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 $T:C^m(E)⟶C^m({\mathbb{R}}^n)$ 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.