The Lusin Theorem and Horizontal Graphs in the Heisenberg Group

Piotr Hajłasz; Jacob Mirra

Analysis and Geometry in Metric Spaces (2013)

  • Volume: 1, page 295-301
  • ISSN: 2299-3274

Abstract

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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.

How to cite

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Piotr Hajłasz, and Jacob Mirra. "The Lusin Theorem and Horizontal Graphs in the Heisenberg Group." Analysis and Geometry in Metric Spaces 1 (2013): 295-301. <http://eudml.org/doc/267034>.

@article{PiotrHajłasz2013,
abstract = {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.},
author = {Piotr Hajłasz, Jacob Mirra},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Lusin theorem; Heisenberg group; characteristic points},
language = {eng},
pages = {295-301},
title = {The Lusin Theorem and Horizontal Graphs in the Heisenberg Group},
url = {http://eudml.org/doc/267034},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Piotr Hajłasz
AU - Jacob Mirra
TI - The Lusin Theorem and Horizontal Graphs in the Heisenberg Group
JO - Analysis and Geometry in Metric Spaces
PY - 2013
VL - 1
SP - 295
EP - 301
AB - 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.
LA - eng
KW - Lusin theorem; Heisenberg group; characteristic points
UR - http://eudml.org/doc/267034
ER -

References

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  1. [1] G. Alberti, A Lusin type theorem for gradients. J. Funct. Anal. 100 (1991), 110–118. Zbl0752.46025
  2. [2] Z. M. Balogh, Size of characteristic sets and functions with prescribed gradient. J. Reine Angew. Math. 564 (2003), 63–83. Zbl1051.53024
  3. [3] Z. M. Balogh, R. Hoefer-Isenegger, J. T. Tyson, Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group. Ergodic Theory Dynam. Systems 26 (2006), 621–651. Zbl1097.22005
  4. [4] L. Capogna, D. Danielli, S. D. Pauls, J. T. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. Progress in Mathematics, 259. Birkhäuser Verlag, Basel, 2007. Zbl1138.53003
  5. [5] B. Franchi, R.L. Wheeden, Compensation couples and isoperimetric estimates for vector fields. Colloq. Math. 74 (1997), 9–27. Zbl0915.46028
  6. [6] G. Francos, The Luzin theorem for higher-order derivatives. Michigan Math. J. 61 (2012), 507–516. Zbl1256.28006
  7. [7] M. Gromov, Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, pp. 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996. Zbl0864.53025
  8. [8] N. Lusin, Sur la notion de l’integrale. Ann. Mat. Pura Appl. 26 (1917), 77-129. Zbl46.0391.01
  9. [9] L. Moonens, W. F. Pfeffer, The multidimensional Luzin theorem. J. Math. Anal. Appl. 339 (2008), 746–752. Zbl1141.28008

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