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The trace of jet space

Authors:	Yuri Brudnyi Pavel Shvartsman

Institution:	Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel ; Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel

Abstract:	The classical Whitney extension theorem describes the trace of the space of -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.

Keywords:	Trace spaces of smooth functions Whitney's extension theorem finiteness property Lipschitz selections of multivalued mappings

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