Extension theory for Sobolev spaces on open sets with Lipschitz boundaries (Q2707687)

The aim of the author is to present a survey of the extension theory for Sobolev spaces \(W^{\ell }_p(\Omega)\), \(\ell \in \mathbb N\), \(1\leq p \leq \infty \), given on an open set \(\Omega \) in \(\mathbb R^n\) with a Lipschitz boundary. The restriction to such a kind of sets \(\Omega \) results from its importance in applications. The exposition is divided into three parts. The first, main part consists of a construction of extension operator NEWLINE\[NEWLINE T\: W^{\ell }_p (\Omega) \rightarrow W^{\ell }_p (\mathbb R^n), \quad (Tf) (x) = f(x), \quad x\in \Omega , NEWLINE\]NEWLINE which is linear and bounded. The second one represents a construction of such operators for various modifications of Sobolev spaces \(W^{\ell }_p(\Omega)\). Finally, the last part is dedicated to sharp estimates of norms of extension operators.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00033].