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Transactions of the American Mathematical Society

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The trace of jet space $J^k\Lambda ^\omega$ to an arbitrary closed subset of $\mathbb {R}^n$
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by Yuri Brudnyi and Pavel Shvartsman PDF
Trans. Amer. Math. Soc. 350 (1998), 1519-1553 Request permission

Abstract:

The classical Whitney extension theorem describes the trace $J^k|_X$ of the space of $k$-jets generated by functions from $C^k(\mathbb R^n)$ to an arbitrary closed subset $X\subset \mathbb R^n$. It establishes existence of a bounded linear extension operator as well. In this paper we investigate a similar problem for the space $C^k\Lambda$ of functions whose higher derivatives satisfy the Zygmund condition with majorant $\omega$. The main result states that the vector function $\vec f=(f_\alpha \colon X\to \mathbb R)_{|\alpha |\le k}$ belongs to the corresponding trace space if the trace $\vec f|_Y$ to every subset $Y\subset X$ of cardinality $3\cdot 2^\ell$, where $\ell =(\begin {smallmatrix}n+k-1 k+1\end {smallmatrix})$, can be extended to a function $f_Y\in C^k\Lambda$ and $\sup _Y|f_Y|_{C^k\Lambda ^\omega }<\infty$. The number $3\cdot 2^l$ generally speaking cannot be reduced. The Whitney theorem can be reformulated in this way as well, but with a two-pointed subset $Y\subset X$. The approach is based on the theory of local polynomial approximations and a result on Lipschitz selections of multivalued mappings.
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Additional Information
  • Yuri Brudnyi
  • Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
  • Email: ybrudnyi@techunix.technion.ac.il
  • Pavel Shvartsman
  • Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
  • Email: pshv@techunix.technion.ac.il
  • Received by editor(s): February 28, 1995
  • Received by editor(s) in revised form: July 25, 1996
  • Additional Notes: The first-named author was supported by the Fund for Promotion of Research at the Technion and the J. & S. Frankel Research Fund. The second-named author was supported by the Center for Absorption in Science, Israel Ministry of Immigrant Absorption.
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 1519-1553
  • MSC (1991): Primary 46E35
  • DOI: https://doi.org/10.1090/S0002-9947-98-01872-8
  • MathSciNet review: 1407483