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

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

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]**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 to an arbitrary closed subset of ,*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]**-, ``-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