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Whitney's extension problem for multivariate -functions

Authors:	Yuri Brudnyi Pavel Shvartsman

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

Abstract:	We prove that the trace of the space $C^{1,omega}({mathbb R}^n)$ to an arbitrary closed subset $Xsubset{mathbb R}^n$ is characterized by the following ``finiteness' property. A function belongs to the trace space if and only if the restriction to an arbitrary subset consisting of at most $3cdot 2^{n-1}$ can be extended to a function $f_Yin C^{1,omega}({mathbb R}^n)$ such that $begin{displaymath}sup{Vert f_YVert _{C^{1,omega}}:~Ysubset X, ~operatorname{card} Yle 3cdot 2^{n-1}}<infty. end{displaymath}$ The constant $3cdot 2^{n-1}$ is sharp. The proof is based on a Lipschitz selection result which is interesting in its own right.

Keywords:	Extension of smooth functions Whitney's extension problem finiteness property Lipschitz selection

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