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.

References

Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava, Twice-Ramanujan sparsifiers, STOC’09—Proceedings of the 2009 ACM International Symposium on Theory of Computing, ACM, New York, 2009, pp. 255–262. MR 2780071
Edward Bierstone, Pierre D. Milman, and Wiesław Pawłucki, Differentiable functions defined in closed sets. A problem of Whitney, Invent. Math. 151 (2003), no. 2, 329–352. MR 1953261, DOI 10.1007/s00222-002-0255-6
Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney’s extension theorem, Internat. Math. Res. Notices 3 (1994), 129 ff., approx. 11. MR 1266108, DOI 10.1155/S1073792894000140
Yuri Brudnyi and Pavel Shvartsman, The Whitney problem of existence of a linear extension operator, J. Geom. Anal. 7 (1997), no. 4, 515–574. MR 1669235, DOI 10.1007/BF02921632
Paul B. Callahan and S. Rao Kosaraju, A decomposition of multidimensional point sets with applications to $k$-nearest-neighbors and $n$-body potential fields, J. Assoc. Comput. Mach. 42 (1995), no. 1, 67–90. MR 1370371, DOI 10.1145/200836.200853
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845, DOI 10.1090/gsm/019
Charles L. Fefferman, A sharp form of Whitney’s extension theorem, Ann. of Math. (2) 161 (2005), no. 1, 509–577. MR 2150391, DOI 10.4007/annals.2005.161.509
Charles Fefferman, Whitney’s extension problem for $C^m$, Ann. of Math. (2) 164 (2006), no. 1, 313–359. MR 2233850, DOI 10.4007/annals.2006.164.313
Charles Fefferman, $C^m$ extension by linear operators, Ann. of Math. (2) 166 (2007), no. 3, 779–835. MR 2373373, DOI 10.4007/annals.2007.166.779
Charles Fefferman, The structure of linear extension operators for $C^m$, Rev. Mat. Iberoam. 23 (2007), no. 1, 269–280. MR 2351135, DOI 10.4171/RMI/495
Charles Fefferman, Fitting a $C^m$-smooth function to data. III, Ann. of Math. (2) 170 (2009), no. 1, 427–441. MR 2521121, DOI 10.4007/annals.2009.170.427
Charles Fefferman and Arie Israel, The jet of an interpolant on a finite set, Rev. Mat. Iberoam. 27 (2011), no. 1, 355–360. MR 2815741, DOI 10.4171/RMI/639
Charles Fefferman and Bo’az Klartag, Fitting a $C^m$-smooth function to data. I, Ann. of Math. (2) 169 (2009), no. 1, 315–346. MR 2480607, DOI 10.4007/annals.2009.169.315
Charles Fefferman and Bo’az Klartag, Fitting a $C^m$-smooth function to data. II, Rev. Mat. Iberoam. 25 (2009), no. 1, 49–273. MR 2514338, DOI 10.4171/RMI/569
Georges Glaeser, Étude de quelques algèbres tayloriennes, J. Analyse Math. 6 (1958), 1–124; erratum, insert to 6 (1958), no. 2 (French). MR 101294, DOI 10.1007/BF02790231
A. Israel, A bounded linear extension operator for ${L}^{2,p}(\mathbb {R}^2)$, Annals of Math. (to appear).
Garving K. Luli, $C^{m,\omega }$ extension by bounded-depth linear operators, Adv. Math. 224 (2010), no. 5, 1927–2021. MR 2646115, DOI 10.1016/j.aim.2010.01.027
G.K. Luli, Sobolev extension in one-dimension (2008), notes available at http://www.math.princeton.edu/~gluli/TH/notes.pdf.
P. Shvartsman, Sobolev $W^1_p$-spaces on closed subsets of $\textbf {R}^n$, Adv. Math. 220 (2009), no. 6, 1842–1922. MR 2493183, DOI 10.1016/j.aim.2008.09.020
P. Shvartsman, Lipschitz spaces generated by the Sobolev-Poincaré inequality and extensions of Sobolev functions (preprint) (2011).
P. Shvartsman, On the sum of a Sobolev space and a weighted $L_p$-space (preprint) (2011).
Daniel A. Spielman and Nikhil Srivastava, Graph sparsification by effective resistances, STOC’08, ACM, New York, 2008, pp. 563–568. MR 2582687, DOI 10.1145/1374376.1374456
Daniel A. Spielman and Shang-Hua Teng, Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 81–90. MR 2121588, DOI 10.1145/1007352.1007372
Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3
Hassler Whitney, Differentiable functions defined in closed sets. I, Trans. Amer. Math. Soc. 36 (1934), no. 2, 369–387. MR 1501749, DOI 10.1090/S0002-9947-1934-1501749-3
Hassler Whitney, Functions differentiable on the boundaries of regions, Ann. of Math. (2) 35 (1934), no. 3, 482–485. MR 1503174, DOI 10.2307/1968745