On Whitney-type problem for weighted Sobolev spaces on $d$-thick closed sets

A. I. Tyulenev, S. K. Vodop'yanov

Published 2016-06-21Version 1

A~complete intrinsic description of the traces of weighted Sobolev space $W^{l}_{p}(\mathbb{R}^{n}, \gamma)$ on~$d$-thick closed weakly regular subsets $F$ of~$\mathbb{R}^{n}$ with $\gamma \in A_{\frac{p}{r}}(\mathbb{R}^{n})$, $p \in (1,\infty)$, $r \in (\max\{1,n-d\},p)$, $l \in \mathbb{N}$, $0 < d \le n$, is given. The results obtained supplement, on one side, the studies of P.~Shvartsman, who described in~\cite{Shv2}, the traces of the spaces $W^{1}_{p}(\mathbb{R}^{n})$, $p > n$, on arbitrary closed sets, and in~\cite{Shv1}, the traces of the Besov and Lizorkin--Triebel spaces on Ahlfors regular closed subsets of $\mathbb{R}^{n}$ with $l \in \mathbb{N}$, $p \in (1,\infty)$. On the other side, our result supplement the results of V.\,S.~Rychkov~\cite{Ry2}, who described the traces of Sobolev spaces on $d$-thick sets under the condition $d > n-1$.