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 
@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},
     mrnumber = {2075987},
     zbl = {1170.35352},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2004_5_3_2_349_0/}
}
TY  - JOUR
AU  - Gutiérrez, Cristian E.
AU  - Montanari, Annamaria
TI  - On the second order derivatives of convex functions on the Heisenberg group
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2004
SP  - 349
EP  - 366
VL  - 3
IS  - 2
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2004_5_3_2_349_0/
LA  - en
ID  - ASNSP_2004_5_3_2_349_0
ER  - 
%0 Journal Article
%A Gutiérrez, Cristian E.
%A Montanari, Annamaria
%T On the second order derivatives of convex functions on the Heisenberg group
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2004
%P 349-366
%V 3
%N 2
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2004_5_3_2_349_0/
%G en
%F 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://archive.numdam.org/item/ASNSP_2004_5_3_2_349_0/

[1] L. Ambrosio - V. Magnani, Weak diferentiability of BV functions on stratified groups, http://cvgmt.sns.it/papers/ambmag02/ | Zbl

[2] Z. M. Balogh - M. Rickly, Regularity of convex functions on Heisenberg groups, http://cvgmt.sns.it/papers/balric/convex.pdf | Numdam | MR

[3] D. Danielli - N. Garofalo - D. M. Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), 263-341. | MR | Zbl

[4] L. C. Evans - R. Gariepy, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992. | MR | Zbl

[5] C. E. Gutiérrez - A. Montanari, Maximum and comparison principles for convex functions on the Heisenberg group, Comm. Partial Differential Equations, to appear. | MR | Zbl

[6] P. Juutinen - G. Lu - J. Manfredi - B. Stroffolini, Convex functions on Carnot groups, Preprint. | MR

[7] Guozhen Lu - J. Manfredi - B. Stroffolini, Convex functions on the Heisenberg group, Calc. Var., Partial Differential Equations, to appear. | MR | Zbl

[8] V. Magnani, Lipschitz continuity

[9] E. M. Stein, “Harmonic Analysis: Real Variable methods, Orthogonality and Oscillatory Integrals”, Vol. 43 of the Princeton Math. Series. Princeton U. Press. Princeton, NJ, 1993. | MR | Zbl

[10] N. S. Trudinger - Xu-Jia Wang, Hessian measures I, Topol. Methods Nonlinear Anal. 10 (1997), 225-239. | MR | Zbl

[11] C. Y. Wang, Viscosity convex functions on Carnot groups, http://arxiv.org/abs/math.AP/0309079, | Zbl