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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Whitney's extension problem for multivariate $C^{1,\omega}$-functions


Authors: Yuri Brudnyi and Pavel Shvartsman
Journal: Trans. Amer. Math. Soc. 353 (2001), 2487-2512
MSC (1991): Primary 46E35
Published electronically: February 7, 2001
MathSciNet review: 1814079
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Abstract:

We prove that the trace of the space $C^{1,\omega}({\mathbb R}^n)$to an arbitrary closed subset $X\subset{\mathbb R}^n$is characterized by the following ``finiteness'' property. A function $f:X\rightarrow{\mathbb R}$belongs to the trace space if and only if the restriction $f\vert _Y$ to an arbitrary subset $Y\subset X$ consisting of at most $3\cdot 2^{n-1}$ can be extended to a function $f_Y\in C^{1,\omega}({\mathbb R}^n)$ such that

\begin{displaymath}\sup\{\Vert f_Y\Vert _{C^{1,\omega}}:~Y\subset X, ~\operatorname{card} Y\le 3\cdot 2^{n-1}\}<\infty. \end{displaymath}

The constant $3\cdot 2^{n-1}$ is sharp.

The proof is based on a Lipschitz selection result which is interesting in its own right.


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Additional Information

Yuri Brudnyi
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
Email: ybrudnyi@tx.technion.ac.il

Pavel Shvartsman
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
Email: pshv@tx.technion.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02756-8
PII: S 0002-9947(01)02756-8
Keywords: Extension of smooth functions, Whitney's extension problem, finiteness property, Lipschitz selection
Received by editor(s): June 26, 2000
Published electronically: February 7, 2001
Additional Notes: The research was supported by Grant No. 95-00225 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel and by Technion V. P. R. Fund - M. and M. L. Bank Mathematics Research Fund. The second named author was also supported by the Center for Absorption in Science, Israel Ministry of Immigrant Absorption.
Dedicated: Dedicated to the memory of Evsey Dyn’kin
Article copyright: © Copyright 2001 American Mathematical Society