# On the second order derivatives of convex functions on the Heisenberg group

• [1] Department of Mathematics Temple University Philadelphia, PA 19122
• [2] Dipartimento di Matematica Università di Bologna Piazza Porta San Donato, 5 40127 Bologna, Italy
• Volume: 3, Issue: 2, page 349-366
• ISSN: 0391-173X

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## Abstract

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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.

## How to cite

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Gutiérrez, Cristian E., and Montanari, Annamaria. "On the second order derivatives of convex functions on the Heisenberg group." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.2 (2004): 349-366. <http://eudml.org/doc/84533>.

@article{Gutiérrez2004,
abstract = {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 $\mathcal \{H\}$–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 $\mathcal \{H\}$–convex functions in the Heisenberg group.},
affiliation = {Department of Mathematics Temple University Philadelphia, PA 19122; Dipartimento di Matematica Università di Bologna Piazza Porta San Donato, 5 40127 Bologna, Italy},
author = {Gutiérrez, Cristian E., Montanari, Annamaria},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {349-366},
publisher = {Scuola Normale Superiore, Pisa},
title = {On the second order derivatives of convex functions on the Heisenberg group},
url = {http://eudml.org/doc/84533},
volume = {3},
year = {2004},
}

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
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 2
SP - 349
EP - 366
AB - 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 $\mathcal {H}$–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 $\mathcal {H}$–convex functions in the Heisenberg group.
LA - eng
UR - http://eudml.org/doc/84533
ER -

## References

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1. [1] L. Ambrosio – V. Magnani, Weak diferentiability of BV functions on stratified groups, http://cvgmt.sns.it/papers/ambmag02/ Zbl1048.49030
2. [2] Z. M. Balogh – M. Rickly, Regularity of convex functions on Heisenberg groups, http://cvgmt.sns.it/papers/balric/convex.pdf Zbl1121.43007MR2040646
3. [3] D. Danielli – N. Garofalo – D. M. Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), 263-341. Zbl1077.22007MR2014879
4. [4] L. C. Evans – R. Gariepy, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992. Zbl0804.28001MR1158660
5. [5] C. E. Gutiérrez – A. Montanari, Maximum and comparison principles for convex functions on the Heisenberg group, Comm. Partial Differential Equations, to appear. Zbl1056.35033MR2103838
6. [6] P. Juutinen – G. Lu – J. Manfredi – B. Stroffolini, Convex functions on Carnot groups, Preprint. Zbl1124.49024MR2351130
7. [7] Guozhen Lu – J. Manfredi – B. Stroffolini, Convex functions on the Heisenberg group, Calc. Var., Partial Differential Equations, to appear. Zbl1072.49019MR2027845
8. [8] V. Magnani, Lipschitz continuity, Aleksandrov’s Theorem and characterization of $H$-convex functions, http://cvgmt.sns.it/onthefly.cgi/papers/mag03a/hconvex.pdf Zbl1115.49004
9. [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. Zbl0821.42001MR1232192
10. [10] N. S. Trudinger – Xu-Jia Wang, Hessian measures I, Topol. Methods Nonlinear Anal. 10 (1997), 225-239. Zbl0915.35039MR1634570
11. [11] C. Y. Wang, Viscosity convex functions on Carnot groups, http://arxiv.org/abs/math.AP/0309079, Zbl1057.22012

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