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Sobolev extension by linear operators


Authors: Charles L. Fefferman, Arie Israel and Garving K. Luli
Journal: J. Amer. Math. Soc. 27 (2014), 69-145
MSC (2010): Primary 42B99
DOI: https://doi.org/10.1090/S0894-0347-2013-00763-8
Published electronically: February 28, 2013
MathSciNet review: 3110796
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Abstract: Let $ L^{m,p}(\mathbb{R}^n)$ be the Sobolev space of functions with $ m^{\textup {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 $ \Vert f\Vert _{L^{m,p}(E)}$ for a given $ f : E \rightarrow \mathbb{R}$ when $ E$ is finite.


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Additional Information

Charles L. Fefferman
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544

Arie Israel
Affiliation: Department of Mathematics, New York University-Courant Institute, Warren Weaver Hall, 251 Mercer Street, New York, NY 10012-1185

Garving K. Luli
Affiliation: Department of Mathematics, Yale University, New Haven, CT 06520

DOI: https://doi.org/10.1090/S0894-0347-2013-00763-8
Received by editor(s): May 11, 2012
Received by editor(s) in revised form: November 12, 2012
Published electronically: February 28, 2013
Additional Notes: The first author was partially supported by NSF and ONR grants DMS 09-01040 and N00014-08-1-0678
The second author was partially supported by NSF postdoctoral fellowship DMS-1103978
The third author was partially supported by NSF and ONR grants DMS 09-01040 and N00014-08-1-0678
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.