no. 2

On the second order derivatives of convex functions on the Heisenberg group
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 2, pp. 349-366.

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 ℋ-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 ℋ-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},
     pages = {349--366},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {2},
     year = {2004},
     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, Série 5, Tome 3 (2004) no. 2, pp. 349-366. http://www.numdam.org/item/ASNSP_2004_5_3_2_349_0/

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