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

Shvartsman, Pavel

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

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

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