems that I will list below. But I will do it later, because first I would like to see your problems.

REPLY [4 votes]: Let $E \subset \mathbb R^n$. For $f : E \to \mathbb R$, let
$$\|f\|_{L^{m,p}(E)} = \inf\{\|F\|_{L^{m,p}(\mathbb R^n)} : F|_E = f\}.$$
Here $\| \cdot \|_{L^{m,p}}$ is the homogeneous Sobolev seminorm
$$\|F\|_{L^{m,p}(\mathbb R^n)} = \max\limits_{|\alpha| = m} \|\partial^\alpha F\|_{L^p(\mathbb R^n)}$$
Fefferman, Israel, and Luli
have shown that in the case $p>n$ there is a linear extension operator $T : L^{m,p}(E) \to L^{m,p}(\mathbb R^n)$ such that $Tf|_E = f$ and $\|Tf\|_{L^{m,p}(\mathbb R^n)} \leq C \|f\|_{L^{m,p}(E)}$, where $C$ depends on $m,n,p$ only. To emphasize the point, $C$ does not depend at all on $E$, which can be completely arbitrary.
In principle, a result of this kind makes sense whenever $p > n/m$, but as far as I know nothing is known about the case $p \leq n$. Fefferman, Israel, and Luli have shown quite a bit more about these operators as well, but even the question of whether linear extension operators of uniformly bounded norm exist is open in the case $p \leq n$.