Whitney's extension problem for multivariate -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 | References | Similar Articles | Additional Information

We prove that the trace of the space to an arbitrary closed subset 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 can be extended to a function such that

The constant is sharp.

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

**[BS1]**Yuri Brudnyi and Pavel Shvartsman,*Generalizations of Whitney’s extension theorem*, Internat. Math. Res. Notices**3**(1994), 129 ff., approx. 11 pp. (electronic). MR**1266108**, 10.1155/S1073792894000140**[BS2]**Yuri Brudnyi and Pavel Shvartsman,*The Whitney problem of existence of a linear extension operator*, J. Geom. Anal.**7**(1997), no. 4, 515–574. MR**1669235**, 10.1007/BF02921632**[BS3]**Yuri Brudnyi and Pavel Shvartsman,*The trace of jet space 𝐽^{𝑘}Λ^{𝜔} to an arbitrary closed subset of 𝐑ⁿ*, Trans. Amer. Math. Soc.**350**(1998), no. 4, 1519–1553. MR**1407483**, 10.1090/S0002-9947-98-01872-8**[G]**Georges Glaeser,*Étude de quelques algèbres tayloriennes*, J. Analyse Math.**6**(1958), 1–124; erratum, insert to 6 (1958), no. 2 (French). MR**0101294****[Sh1]**P. A. Shvartsman,*Lipschitz sections of multivalued mappings*, Studies in the theory of functions of several real variables (Russian), Yaroslav. Gos. Univ., Yaroslavl′, 1986, pp. 121–132, 149 (Russian). MR**878806****[Sh2]**P. A. Shvartsman,*𝐾-functionals of weighted Lipschitz spaces and Lipschitz selections of multivalued mappings*, Interpolation spaces and related topics (Haifa, 1990) Israel Math. Conf. Proc., vol. 5, Bar-Ilan Univ., Ramat Gan, 1992, pp. 245–268. MR**1206505****[St]**Elias M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095****[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