The trace of jet space

to an arbitrary closed subset of

Authors:
Yuri Brudnyi and Pavel Shvartsman

Journal:
Trans. Amer. Math. Soc. **350** (1998), 1519-1553

MSC (1991):
Primary 46E35

DOI:
https://doi.org/10.1090/S0002-9947-98-01872-8

MathSciNet review:
1407483

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Abstract | References | Similar Articles | Additional Information

Abstract: The classical Whitney extension theorem describes the trace of the space of -jets generated by functions from to an arbitrary closed subset . It establishes existence of a bounded linear extension operator as well. In this paper we investigate a similar problem for the space of functions whose higher derivatives satisfy the Zygmund condition with majorant . The main result states that the vector function belongs to the corresponding trace space if the trace to every subset of cardinality , where , can be extended to a function and . The number generally speaking cannot be reduced. The Whitney theorem can be reformulated in this way as well, but with a two-pointed subset . The approach is based on the theory of local polynomial approximations and a result on Lipschitz selections of multivalued mappings.

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Additional Information

**Yuri Brudnyi**

Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel

Email:
ybrudnyi@techunix.technion.ac.il

**Pavel Shvartsman**

Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel

Email:
pshv@techunix.technion.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-98-01872-8

Keywords:
Trace spaces of smooth functions,
Whitney's extension theorem,
finiteness property,
Lipschitz selections of multivalued mappings

Received by editor(s):
February 28, 1995

Received by editor(s) in revised form:
July 25, 1996

Additional Notes:
The first-named author was supported by the Fund for Promotion of Research at the Technion and the J. & S. Frankel Research Fund. The second-named author was supported by the Center for Absorption in Science, Israel Ministry of Immigrant Absorption.

Article copyright:
© Copyright 1998
American Mathematical Society