The Structure of Sobolev Extension Operators

Abstract

Let $L^{m,p}({\mathbb{R}}^n)$ denote the Sobolev space of functions whose $m$-th derivatives lie in $L^p({\mathbb{R}}^n)$, and assume that $p>n$. For $E\subset {\mathbb{R}}^n$, denote by $L^{m,p}(E)$ the space of restrictions to $E$ of functions $F\in L^{m,p}({\mathbb{R}}^n)$. It is known that there exist bounded linear maps $T:L^{m,p}(E)\to L^{m,p}({\mathbb{R}}^n)$ such that $Tf=f$ on $E$ for any $f\in L^{m,p}(E)$. We show that $T$ cannot have a simple form called "bounded depth."