- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:48
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
32
2016
2
Whitney extension operators without loss of derivatives
Leonhard
Frerick
Universität Trier, TRIER, GERMANY
Enrique
Jordá
Universidad Politécnica de Valencia, ALCOY (ALICANTE), SPAIN
Jochen
Wengenroth
Universität Trier, TRIER, GERMANY
Whitney jets, extension operator
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 $n\in\{0,\ldots,d-1\}$.
Operator theory
377
390
10.4171/RMI/888
http://www.ems-ph.org/doi/10.4171/RMI/888