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

Author:
Pavel Shvartsman

Journal:
Trans. Amer. Math. Soc. **360** (2008), 5529-5550

MSC (2000):
Primary 46E35; Secondary 52A35, 54C60, 54C65

Published electronically:
April 9, 2008

MathSciNet review:
2415084

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 set-valued mappings into a certain family of subsets of . 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 due to C. Fefferman.

**[1]**Edward Bierstone and Pierre D. Milman,*𝒞^{𝓂}-norms on finite sets and 𝒞^{𝓂} extension criteria*, Duke Math. J.**137**(2007), no. 1, 1–18. MR**2309142**, 10.1215/S0012-7094-07-13711-6**[2]**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**, 10.1007/s00222-002-0255-6**[3]**Edward Bierstone, Pierre D. Milman, and Wiesław Pawłucki,*Higher-order tangents and Fefferman’s paper on Whitney’s extension problem*, Ann. of Math. (2)**164**(2006), no. 1, 361–370. MR**2233851**, 10.4007/annals.2006.164.361**[4]**Yuri Brudnyi and Pavel Shvartsman,*Generalizations of Whitney’s extension theorem*, Internat. Math. Res. Notices**3**(1994), 129 ff., approx. 11 pp. (electronic). MR**1266108**, 10.1155/S1073792894000140**[5]**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**, 10.1007/BF02921632**[6]**Yuri Brudnyi and Pavel Shvartsman,*The trace of jet space 𝐽^{𝑘}Λ^{𝜔} to an arbitrary closed subset of 𝐑ⁿ*, Trans. Amer. Math. Soc.**350**(1998), no. 4, 1519–1553. MR**1407483**, 10.1090/S0002-9947-98-01872-8**[7]**Yuri Brudnyi and Pavel Shvartsman,*Whitney’s extension problem for multivariate 𝐶^{1,𝜔}-functions*, Trans. Amer. Math. Soc.**353**(2001), no. 6, 2487–2512 (electronic). MR**1814079**, 10.1090/S0002-9947-01-02756-8**[8]**Ludwig Danzer, Branko Grünbaum, and Victor Klee,*Helly’s theorem and its relatives*, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 101–180. MR**0157289****[9]**Charles L. Fefferman,*A sharp form of Whitney’s extension theorem*, Ann. of Math. (2)**161**(2005), no. 1, 509–577. MR**2150391**, 10.4007/annals.2005.161.509**[10]**Charles Fefferman,*Interpolation and extrapolation of smooth functions by linear operators*, Rev. Mat. Iberoamericana**21**(2005), no. 1, 313–348. MR**2155023**, 10.4171/RMI/424**[11]**C. Fefferman,*Whitney's Extension Problem in Certain Function Spaces,*Rev. Mat. Iberoamericana (to appear).**[12]**Charles Fefferman,*A generalized sharp Whitney theorem for jets*, Rev. Mat. Iberoamericana**21**(2005), no. 2, 577–688. MR**2174917**, 10.4171/RMI/430**[13]**Charles Fefferman,*Whitney’s extension problem for 𝐶^{𝑚}*, Ann. of Math. (2)**164**(2006), no. 1, 313–359. MR**2233850**, 10.4007/annals.2006.164.313**[14]**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, 779-835.**[16]**Georges Glaeser,*Étude de quelques algèbres tayloriennes*, J. Analyse Math.**6**(1958), 1–124; erratum, insert to 6 (1958), no. 2 (French). MR**0101294****[17]**P. A. Shvartsman,*Lipschitz sections of set-valued mappings and traces of functions from the Zygmund class on an arbitrary compactum*, Dokl. Akad. Nauk SSSR**276**(1984), no. 3, 559–562 (Russian). MR**752427****[18]**P. A. Shvartsman,*Traces of functions of Zygmund class*, Sibirsk. Mat. Zh.**28**(1987), no. 5, 203–215 (Russian). MR**924998****[19]**P. Shvartsman,*On Lipschitz selections of affine-set valued mappings*, Geom. Funct. Anal.**11**(2001), no. 4, 840–868. MR**1866804**, 10.1007/PL00001687**[20]**Pavel Shvartsman,*Lipschitz selections of set-valued mappings and Helly’s theorem*, J. Geom. Anal.**12**(2002), no. 2, 289–324. MR**1888519**, 10.1007/BF02922044**[21]**P. Shvartsman,*Barycentric selectors and a Steiner-type point of a convex body in a Banach space*, J. Funct. Anal.**210**(2004), no. 1, 1–42. MR**2051631**, 10.1016/S0022-1236(03)00211-8**[22]**Hassler Whitney,*Analytic extensions of differentiable functions defined in closed sets*, Trans. Amer. Math. Soc.**36**(1934), no. 1, 63–89. MR**1501735**, 10.1090/S0002-9947-1934-1501735-3**[23]**Hassler Whitney,*Differentiable functions defined in closed sets. I*, Trans. Amer. Math. Soc.**36**(1934), no. 2, 369–387. MR**1501749**, 10.1090/S0002-9947-1934-1501749-3**[24]**Nahum Zobin,*Whitney’s problem on extendability of functions and an intrinsic metric*, Adv. Math.**133**(1998), no. 1, 96–132. MR**1492787**, 10.1006/aima.1997.1685**[25]**Nahum Zobin,*Extension of smooth functions from finitely connected planar domains*, J. Geom. Anal.**9**(1999), no. 3, 491–511. MR**1757457**, 10.1007/BF02921985

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:
https://doi.org/10.1090/S0002-9947-08-04469-3

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

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.