Geodesic
The Lusin Theorem and Horizontal Graphs in the Heisenberg Group
Analysis and Geometry in Metric Spaces, Tome 1 (2013) no. 1, pp. 295-301.

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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.
Mots-clés : Lusin theorem, Heisenberg group, characteristic points 
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Hajłasz, Piotr; Mirra, Jacob. The Lusin Theorem and Horizontal Graphs in the Heisenberg Group. Analysis and Geometry in Metric Spaces, Tome 1 (2013) no. 1, pp. 295-301. http://geodesic.mathdoc.fr/item/AGMS_2013_1_1_a13/