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.

**[B]**Yu. Brudnyi,*A multidimensional analog of a theorem of Whitney*, Mat. Sbornik,**82**(124) (1970), N2, 175-191; English transl. in Math. USSR Sbornik,**11**(1970), N2, 157-170.**[BS]**Yuri Brudnyi and Pavel Shvartsman,*Generalizations of Whitney’s extension theorem*, Internat. Math. Res. Notices**3**(1994), 129 ff., approx. 11 pp.}, issn=1073-7928, review=\MR{1266108}, doi=10.1155/S1073792894000140,.**[BS1]**Yu. A. Brudnyĭ and P. A. Shvartsman,*A linear extension operator for a space of smooth functions defined on a closed subset in 𝑅ⁿ*, Dokl. Akad. Nauk SSSR**280**(1985), no. 2, 268–272 (Russian). MR**775048****[BS2]**-,*The Whitney Problem of Existence of a Linear Extension Operator*, The Journal of Geometric Analysis (to appear).**[BS3]**-, ``A description of the trace of a function from the generalized Lipschitz space to an arbitrary compact'' in*Studies in the Theory of Functions of Several Real Variables*, Yaroslav State Univ., Yaroslavl, 1982, 16-24 (Russian).**[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**, https://doi.org/10.1007/BF02790231**[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, 128-144 (Russian).**[JW]**Alf Jonsson and Hans Wallin,*Function spaces on subsets of 𝑅ⁿ*, Math. Rep.**2**(1984), no. 1, xiv+221. MR**820626****[Sh]**P. A. Shvartsman,*Lipschitz sections of set-valued mappings and traces of functions from the Zygmund class on an arbitrary compactum*, Dokl. Akad. Nauk SSSR**276**(1984), no. 3, 559–562 (Russian). MR**752427****[Sh1]**P. A. Shvartsman,*Traces of functions of Zygmund class*, Sibirsk. Mat. Zh.**28**(1987), no. 5, 203–215 (Russian). MR**924998****[Sh2]**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****[Sh3]**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****[W]**H. Whitney,*Analytic extension of differentiable function defined in closed sets*, Trans. Amer. Math. Soc.**36**(1934), 63-89.

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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