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

Cristian E. Gutiérrez[1]; Annamaria Montanari[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

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2004)

  • Volume: 3, Issue: 2, page 349-366
  • ISSN: 0391-173X

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 –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.

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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  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
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  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 -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
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