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On the second order derivatives of convex functions on the Heisenberg group. (English) Zbl 1170.35352
Summary: 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.

##### MSC:
 35B50 Maximum principles in context of PDEs 35B45 A priori estimates in context of PDEs 35H20 Subelliptic equations 22E30 Analysis on real and complex Lie groups
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##### References:
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