Whitney-type extension theorems for jets generated by Sobolev functions

TL;DR

This paper characterizes the restrictions of jets generated by Sobolev functions to closed sets, extending Whitney's classical theorem to Sobolev spaces using a novel approach involving metrics and Lipschitz conditions.

Contribution

It provides an intrinsic trace criterion for Sobolev-generated jets and shows that classical Whitney extension operators are nearly optimal for these jets.

Findings

Characterization of Sobolev jet restrictions on closed sets.

Extension operators are nearly optimal for Sobolev jets.

Unified approach using metrics and Lipschitz spaces.

Abstract

Let $L^m_p(R^n)$, $p\in [1,\infty]$, be the homogeneous Sobolev space, and let $E\subset R^n$ be a closed set. For each $p>n$ and each non-negative integer $m$ we give an intrinsic characterization of the restrictions to $E$ of $m$-jets generated by functions $F\in L^{m+1}_p(R^n)$. 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 $p=\infty$ this result coincides with the classical Whitney-Glaeser extension theorem for $m$-jets. Our approach is based on a representation of the Sobolev space $L^{m+1}_p(R^n)$, $p>n$, as a union of $C^{m,(d)}(R^n)$-spaces where $d$ belongs to a family of metrics on $R^n$ with certain "nice" properties. Here $C^{m,(d)}(R^n)$ is the space of $C^m$-functions on $R^n$ 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 $p\in (n,\infty)$, the very same classical linear Whitney extension operator provides an almost optimal extension of $m$-jets generated by $L^{m+1}_p$-functions.