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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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