Whitney-type extension theorems for jets generated by Sobolev functions

Abstract

Let Lpm(Rn), 1≤p≤∞, be the homogeneous Sobolev space, and let E⊂Rn be a closed set. For each p>n and each non-negative integer m we give an intrinsic characterization of the restrictions {DαF|E:|α|≤m} to E of m-jets generated by functions F∈Lpm+1(Rn). Our trace criterion is expressed in terms of variations of corresponding Taylor remainders of m-jets evaluated on a certain family of “well separated” two point subsets of E. For p=∞ this result coincides with the classical Whitney–Glaeser extension theorem for m-jets.Our approach is based on a representation of the Sobolev space Lpm+1(Rn), p>n, as a union of Cm,(d)(Rn)-spaces where d belongs to a family of metrics on Rn with certain “nice” properties. Here Cm,(d)(Rn) is the space of Cm-functions on Rn whose partial derivatives of order m are Lipschitz functions with respect to d. This enables us to show that, for every non-negative integer m and every p∈(n,∞), the very same classical linear Whitney extension operator as in [31] provides an almost optimal extension of m-jets generated by Lpm+1-functions.