Inserted: 3 sep 2003
Last Updated: 24 apr 2006
Journal: Math. Ann.
In the geometries of stratified groups, we show that H-convex functions locally bounded from above are locally Lipschitz continuous and that the class of v-convex functions exactly corresponds to the class of upper semicontinuous H-convex functions. As a consequence, v-convex functions are locally Lipschitz continuous in every stratified group. In the class of step 2 groups we characterize locally Lipschitz H-convex functions as measures whose distributional horizontal Hessian is positive semidefinite. In Euclidean space the same results were obtained by Dudley and Reshetnyak. We prove that a continuous H-convex function is a.e. twice differentiable whenever its second horizontal derivatives are Radon measures. Due to recent results by Garofalo-Tournier and Gutiérrez-Montanari the assumptions of the previous differentiability theorem are satisfied by H-convex functions in the Heisenberg group. Then H-convex functions in the Heisenberg group satisfy the sub-Riemannian version of the classical Aleksandrov-Busemann-Feller theorem.
Keywords: convex functions, lipschitz continuity, Dudley and Reshetnyak theorems, Aleksandrov theorem