Whitney extension operators without loss of derivatives

Abstract

For a compact set $K\subseteq {\mathbb{R}}^d$ we characterize the existence of a linear extension operator $E:\mathcal{E}(K)\to {\mathcal{C}}^{\infty }({\mathbb{R}}^d)$ for the space of Whitney jets $\mathcal{E}(K)$ without loss of derivatives, that is, it satisfies the best possible continuity estimates

where $\Vert \cdot {\Vert }_n$ denotes the $n$-th Whitney norm. The characterization is by a surprisingly simple purely geometric condition introduced by Jonsson, Sjögren, and Wallis: there is $\varrho \in (0,1)$ such that, for every $x_0\in K$ and $ϵ\in (0,1)$, there are $d$ points $x_1\dots ,x_d$ in $K\cap B(x_0,ϵ)$ satisfying $\mathrm{dist}(x_{n+1},\text{affine hull}\{x_0,\dots ,x_n\})\ge \varrho ϵ$ for all $n\in \{0,\dots ,d-1\}$.