  Let $S \subset \mathbb{R}^{n}$ be an arbitrary nonempty compact set such that
the $d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S) > 0$ for some $d \in
(0,n]$. For each $p \in (\max\{1,n-d\},n]$, an almost sharp intrinsic
description of the trace space $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ of the Sobolev
space $W_{p}^{1}(\mathbb{R}^{n})$ to the set $S$ is obtained. Furthermore, for
each $p \in (\max\{1,n-d\},n]$ and $\varepsilon \in (0, \min\{p-(n-d),p-1\})$,
new bounded linear extension operators from the trace space
$W_{p}^{1}(\mathbb{R}^{n})|_{S}$ into the space
$W_{p-\varepsilon}^{1}(\mathbb{R}^{n})$ are constructed.

Functional Analysis