Whitney’s extension problem for multivariate $C^{1,\omega }$-functions
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- by Yuri Brudnyi and Pavel Shvartsman PDF
- Trans. Amer. Math. Soc. 353 (2001), 2487-2512 Request permission
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 $f|_Y$ 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 \[ \sup \{\|f_Y\|_{C^{1,\omega }}:~Y\subset X, ~\operatorname {card} Y\le 3\cdot 2^{n-1}\}<\infty . \] 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
- 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.
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 2487-2512
- MSC (1991): Primary 46E35
- DOI: https://doi.org/10.1090/S0002-9947-01-02756-8
- MathSciNet review: 1814079
Dedicated: Dedicated to the memory of Evsey Dyn’kin