Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The trace of jet space $\protect{J^{k}\Lambda^\omega}$
to an arbitrary closed subset of $\protect{\mathbb{R}^n}$


Authors: Yuri Brudnyi and Pavel Shvartsman
Journal: Trans. Amer. Math. Soc. 350 (1998), 1519-1553
MSC (1991): Primary 46E35
MathSciNet review: 1407483
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The classical Whitney extension theorem describes the trace $J^k|_X$ of the space of $k$-jets generated by functions from $C^k(\mathbb R^n)$ to an arbitrary closed subset $X\subset\mathbb R^n$. It establishes existence of a bounded linear extension operator as well. In this paper we investigate a similar problem for the space $C^k\Lambda^\omega(\mathbb R^n)$ of functions whose higher derivatives satisfy the Zygmund condition with majorant $\omega $. The main result states that the vector function $\vec f=(f_\alpha \colon X\to\mathbb R)_{|\alpha |\le k}$ belongs to the corresponding trace space if the trace $\vec f|_Y$ to every subset $Y\subset X$ of cardinality $3\cdot 2^\ell$, where $\ell=(\begin{smallmatrix}n+k-1\\ k+1\end{smallmatrix})$, can be extended to a function $f_Y\in C^k\Lambda^\omega(\mathbb R^n)$ and $\sup _Y|f_Y|_{C^k\Lambda^\omega}<\infty$. The number $3\cdot 2^l$ generally speaking cannot be reduced. The Whitney theorem can be reformulated in this way as well, but with a two-pointed subset $Y\subset X$. The approach is based on the theory of local polynomial approximations and a result on Lipschitz selections of multivalued mappings.


References [Enhancements On Off] (What's this?)

  • [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.\ (electronic). MR 1266108 (95c:58018), http://dx.doi.org/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 (86f:46031)
  • [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 (21 #107)
  • [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 (87f:46056)
  • [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 (85j:46057)
  • [Sh1] P. A. Shvartsman, Traces of functions of Zygmund class, Sibirsk. Mat. Zh. 28 (1987), no. 5, 203–215 (Russian). MR 924998 (89a:46081)
  • [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 (88e:46032)
  • [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 (94c:46069)
  • [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 (44 #7280)
  • [W] H. Whitney, Analytic extension of differentiable function defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 46E35

Retrieve articles in all journals with MSC (1991): 46E35


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: http://dx.doi.org/10.1090/S0002-9947-98-01872-8
PII: S 0002-9947(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