The Structure of Linear Extension Operators for $C^m$

Abstract

For any subset $E \subset \mathbb{R}^n$, let $C^m (E)$ denote the Banach space of restrictions to $E$ of functions $F \in C^m (\mathbb{R}^n)$. It is known that there exist bounded linear maps $T:C^m(E)\longrightarrow C^m(\mathbb{R}^n)$ such that $Tf = f$ on $E$ for any $f \in C^m (E)$. We show that $T$ can be taken to have a simple form, but cannot be taken to have an even simpler form.