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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 $ {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.


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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.

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