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

DOI:
https://doi.org/10.1090/S0002-9947-01-02756-8

Published electronically:
February 7, 2001

MathSciNet review:
1814079

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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