Mathematics

Let $S$ be an arbitrary $d$-thick closed subset of $n$-dimensional Euclidean space. This means that $d$-Hausdorff content of intersection of arbitrary ball (centered in $S$) with $S$ is comparable to the $r^{d}$. For every $p \in (\max\{1,n-d\},\infty)$ we obtain characterization of the trace space of the classical Sobolev space $W_{p}^{1}(\mathbb{R}^{n})$ to the set $S$. Furthermore, we give explicit formula for bounded linear extension operator which is right-inverse for the usual trace operator.