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

HTML articles powered by AMS MathViewer

- by Pavel Shvartsman PDF
- Trans. Amer. Math. Soc.
**360**(2008), 5529-5550 Request permission

## 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

- Edward Bierstone and Pierre D. Milman,
*$\scr C^m$-norms on finite sets and $\scr C^m$ extension criteria*, Duke Math. J.**137**(2007), no. 1, 1–18. MR**2309142**, DOI 10.1215/S0012-7094-07-13711-6 - 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 - 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**, DOI 10.4007/annals.2006.164.361 - 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 - Yuri Brudnyi and Pavel 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), no. 4, 1519–1553. MR**1407483**, DOI 10.1090/S0002-9947-98-01872-8 - Yuri Brudnyi and Pavel Shvartsman,
*Whitney’s extension problem for multivariate $C^{1,\omega }$-functions*, Trans. Amer. Math. Soc.**353**(2001), no. 6, 2487–2512. MR**1814079**, DOI 10.1090/S0002-9947-01-02756-8 - 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** - 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,
*Interpolation and extrapolation of smooth functions by linear operators*, Rev. Mat. Iberoamericana**21**(2005), no. 1, 313–348. MR**2155023**, DOI 10.4171/RMI/424 - C. Fefferman,
*Whitney’s Extension Problem in Certain Function Spaces,*Rev. Mat. Iberoamericana (to appear). - Charles Fefferman,
*A generalized sharp Whitney theorem for jets*, Rev. Mat. Iberoamericana**21**(2005), no. 2, 577–688. MR**2174917**, DOI 10.4171/RMI/430 - 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 - C. Fefferman,
*Extension of $C^{m,w}$-Smooth Functions by Linear Operators,*Rev. Mat. Iberoamericana (to appear). - C. Fefferman,
*$C^m$ Extension by Linear Operators,*Ann. of Math. (2) 166 (2007), no. 3, 779–835. - 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 - 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** - P. A. Shvartsman,
*Traces of functions of Zygmund class*, Sibirsk. Mat. Zh.**28**(1987), no. 5, 203–215 (Russian). MR**924998** - P. Shvartsman,
*On Lipschitz selections of affine-set valued mappings*, Geom. Funct. Anal.**11**(2001), no. 4, 840–868. MR**1866804**, DOI 10.1007/PL00001687 - Pavel Shvartsman,
*Lipschitz selections of set-valued mappings and Helly’s theorem*, J. Geom. Anal.**12**(2002), no. 2, 289–324. MR**1888519**, DOI 10.1007/BF02922044 - 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**, DOI 10.1016/S0022-1236(03)00211-8 - 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 - Nahum Zobin,
*Whitney’s problem on extendability of functions and an intrinsic metric*, Adv. Math.**133**(1998), no. 1, 96–132. MR**1492787**, DOI 10.1006/aima.1997.1685 - Nahum Zobin,
*Extension of smooth functions from finitely connected planar domains*, J. Geom. Anal.**9**(1999), no. 3, 491–511. MR**1757457**, DOI 10.1007/BF02921985

## Additional Information

**Pavel Shvartsman**- Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
- Email: pshv@tx.technion.ac.il
- Received by editor(s): March 20, 2006
- Received by editor(s) in revised form: November 29, 2006
- Published electronically: April 9, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2415084