Whitney extension operators with arbitrary loss of differentiability

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) with a certain loss of derivatives sigma, that is, the operator satisfies the following continuity estimates for all n is an element of N-0 and all F is an element of E(K)

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

where parallel to.parallel to(sigma(n)) denotes the Whitney norm and the map s: N-0 -> N-0 is monotonically increasing with sigma(n) >= n and sigma(0) = 0. From our main result it follows directly that if a compact set (K) over bar admits an extension operator, then it is always possible to construct a second extension operator resembling the original Whitney operators E-n: E-n (K) -> C-n(R-d) where the evaluations of the jet occurring in the Taylor polynomials are approximated by measures. (C) 2020 Elsevier Inc. All rights reserved.