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}, pages = {349--366}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 3}, number = {2}, year = {2004}, zbl = {1170.35352}, mrnumber = {2075987}, language = {en}, url = {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/
[1] Weak diferentiability of BV functions on stratified groups, http://cvgmt.sns.it/papers/ambmag02/ | Zbl 1048.49030
- ,[2] Regularity of convex functions on Heisenberg groups, http://cvgmt.sns.it/papers/balric/convex.pdf | Numdam | MR 2040646
- ,[3] Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), 263-341. | MR 2014879 | Zbl 1077.22007
- - ,[4] “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992. | MR 1158660 | Zbl 0804.28001
- ,[5] Maximum and comparison principles for convex functions on the Heisenberg group, Comm. Partial Differential Equations, to appear. | MR 2103838 | Zbl 1056.35033
- ,[6] Convex functions on Carnot groups, Preprint. | MR 2351130
- - - ,[7] Convex functions on the Heisenberg group, Calc. Var., Partial Differential Equations, to appear. | MR 2027845 | Zbl 1072.49019
- - ,[8] Lipschitz continuity
,[9] “Harmonic Analysis: Real Variable methods, Orthogonality and Oscillatory Integrals”, Vol. 43 of the Princeton Math. Series. Princeton U. Press. Princeton, NJ, 1993. | MR 1232192 | Zbl 0821.42001
,[10] Hessian measures I, Topol. Methods Nonlinear Anal. 10 (1997), 225-239. | MR 1634570 | Zbl 0915.35039
- ,[11] Viscosity convex functions on Carnot groups, http://arxiv.org/abs/math.AP/0309079, | Zbl 1057.22012
,