Lipschitz spaces generated by the Sobolev-Poincaré inequality and extensions of Sobolev functions

Abstract

Let $d$ be a metric on $R^n$ and let $C^{m,(d)}(R^n)$ be the space of $C^m$-function on $R^n$ whose partial derivatives of order $m$ belong to the space $Lip(R^n;d)$. We show that the homogeneous Sobolev space $L_p^{m+1}(R^n),p>n,$ can be represented 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. This enables us in several important cases to give intrinsic characterizations of the restrictions of Sobolev spaces to arbitrary closed subsets of $R^n$. In particular, we generalize the classical Whitney extension theorem for the space $C^m(R^n)$ to the case of the Sobolev space $L_p^m(R^n)$ whenever $m\ge 1$ and $p>n$.

49 pages

Cited by in corpus (2)

• Sobolev Extension By Linear Operators
• Sobolev $L_p^2$-functions on closed subsets of $R^2$