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

2003/09/09 by Cristian E. Gutierrez, Gutierrez, Cristian E., Annamaria Montanari +1
Mathematics · #35B45 #35B50 #35H20 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B45 #msc:35B50 #msc:35H20

paper · pdf · doi:10.48550/arxiv.math/0309167

arxiv created 2003/09/09 · arxiv updated 2009/12/01

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 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.

Related

• Horizontal normal map on the Heisenberg group
• Geodetically convex sets in the Heisenberg group \mathbb Hⁿ
• Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations
• Note on the Fenchel transform in the Heisenberg group
• Characterizations of differentiability for h-convex functions in stratified groups