# The Lusin Theorem and Horizontal Graphs in the Heisenberg Group

Analysis and Geometry in Metric Spaces (2013)

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

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topPiotr 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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- [5] B. Franchi, R.L. Wheeden, Compensation couples and isoperimetric estimates for vector fields. Colloq. Math. 74 (1997), 9–27. Zbl0915.46028
- [6] G. Francos, The Luzin theorem for higher-order derivatives. Michigan Math. J. 61 (2012), 507–516. Zbl1256.28006
- [7] M. Gromov, Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, pp. 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996. Zbl0864.53025
- [8] N. Lusin, Sur la notion de l’integrale. Ann. Mat. Pura Appl. 26 (1917), 77-129. Zbl46.0391.01
- [9] L. Moonens, W. F. Pfeffer, The multidimensional Luzin theorem. J. Math. Anal. Appl. 339 (2008), 746–752. Zbl1141.28008

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