The Lusin Theorem and Horizontal Graphs in the Heisenberg Group
Piotr Hajłasz ; Jacob Mirra
Analysis and Geometry in Metric Spaces, Tome 1 (2013), p. 295-301 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 2013-01-01
EUDML-ID : urn:eudml:doc:267034 
@article{bwmeta1.element.doi-10_2478_agms-2013-0008,
     author = {Piotr Haj\l asz and Jacob Mirra},
     title = {The Lusin Theorem and Horizontal Graphs in the Heisenberg Group},
     journal = {Analysis and Geometry in Metric Spaces},
     volume = {1},
     year = {2013},
     pages = {295-301},
     zbl = {1286.46036},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_agms-2013-0008}
}

Piotr Hajłasz; Jacob Mirra. The Lusin Theorem and Horizontal Graphs in the Heisenberg Group. Analysis and Geometry in Metric Spaces, Tome 1 (2013) pp. 295-301. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_agms-2013-0008/

[1] G. Alberti, A Lusin type theorem for gradients. J. Funct. Anal. 100 (1991), 110–118. | Zbl 0752.46025

[2] Z. M. Balogh, Size of characteristic sets and functions with prescribed gradient. J. Reine Angew. Math. 564 (2003), 63–83. | Zbl 1051.53024

[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. | Zbl 1097.22005

[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. | Zbl 1138.53003

[5] B. Franchi, R.L. Wheeden, Compensation couples and isoperimetric estimates for vector fields. Colloq. Math. 74 (1997), 9–27. | Zbl 0915.46028

[6] G. Francos, The Luzin theorem for higher-order derivatives. Michigan Math. J. 61 (2012), 507–516. | Zbl 1256.28006

[7] M. Gromov, Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, pp. 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996. | Zbl 0864.53025

[8] N. Lusin, Sur la notion de l’integrale. Ann. Mat. Pura Appl. 26 (1917), 77-129. | Zbl 46.0391.01

[9] L. Moonens, W. F. Pfeffer, The multidimensional Luzin theorem. J. Math. Anal. Appl. 339 (2008), 746–752. | Zbl 1141.28008