Tame linear extension operators for smooth Whitney functions (Q544013)

A linear operator \( T : G \to H \) between Fréchet spaces is linear tame if there is \( s \in \mathbb N \) such that \( \| Tx \|_k \leq s \| x \|_{sk} \) for all \( k \in \mathbb N \). For a compact \( K \subset \mathbb R^d \) the space of Whitney jets on~\(K\) is denoted by \( \mathcal E(K) \). The authors prove that there is a linear tame extension operator \( E : \mathcal E(K) \to C^\infty(\mathbb R^d) \) if and only if \( K \) satisfies the local Markov inequality introduced by \textit{L. P. Bos} and \textit{P. D. Milman} [Geom. Funct. Anal. 5, No. 6, 853--923 (1995; Zbl 0848.46022)]. The extension operator is constructed by an interpolation formula using function values only. A careful choice of the parameters in this formula guarantees the linear tame continuity estimates.