Sobolev extension by linear operators

Fefferman, Charles¹ ; Israel, Arie² ; Luli, Garving³

¹ Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544
² Department of Mathematics, New York University-Courant Institute, Warren Weaver Hall, 251 Mercer Street, New York, NY 10012-1185
³ Department of Mathematics, Yale University, New Haven, CT 06520

Journal of the American Mathematical Society, Tome 27 (2014) no. 1, pp. 69-145

Résumé

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.

Détail

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

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