Whitney-type problems for Sobolev spaces

In 1934 H. Whitney posed the following problem. Let $m$ be natural and let $S$ be a nonempty subset of $\mathbb{R}^n$. Given a real valued function $f$ on $S$ find necessary and sufficient conditions for the existence of the extension $F \in C^m(\mathbb{R}^n)$ of $f$, i.е. $F|_S=f$. In the full generality such a problem was solved by C. Fefferman in the middle of 2000-th. A similar problem formulated in the context of Sobolev spaces $W_p^m(\mathbb{R}^n), p \in [1,\infty]$ is of a great importance. Such a problem is very far from the complete solution. The talk will be devoted to some recent results in this direction.