Convex functions on the Heisenberg group

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Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex functions (2006, methodology): “It has been proved in [9] and [ 23 ] that for every locally summable H-convex function u the matrix of distributions D 2 u is nonnegative,”
Viscosity convex functions on carnot groups (2003, methodology): “5.11) or [ LMS ](§4, Proposition 4.1) implies that uǫδ is h-convex on (2R0+1)ǫ+δ, i.e.”
Hardy-type inequalities related to degenerate elliptic differential operators (2006, background): “have −Lp(φ) ≤ 0, thus in order to obtain Hardy inequalities involving thefunction φwe can applythe results ofprevious section. ForH-convex function on Carnot groups, we refer the interested reader to [22, 38, 39]. For instance, in [22] the authors prove that in an H-type group the gauge Ndeﬁned in (”
Convex functions on Carnot groups. (2007, methodology): “The point of view adapted in [13] is to use the fact that convex functions are subsolutions of all homogeneous elliptic equations.”
Geodetically Convex Sets in the Heisenberg Group (2005, background): “Relations between convexity in the viscosity sense and horizontal convexity as well as sharp regularity properties of convex functions in Heisenberg groups are discussed in [9] and [3].”
Notions of Convexity in Carnot Groups (2003, background): “The notion of convexity adopted in [ LMS ] is a reformulation of Caffarelli’s, and thereby equivalent to our notion of weak H-convexity.”