The trace of jet space to an arbitrary closed subset of
Authors:
Yuri Brudnyi and Pavel Shvartsman
Journal:
Trans. Amer. Math. Soc. 350 (1998), 15191553
MSC (1991):
Primary 46E35
MathSciNet review:
1407483
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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 twopointed 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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 [BS]
 Yu. Brudnyi and P. Shvartsman, Generalizations of Whitney's Extension Theorem, Internat. Math. Res. Notices, N3 (1994), 129139. MR 95c:58018
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 , A linear extension operator for a space of smooth functions defined on a closed subset of , Dokl. Akad. Nauk SSSR 280:2 (1985), 268272; English transl. in Soviet Math. Dokl. 31 (1985), 4851. MR 86f:46031
 [BS2]
 , The Whitney Problem of Existence of a Linear Extension Operator, The Journal of Geometric Analysis (to appear).
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 [H]
 L. G. Hanin, ``A trace description of functions with high order derivatives from generalized Zygmund space to arbitrary closed subsets'' in Studies in the Theory of Functions of Several Real Variables, Yaroslav. State Univ., Yaroslavl, 1987, 128144 (Russian).
 [JW]
 A Jonsson and H. Wallin, Function spaces on subsets of , Math. Rep. 2 (1984), Part 1. MR 87f:46056
 [Sh]
 P. Shvartsman, Lipschitz sections of setvalued mappings and traces of functions from the Zygmund class on an arbitrary compactum, Dokl. Akad. Nauk SSSR 276 (1984), 559562; English transl. in Soviet Math. Dokl., 29 (1984), 565568. MR 85j:46057
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 , ``Lipschitz sections of multivalued mappings'' in Studies in the Theory of Functions of Several Real Variables, Yaroslav. State Univ., Yaroslavl, 1986, 121132 (Russian). MR 88e:46032
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 , ``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, 245268. MR 94c:46069
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 E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970. MR 44:7280
 [W]
 H. Whitney, Analytic extension of differentiable function defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 6389.
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Additional Information
Yuri Brudnyi
Affiliation:
Department of Mathematics, TechnionIsrael Institute of Technology, Haifa 32000, Israel
Email:
ybrudnyi@techunix.technion.ac.il
Pavel Shvartsman
Affiliation:
Department of Mathematics, TechnionIsrael Institute of Technology, Haifa 32000, Israel
Email:
pshv@techunix.technion.ac.il
DOI:
http://dx.doi.org/10.1090/S0002994798018728
PII:
S 00029947(98)018728
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 firstnamed author was supported by the Fund for Promotion of Research at the Technion and the J. & S. Frankel Research Fund. The secondnamed author was supported by the Center for Absorption in Science, Israel Ministry of Immigrant Absorption.
Article copyright:
© Copyright 1998
American Mathematical Society
