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Let $L^m_p(R^n)$, $p\in [1,\infty]$, be the homogeneous Sobolev space, andlet $E\subset R^n$ be a closed set. For each $p>n$ and each non-negativeinteger $m$ 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 isexpressed in terms of variations of corresponding Taylor remainders of $m$-jetsevaluated on a certain family of "well separated" two point subsets of $E$. For$p=\infty$ this result coincides with the classical Whitney-Glaeser extensiontheorem 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 partialderivatives 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 operatorprovides an almost optimal extension of $m$-jets generated by$L^{m+1}_p$-functions.

Whitney-type extension theorems for jets generated by Sobolev functions