In the euclidean setting the celebrated Aleksandrov-Busemann-Feller theorem states that convex functions are a.e. twice differentiable. In this paper we prove that a similar result holds in the Heisenberg group, by showing that every continuous -convex function belongs to the class of functions whose second order horizontal distributional derivatives are Radon measures. Together with a recent result by Ambrosio and Magnani, this proves the existence a.e. of second order horizontal derivatives for the class of continuous -convex functions in the Heisenberg group.
@article{ASNSP_2004_5_3_2_349_0,
author = {Guti\'errez, Cristian E. and Montanari, Annamaria},
title = {On the second order derivatives of convex functions on the Heisenberg group},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 3},
number = {2},
year = {2004},
pages = {349-366},
zbl = {1170.35352},
mrnumber = {2075987},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2004_5_3_2_349_0}
}
Gutiérrez, Cristian E.; Montanari, Annamaria. On the second order derivatives of convex functions on the Heisenberg group. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 2, pp. 349-366. http://www.numdam.org/item/ASNSP_2004_5_3_2_349_0/
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