Sobolev extension by linear operators

Fefferman, Charles; Israel, Arie; Luli, Garving

doi:10.1090/S0894-0347-2013-00763-8

Sobolev extension by linear operators
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by Charles L. Fefferman, Arie Israel and Garving K. Luli

J. Amer. Math. Soc. 27 (2014), 69-145

DOI: https://doi.org/10.1090/S0894-0347-2013-00763-8

Published electronically: February 28, 2013

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Abstract:

Let $L^{m,p}(\mathbb {R}^n)$ be the Sobolev space of functions with $m^{\mathrm {th}}$ derivatives lying in $L^p(\mathbb {R}^n)$. Assume that $n< p < \infty$. For $E \subset \mathbb {R}^n$, let $L^{m,p}(E)$ denote the space of restrictions to $E$ of functions in $L^{m,p}(\mathbb {R}^n)$. We show that there exists a bounded linear map $T : L^{m,p}(E) \rightarrow L^{m,p}(\mathbb {R}^n)$ 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 $f : E \rightarrow \mathbb {R}$ when $E$ is finite.

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