Extension of $C^{m, \omega}$-Smooth Functions by Linear Operators

Abstract

Let $C^{m, \omega} ( \mathbb{R}^n)$ be the space of functions on $\mathbb{R}^n$ whose $m^{\sf th}$ derivatives have modulus of continuity $\omega$. For $E \subset \mathbb{R}^n$, let $C^{m , \omega} (E)$ be the space of all restrictions to $E$ of functions in $C^{m , \omega} ( \mathbb{R}^n)$. We show that there exists a bounded linear operator $T: C^{m , \omega} ( E ) \rightarrow C^{m , \omega } ( \mathbb{R}^n)$ such that, for any $f \in C^{m , \omega} ( E )$, we have $T f = f$ on $E$.