Sobolev Extension By Linear Operators

Let $L^{m,p}(\R^n)$ be the Sobolev space of functions with $m^{th}$ derivatives lying in $L^p(\R^n)$. Assume that $n< p < \infty$. For $E \subset \R^n$, let $L^{m,p}(E)$ denote the space of restrictions to $E$ of functions in $L^{m,p}(\R^n)$. We show that there exists a bounded linear map $T : L^{m,p}(E) \rightarrow L^{m,p}(\R^n)$ such that, for any $f \in L^{m,p}(E)$, we have $Tf = f$ on $E$. We also give a formula for the order of magnitude of $\|f\|_{L^{m,p}(E)}$ for a given $f : E \rightarrow \R$ when $E$ is finite.

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