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 $p \in (1,n]$
Accepted Paper
Inserted: 27 mar 2023
Last Updated: 26 oct 2023
Journal: Ann.~Sc.~Norm.~Super.~Pisa Cl.~Sci.~(5)
Volume: 26
Number: 1
Year: 2023
Abstract:
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
Download: