Whitney-type extension theorems for jets generated by Sobolev functions

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Let Lᵐₚ(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∈ Lᵐ⁺¹ₚ(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 Lᵐ⁺¹ₚ(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 provides an almost optimal extension of m-jets generated by Lᵐ⁺¹ₚ-functions.