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

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

Published electronically:
April 9, 2008

MathSciNet review:
2415084

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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 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]**E. Bierstone and P.D. Milman,*norms on finite sets and 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 to an arbitrary closed subset of ,*Trans. Amer. Math. Soc.**350**(1998) 1519-1553. 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, 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 ,*Ann. of Math. (2) 164 (2006), no. 1, 313-359. MR**2233850 (2007g:58013)****[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]**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)**

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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.