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

Shvartsman, Pavel

doi:10.1090/S0002-9947-08-04469-3

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ISSN 1088-6850(online) ISSN 0002-9947(print)

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
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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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