The Whitney extension problem and Lipschitz selections of setvalued mappings in jetspaces
Author:
Pavel Shvartsman
Journal:
Trans. Amer. Math. Soc. 360 (2008), 55295550
MSC (2000):
Primary 46E35; Secondary 52A35, 54C60, 54C65
Published electronically:
April 9, 2008
MathSciNet review:
2415084
Fulltext PDF Free Access
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Abstract: We study a variant of the Whitney extension problem (1934) for the space . We identify with a space of Lipschitz mappings from into the space of polynomial fields on equipped with a certain metric. This identification allows us to reformulate the Whitney problem for as a Lipschitz selection problem for setvalued mappings into a certain family of subsets of . We prove a Hellytype criterion for the existence of Lipschitz selections for such setvalued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for due to C. Fefferman.
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 [1]
 E. Bierstone and P.D. Milman, norms on finite sets and extension criteria, Duke Math. J. 137 (2007), 118. MR 2309142
 [2]
 E. Bierstone, P. Milman and W. Pawlucki, Differentiable functions defined in closed sets. A problem of Whitney. Invent. Math. 151 (2003), no. 2, 329352. MR 1953261 (2004h:58009)
 [3]
 E. Bierstone, P. Milman and W. Pawlucki, Higherorder tangents and Fefferman's paper on Whitney's extension problem, Ann. of Math. (2) 164 (2006), no. 1, 361370. MR 2233851 (2007g:58012)
 [4]
 Yu. Brudnyi and P. Shvartsman, Generalizations of Whitney's Extension Theorem, Intern. Math. Research Notices (1994), no. 3, 129139. MR 1266108 (95c:58018)
 [5]
 Yu. Brudnyi and P. Shvartsman, The Whitney Problem of Existence of a Linear Extension Operator, J. Geom. Anal., 7, no. 4 (1997), 515574. MR 1669235 (2000a:46051)
 [6]
 Yu. Brudnyi and P. Shvartsman, The Trace of Jet Space to an arbitrary closed subset of , Trans. Amer. Math. Soc. 350 (1998) 15191553. MR 1407483 (98i:58010)
 [7]
 Yu. Brudnyi and P. Shvartsman, Whitney's Extension Problem for Multivariate functions, Trans. Amer. Math. Soc. 353 (2001), no. 6, 24872512. MR 1814079 (2002b:46052)
 [8]
 L. Danzer, B. Grünbaum and V. Klee, Helly's Theorem and Its Relatives, in ``Am. Math. Soc. Symp. on Convexity,'' Seattle, Proc. Symp. Pure Math., Vol. 7, pp. 101180, Amer. Math. Soc., Providence, R.I., 1963. MR 0157289 (28:524)
 [9]
 C. Fefferman, A sharp form of Whitney's extension theorem, Ann. of Math. (2) 161 (2005), no. 1, 509577. MR 2150391 (2006h:58008)
 [10]
 C. Fefferman, Interpolation and Extrapolation of Smooth Functions by Linear Operators, Rev. Mat. Iberoamericana 21 (2005), no. 1, 313348. MR 2155023 (2006h:58009)
 [11]
 C. Fefferman, Whitney's Extension Problem in Certain Function Spaces, Rev. Mat. Iberoamericana (to appear).
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 C. Fefferman, A Generalized Sharp Whitney Theorem for Jets, Rev. Mat. Iberoamericana 21 (2005), no. 2, 577688. MR 2174917 (2007a:58009)
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 C. Fefferman, Whitney's Extension Problem for , Ann. of Math. (2) 164 (2006), no. 1, 313359. MR 2233850 (2007g:58013)
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 C. Fefferman, Extension of Smooth Functions by Linear Operators, Rev. Mat. Iberoamericana (to appear).
 [15]
 C. Fefferman, Extension by Linear Operators, Ann. of Math. (2) 166 (2007), no. 3, 779835.
 [16]
 G. Glaeser, Étude de quelques algebres Tayloriennes, J. d'Analyse Math. 6 (1958), 1125. MR 0101294 (21:107)
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 P. Shvartsman, Lipschitz selections of multivalued mappings and the traces of the Zygmund class functions to an arbitrary compact, Dokl. Akad. Nauk SSSR 276 (1984), no. 3, 559562; English transl. in Soviet. Math. Dokl. 29 (1984), no. 3, 565568. MR 752427 (85j:46057)
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 P. Shvartsman, On the traces of functions of the Zygmund class, Sib. Mat. Zh. 28 (1987), no. 5, 203215; English transl. in Sib. Math. J. 28 (1987) 853863. MR 924998 (89a:46081)
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 P. Shvartsman, Lipschitz Selections of SetValued Mappings and Helly's Theorem, J. Geom. Anal. 12 (2002) 289324. MR 1888519 (2002m:52006)
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 P. Shvartsman, Barycentric Selectors and a Steinertype Point of a Convex Body in a Banach Space, J. Func. Anal. 210 (2004), no. 1, 142. MR 2051631 (2005b:46036)
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Additional Information
Pavel Shvartsman
Affiliation:
Department of Mathematics, TechnionIsrael Institute of Technology, Haifa 32000, Israel
Email:
pshv@tx.technion.ac.il
DOI:
http://dx.doi.org/10.1090/S0002994708044693
PII:
S 00029947(08)044693
Keywords:
Whitney's extension problem,
smooth functions,
finiteness,
metric,
jetspace,
setvalued mapping,
Lipschitz selection
Received by editor(s):
March 20, 2006
Received by editor(s) in revised form:
November 29, 2006
Published electronically:
April 9, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
