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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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