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The trace of jet space $\protect{J^{k}\Lambda^\omega}$
to an arbitrary closed subset of $\protect{\mathbb{R}^n}$

Authors: Yuri Brudnyi and Pavel Shvartsman
Journal: Trans. Amer. Math. Soc. 350 (1998), 1519-1553
MSC (1991): Primary 46E35
MathSciNet review: 1407483
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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 $C^k\Lambda^\omega(\mathbb R^n)$ 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 $f_Y\in C^k\Lambda^\omega(\mathbb R^n)$ 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

Pavel Shvartsman
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel

Keywords: Trace spaces of smooth functions, Whitney's extension theorem, finiteness property, Lipschitz selections of multivalued mappings
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.
Article copyright: © Copyright 1998 American Mathematical Society