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Manuscripta Mathematica

, Volume 142, Issue 3–4, pp 475–489 | Cite as

Density rate of a set, application to rectifiability results for measurable jets

  • Silvano DelladioEmail author
Article
  • 140 Downloads

Abstract

In this work, we present some new results concerning the sets of h-density points and their application in the context of the Whitney extension theorem. In particular, we establish a simple relation between the amount of density of a set E at x and the property that u φ E has derivative of order h at x in the L p sense whenever u is continuous at x. Moreover we prove Whitney-type rectifiability results for measurable jets restricted to sets of high density points in its domain. It’s worth mentioning the case when the domain is a locally finite perimeter set.

Mathematics Subject Classifications

28A75 26A24 49Q15 

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References

  1. 1.
    Ambrosio L., Caselles V., Masnou S., Morel J.-M.: Connected components of sets of finite perimeter and applications to image processing. J. Eur. Math. Soc 3, 39–92 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anzellotti G., Serapioni R.: \({\mathcal{C}^k}\) -rectifiable sets. J. Reine Angew. Math 453, 1–20 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bojarski, B.: Differentiation of measurable functions and Whitney-Luzin type structure theorems. Helsinki University of Technology Institute of Mathematics Research Reports A572. http://math.tkk.fi/reports/a572.pdf
  4. 4.
    Calderón P.A., Zygmund A.: Local properties of solutions of elliptic partial differential equations. Studia. Math. 20, 171–225 (1961)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Delladio S.: Taylor’s polynomials and non-homogeneous blow-ups. Manuscripta Math. 113(3), 383–396 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Delladio, S.: Dilatations of graphs and Taylor’s formula: some results about convergence. Real Anal. Exch. 29(2), 687–713 (2003/2004)Google Scholar
  7. 7.
    Delladio S.: Functions of class C 1 subject to a Legendre condition in an enhanced density set. Rev. Matem. Iberoam 28(1), 127–140 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Delladio S.: A short note on enhanced density sets. Glasgow Math. J 53(3), 631–635 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Delladio, S.: A Whitney-type result about rectifiability of graphs. Preprint (2013)Google Scholar
  10. 10.
    Evans, C.L., Gariepy, F.R.: Lecture notes on measure theory and fine properties of functions. (Studies in Advanced Mathematics) CRC Press, New York (1992)Google Scholar
  11. 11.
    Federer H.: Geometric measure theory. Springer, Berlin (1969)zbMATHGoogle Scholar
  12. 12.
    Malgrange B.: Ideals of differentiable functions. Oxford University Press, London (1966)zbMATHGoogle Scholar
  13. 13.
    Mattila P.: Geometry of sets and measures in Euclidean spaces. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  14. 14.
    Ziemer, W.P.: Weakly differentiable functions. GTM 120, Springer, Berlin (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TrentoTrentoItaly

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