Whitney-type extension theorems for jets generated by Sobolev functions

2016/07/06 by Shvartsman, Pavel · 1 citation
#46E35 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1607.01660

Abstract

Let L^m_p(Rⁿ), p∈ [1,∞], 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 to E of m-jets generated by functions F∈ L^m+1_p(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 L^m+1_p(Rⁿ), p>n, as a union of C^m,(d)(Rⁿ)-spaces where d belongs to a family of metrics on Rⁿ with certain "nice" properties. Here C^m,(d)(Rⁿ) is the space of C^m-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 provides an almost optimal extension of m-jets generated by L^m+1_p-functions.

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• Differentiable convex extensions with sharp Lipschitz constants

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