Calculus of Variations and Geometric Measure Theory

A. Tyulenev

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

created by tyulenev on 27 Mar 2023


Accepted Paper

Inserted: 27 mar 2023
Last Updated: 27 mar 2023

Journal: Ann.~Sc.~Norm.~Super.~Pisa Cl.~Sci.~(5)
Volume: 26
Number: 1
Year: 2023


Let $S \subset \mathbb{R}^{n}$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S) > 0$ 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 $\varepsilon \in (0, \min\{p-(n-d),p-1\})$, new bounded linear extension operators from the trace space $W_{p}^{1}(\mathbb{R}^{n})
_{S}$ into the space $W_{p-\varepsilon}^{1}(\mathbb{R}^{n})$ are constructed.

Keywords: extensions, traces, Frostman measures