Whitney extension operators without loss of derivatives

Abstract

For a compact set K⊆Rd we characterize the existence of a linear extension operator E:E(K)→C∞(Rd) for the space of Whitney jets E(K) without loss of derivatives, that is, it satisfies the best possible continuity estimates sup{|∂αE(f)(x)|:|α|≤n,x∈Rd}≤Cn∥f∥n, where ∥⋅∥n denotes the nn-th Whitney norm. The characterization is by a surprisingly simple purely geometric condition introduced by Jonsson, Sjogren, and Wallis: there is ϱ∈(0,1) such that, for every x0∈K and ϵ∈(0,1), there are dd points x1…,xd in K∩B(x0,ϵ) satisfying dist(xn+1,\rm affine hull{x0,…,xn})≥ϱϵ for all n∈{0,…,d−1}.