Sobolev $W_{p}^{1}(\mathbb{R}^{n})$ spaces on $d$-thick closed subsets of $\mathbb{R}^{n}$

TL;DR

This paper characterizes the restriction of Sobolev spaces to certain fractal-like sets with positive Hausdorff content and constructs extension operators, broadening understanding of Sobolev traces on irregular sets.

Contribution

It provides an intrinsic characterization of Sobolev space restrictions to $d$-thick sets and constructs bounded extension operators, extending previous results to less restrictive sets.

Findings

Characterization of Sobolev restrictions on $d$-thick sets.

Existence of bounded extension operators for these restrictions.

Extension of trace results to more general sets with positive Hausdorff content.

Abstract

Let $S \subset \mathbb{R}^{n}$ be a~closed 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 every $p \in (1,\infty)$, denote by $W_{p}^{1}(\mathbb{R}^{n})$ the classical Sobolev space on $\mathbb{R}^{n}$. We give an~intrinsic characterization of the restriction $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ of the space $W_{p}^{1}(\mathbb{R}^{n})$ to~the set $S$ provided that $p > \max\{1,n-d\}$. 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. In particular, for $p > n-1$ we characterize the trace space of the Sobolev space $W_{p}^{1}(\mathbb{R}^{n})$ to the closure $\overline{\Omega}$ of an arbitrary open path-connected set~$\Omega$. Our results extend those available for $p \in (1,n]$ with much more stringent restrictions on~$S$.