Available in electronic format
Available in print format		 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

Author(s): Pavel Shvartsman
Journal: Trans. Amer. Math. Soc. 360 (2008), 5529-5550.
MSC (2000): Primary 46E35; Secondary 52A35, 54C60, 54C65
Posted: April 9, 2008
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We study a variant of the Whitney extension problem (1934) for the space $ {C^{k,\omega }(\mathbf{R}^{n})}$. We identify $ {C^{k,\omega }(\mathbf{R}^{n})}$ with a space of Lipschitz mappings from $ \mathbf{R}^n$ into the space $ \mathcal{P}_k\times\mathbf{R}^n$ of polynomial fields on $ \mathbf{R}^n$ equipped with a certain metric. This identification allows us to reformulate the Whitney problem for $ {C^{k,\omega } (\mathbf{R}^{n})}$ as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of $ \mathcal{P}_k\times\mathbf{R}^n$. We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for $ {C^{k,\omega }(\mathbf{R}^{n})}$ due to C. Fefferman.

References:

[1]
E. Bierstone and P.D. Milman, $ {\mathcal C}^m$ norms on finite sets and $ {\mathcal C}^m$ extension criteria, Duke Math. J. 137 (2007), 1-18. 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, 329-352. MR 1953261 (2004h:58009)

[3]
E. Bierstone, P. Milman and W. Pawlucki, Higher-order tangents and Fefferman's paper on Whitney's extension problem, Ann. of Math. (2) 164 (2006), no. 1, 361-370. MR 2233851 (2007g:58012)

[4]
Yu. Brudnyi and P. Shvartsman, Generalizations of Whitney's Extension Theorem, Intern. Math. Research Notices (1994), no. 3, 129-139. 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), 515-574. MR 1669235 (2000a:46051)

[6]
Yu. Brudnyi and P. Shvartsman, The Trace of Jet Space $ J^{k}\Lambda^{\omega}$ to an arbitrary closed subset of $ \mathbf{R}^n$, Trans. Amer. Math. Soc. 350 (1998) 1519-1553. MR 1407483 (98i:58010)

[7]
Yu. Brudnyi and P. Shvartsman, Whitney's Extension Problem for Multivariate $ C^{1,\omega}$-functions, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2487-2512. 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. 101-180, 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, 509-577. MR 2150391 (2006h:58008)

[10]
C. Fefferman, Interpolation and Extrapolation of Smooth Functions by Linear Operators, Rev. Mat. Iberoamericana 21 (2005), no. 1, 313-348. MR 2155023 (2006h:58009)

[11]
C. Fefferman, Whitney's Extension Problem in Certain Function Spaces, Rev. Mat. Iberoamericana (to appear).

[12]
C. Fefferman, A Generalized Sharp Whitney Theorem for Jets, Rev. Mat. Iberoamericana 21 (2005), no. 2, 577-688. MR 2174917 (2007a:58009)

[13]
C. Fefferman, Whitney's Extension Problem for $ C^m$, Ann. of Math. (2) 164 (2006), no. 1, 313-359. MR 2233850 (2007g:58013)

[14]
C. Fefferman, Extension of $ C^{m,w}$-Smooth Functions by Linear Operators, Rev. Mat. Iberoamericana (to appear).

[15]
C. Fefferman, $ C^m$ Extension by Linear Operators, Ann. of Math. (2) 166 (2007), no. 3, 779-835.

[16]
G. Glaeser, Étude de quelques algebres Tayloriennes, J. d'Analyse Math. 6 (1958), 1-125. MR 0101294 (21:107)

[17]
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, 559-562; English transl. in Soviet. Math. Dokl. 29 (1984), no. 3, 565-568. MR 752427 (85j:46057)

[18]
P. Shvartsman, On the traces of functions of the Zygmund class, Sib. Mat. Zh. 28 (1987), no. 5, 203-215; English transl. in Sib. Math. J. 28 (1987) 853-863. MR 924998 (89a:46081)

[19]
P. Shvartsman, On Lipschitz selections of affine-set valued mappings, GAFA, Geom. Funct. Anal. 11 (2001), no. 4, 840-868. MR 1866804 (2002m:52005)

[20]
P. Shvartsman, Lipschitz Selections of Set-Valued Mappings and Helly's Theorem, J. Geom. Anal. 12 (2002) 289-324. MR 1888519 (2002m:52006)

[21]
P. Shvartsman, Barycentric Selectors and a Steiner-type Point of a Convex Body in a Banach Space, J. Func. Anal. 210 (2004), no. 1, 1-42. MR 2051631 (2005b:46036)

[22]
H. Whitney, Analytic extension of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934) 63-89. MR 1501735

[23]
H. Whitney, Differentiable functions defined in closed sets. I., Trans. Amer. Math. Soc. 36 (1934) 369-387. MR 1501749

[24]
N. Zobin, Whitney's problem on extendability of functions and an intrinsic metric, Advances in Math. 133 (1998) 96-132. MR 1492787 (98k:46058)

[25]
N. Zobin, Extension of smooth functions from finitely connected planar domains, J. Geom. Anal. 9 (1999), 489-509. MR 1757457 (2001d:46048)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46E35, 52A35, 54C60, 54C65

Retrieve articles in all Journals with MSC (2000): 46E35, 52A35, 54C60, 54C65

Additional Information:

Pavel Shvartsman
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
Email: pshv@tx.technion.ac.il

DOI: 10.1090/S0002-9947-08-04469-3
PII: S 0002-9947(08)04469-3
Keywords: Whitney's extension problem, smooth functions, finiteness, metric, jet-space, set-valued mapping, Lipschitz selection
Received by editor(s): March 20, 2006
Received by editor(s) in revised form: November 29, 2006
Posted: April 9, 2008
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google