Revista Matemática Iberoamericana

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Volume 32, Issue 2, 2016, pp. 377–390
DOI: 10.4171/RMI/888

Published online: 2016-06-08

Whitney extension operators without loss of derivatives

Leonhard Frerick[1], Enrique Jordá[2] and Jochen Wengenroth[3]

(1) Universität Trier, Germany
(2) Universidad Politécnica de Valencia, Alcoy (Alicante), Spain
(3) Universität Trier, Germany

For a compact set $K\subseteq \mathbb R^d$ we characterize the existence of a linear extension operator $E\colon \mathcal E(K) \to \mathcal C^\infty(\mathbb R^d)$ for the space of Whitney jets $\mathcal E(K)$ without loss of derivatives, that is, it satisfies the best possible continuity estimates \[ \sup\{|\partial^\alpha E(f)(x)|: |\alpha|\le n, x\in\mathbb R^d\} \le C_n \|f\|_{n}, \] where $\|\cdot\|_n$ 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 $\epsilon \in (0,1)$, there are $d$ points $x_1\ldots,x_d$ in $K\cap B(x_0,\epsilon)$ satisfying $\mathrm {dist}(x_{n+1}, \text{\rm affine hull}\{x_0,\ldots,x_n\}) \ge \varrho \epsilon$ for all $n\in\{0,\ldots,d-1\}$.

Keywords: Whitney jets, extension operator

Frerick Leonhard, Jordá Enrique, Wengenroth Jochen: Whitney extension operators without loss of derivatives. Rev. Mat. Iberoam. 32 (2016), 377-390. doi: 10.4171/RMI/888