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

Abstract

Let $S\subset {\mathbb{R}}^n$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content ${\mathcal{H}}_{\infty }^d(S)>0$ 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 $W_p^1({\mathbb{R}}^n)$ to the set $S$ is obtained. Furthermore, for each $p\in (\max \{1,n-d\},n]$ and $\epsilon \in (0,\min \{p-(n-d),p-1\})$, new bounded linear extension operators from the trace space $W_p^1({\mathbb{R}}^n){|}_S$ into the space $W_{p-\epsilon }^1({\mathbb{R}}^n)$ are constructed.

The final version accepted in Ann.~Sc.~Norm.~Super.~Pisa Cl.~Sci.~(5) 26, issue 1 (2025)

References in corpus (2)

• Traces of Newton-Sobolev, Hajlasz-Sobolev and BV functions on metric spaces
• Restrictions of Sobolev $W_p^1({\mathbb{R}}^2)$-spaces to planar rectifiable curves