Abstract
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 order horizontal derivatives are Radon measures.
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Magnani, V. Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex functions. Math. Ann. 334, 199–233 (2006). https://doi.org/10.1007/s00208-005-0717-4
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DOI: https://doi.org/10.1007/s00208-005-0717-4