Tame linear extension operators for smooth Whitney functions

Abstract

For a compact set K subset of R(d) we present a rather easy construction of a linear extension operator E : epsilon (K) -> C(infinity) for the space of Whitney jets epsilon (K) which satisfies linear tame continuity estimates sup{vertical bar partial derivative alpha E(f)(x)vertical bar: vertical bar alpha vertical bar <= m, x is an element of R(d)} <= C(m,epsilon)parallel to f parallel to((r+epsilon)m), where parallel to center dot parallel to(s) denotes the s-th Whitney norm. The construction turns out to be possible if and only if the local Markov inequality LMI(s) introduced by Bos and Milmon holds for every s > r on K. In particular, epsilon(K) admits a tame linear extension operator if and only if the local Markov inequality LMI(s) holds on K for some s >= 1. (C) 2011 Elsevier Inc. All rights reserved.