On the second order derivatives of convex functions on the Heisenberg group
Gutiérrez, Cristian E.1 ; Montanari, Annamaria2
1 Department of Mathematics Temple University Philadelphia, PA 19122
2 Dipartimento di Matematica Università di Bologna Piazza Porta San Donato, 5 40127 Bologna, Italy
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.

Classification : 35B50, 35B45, 35H20
Gutiérrez, Cristian E. 1 ; Montanari, Annamaria 2

1 Department of Mathematics Temple University Philadelphia, PA 19122
2 Dipartimento di Matematica Università di Bologna Piazza Porta San Donato, 5 40127 Bologna, Italy 
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     title = {On the second order derivatives of convex functions on the {Heisenberg} group},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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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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