## 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.## References

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## Bibliographic Information

**Charles L. Fefferman**- Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544
- MR Author ID: 65640
**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
- 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 - © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**27**(2014), 69-145 - MSC (2010): Primary 42B99
- DOI: https://doi.org/10.1090/S0894-0347-2013-00763-8
- MathSciNet review: 3110796