Sobolev $W^1_p$-spaces on $d$-thick closed subsets of $\mathbb R^n$

Let $S\subset\mathbb R^n$ be a nonempty closed set such that for some $d\in[0,n]$ and $\varepsilon>0$ the $d$-Hausdorff content $\mathscr H^d_\infty(S\cap Q(x,r))\geq\varepsilon r^d$ for all cubes $Q(x,r)$ with centre $x\in S$ and edge length $2r\in(0,2]$. For each $p>\max\{1,n-d\}$ we give an intrinsic characterization of the trace space $W_p^1(\mathbb R^n)|_S$ of the Sobolev space $W_p^1(\mathbb R^n)$ to the set $S$. Furthermore, we prove the existence of a bounded linear operator $\operatorname{Ext}\colon W_p^1(\mathbb R^n)|_S\to W_p^1(\mathbb R^n)$ such that $\operatorname{Ext}$ is the right inverse to the standard trace operator. Our results extend those available in the case $p\in(1,n]$ for Ahlfors-regular sets $S$. Bibliography: 36 titles.

Sobolev spaces, Whitney problem, traces, extension operators