Whitney extension operators without loss of derivatives

Abstract

[EN] For a compact set K subset of R-d we characterize the existence of a linear extension operator E: E (K) -> C-infinity(R-d) for the space of Whitney jets E (K) without loss of derivatives, that is, it satisfies the best possible continuity estimates

sup{vertical bar partial derivative(alpha) E(f)(x)vertical bar : vertical bar alpha vertical bar <= n, x is an element of R-d} <= C-n parallel to f parallel to(n),

where parallel to . parallel to(n) denotes the n-th Whitney norm. The characterization is by a surprisingly simple purely geometric condition introduced by Jonsson, Sjogren, and Wallis: there is rho is an element of (0, 1) such that, for every x(0) is an element of K and epsilon is an element of (0, 1), there are d points x(1)..., x(d) in K n B(x(0), epsilon) satisfying dist(x(n+1), affine hull{x(0),..., x(n)}) = >= rho epsilon for all n is an element of {0,..., d - 1}.