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2013 | 1 | 295-301
Tytuł artykułu

The Lusin Theorem and Horizontal Graphs in the Heisenberg Group

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EN
Abstrakty
EN
In this paper we prove that every collection of measurable functions fα , |α| = m, coincides a.e. withmth order derivatives of a function g ∈ Cm−1 whose derivatives of order m − 1 may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.
Twórcy
  • Department of Mathematics, University of Pittsburgh,
    301 Thackeray Hall, Pittsburgh, PA 15260, USA, hajlasz@pitt.edu
autor
  • Department of Mathematics, University of Pittsburgh,
    301 Thackeray Hall, Pittsburgh, PA 15260, USA, jrm152@pitt.edu
Bibliografia
  • [1] G. Alberti, A Lusin type theorem for gradients. J. Funct. Anal. 100 (1991), 110–118.
  • [2] Z. M. Balogh, Size of characteristic sets and functions with prescribed gradient. J. Reine Angew. Math. 564 (2003),63–83.
  • [3] Z. M. Balogh, R. Hoefer-Isenegger, J. T. Tyson, Lifts of Lipschitz maps and horizontal fractals in the Heisenberggroup. Ergodic Theory Dynam. Systems 26 (2006), 621–651.
  • [4] L. Capogna, D. Danielli, S. D. Pauls, J. T. Tyson, An introduction to the Heisenberg group and the sub-Riemannianisoperimetric problem. Progress in Mathematics, 259. Birkhäuser Verlag, Basel, 2007.
  • [5] B. Franchi, R.L. Wheeden, Compensation couples and isoperimetric estimates for vector fields. Colloq. Math. 74(1997), 9–27.
  • [6] G. Francos, The Luzin theorem for higher-order derivatives. Michigan Math. J. 61 (2012), 507–516.
  • [7] M. Gromov, Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, pp. 79–323, Progr. Math., 144,Birkhäuser, Basel, 1996.
  • [8] N. Lusin, Sur la notion de l’integrale. Ann. Mat. Pura Appl. 26 (1917), 77-129.
  • [9] L. Moonens, W. F. Pfeffer, The multidimensional Luzin theorem. J. Math. Anal. Appl. 339 (2008), 746–752.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_agms-2013-0008
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