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

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 $W_{p}^{1}(mathbb{R}^{n})$ is given. 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.