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Let $S \subset \mathbb{R}^{n}$ be an arbitrary nonempty compact set such thatthe $d$-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 intrinsicdescription of the trace space $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ of the Sobolevspace $W_{p}^{1}(\mathbb{R}^{n})$ to the set $S$ is obtained. Furthermore, foreach $p \in (\max\{1,n-d\},n]$ 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.

Almost sharp descriptions of traces of Sobolev  $W_{p}^{1}(\mathbb{R}^{n})$-spaces to arbitrary compact subsets of  $\mathbb{R}^{n}$. The case $p \in (1,n]$