Sobolev extension by linear operators

Abstract

Let $L^{m,p}(\mathbb {R}^n)$ be the Sobolev space of functions with $m^{\mathrm {th}}$ derivatives lying in $L^p(\mathbb {R}^n)$. Assume that $n< p < \infty$. For $E \subset \mathbb {R}^n$, let $L^{m,p}(E)$ denote the space of restrictions to $E$ of functions in $L^{m,p}(\mathbb {R}^n)$. We show that there exists a bounded linear map $T : L^{m,p}(E) \rightarrow L^{m,p}(\mathbb {R}^n)$ such that, for any $f \in L^{m,p}(E)$, we have $Tf = f$ on $E$. We also give a formula for the order of magnitude of $\|f\|_{L^{m,p}(E)}$ for a given $f : E \rightarrow \mathbb {R}$ when $E$ is finite.