Sobolev \(W^1_p\)-spaces on \(d\)-thick closed subsets of \({\mathbb{R}}^n \). (English. Russian original) Zbl 1462.46044
Sb. Math. 211, No. 6, 786-837 (2020); translation from Mat. Sb. 211, No. 6, 40-94 (2020).
Summary: 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 \(S\).
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 28A78 | Hausdorff and packing measures |
| 28A25 | Integration with respect to measures and other set functions |
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