Abstract:
Let $S \subset \mathbb{R}^{n}$ be a closed nonempty set such that for some $d \in [0,n]$ and $\varepsilon > 0$ the $d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S \cap Q(x,r)) \geq \varepsilon r^{d}$ for all cubes $Q(x,r)$ centered in $x \in S$ with side 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 $\operatorname{Ext}:W_{p}^{1}(\mathbb{R}^{n})|_{S} \to W_{p}^{1}(\mathbb{R}^{n})$ such that $\operatorname{Ext}$ is right inverse for the usual trace operator.
Our results extend those available in the case $p \in (1,n]$ for Ahlfors-regular sets $S$.
