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Transactions of the American Mathematical Society

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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
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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.


References [Enhancements On Off] (What's this?)

  • [BS1] Yu. Brudnyi and P. Shvartsman, Generalizations of Whitney's extension theorem, Internat. Math. Res. Notices, N3 (1994), 129-139. MR 95c:58018
  • [BS2] -, The Whitney problem of existence of a linear extension operator, J. Geom. Anal. 7 (1997), no. 4, 515-574. MR 2000a:46051
  • [BS3] -, The trace of jet space $J^{k}\Lambda^{\omega}$ to an arbitrary closed subset of $\mathbf{R}^{n}$, Trans. Amer. Math. Soc. 350 (1998), 1519-1553. MR 98i:58010
  • [G] G. Glaeser, Étude de quelques algèbres Tayloriennes, J. d'Analyse Math. 6 (1958), 1-125. MR 21:107
  • [Sh1] P. Shvartsman, ``Lipschitz sections of multivalued mappings'', in Studies in the Theory of Functions of Several Real Variables, Yaroslav. State. Univ., Yaroslavl, 1986, 121-132 (Russian). MR 88e:46032
  • [Sh2] -, ``$K$-functionals of weighted Lipschitz spaces and Lipschitz selections of multivalued mappings'', in Interpolation Spaces and Related Topics, Israel Math. Conf. Proc. 5, Weizmann, Jerusalem, 1992, 245-268. MR 94c:46069
  • [St] E. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, 1970. MR 44:7280
  • [W1] H. Whitney, Analytic extension of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89.
  • [W2] -, Differentiable functions defined in closed sets. I., Trans. Amer. Math. Soc. 36 (1934), 369-387.

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

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