Interpolation and extrapolation of smooth functions by linear operators

Abstract

Let $C^{m , 1} ( \mathbb{R}^n)$ be the space of functions on $\mathbb{R}^n$ whose $m^{\sf th}$ derivatives are Lipschitz 1. For $E \subset \mathbb{R}^n$, let $C^{m , 1} (E)$ be the space of all restrictions to $E$ of functions in $C^{m,1} ( \mathbb{R}^n)$. We show that there exists a bounded linear operator $T: C^{m , 1} (E) \rightarrow C^{m , 1} ( \mathbb{R}^n)$ such that, for any $f \in C^{m , 1} ( E )$, we have $T f = f$ on $E$.