The Theorem of Busemann-Feller-Alexandrov in Carnot Groups

On Hessian Measures for Non-Commuting Vector Fields

“…The identity (2.5) was discovered by Gutiérrez and Montanari [8], [9] for the Heisenberg groups H n , (see also [5], [7]). From (2.5), we infer the monotonicity formula for the operator F 2 defined by (2.7)…”

“…More general version of Lemma 2.1 are presented in [5]. For weak continuity with respect to C 0 (Ω) and for groups of Heisenberg type we may proceed exactly as in [18].…”

“…See [5] for a more thorough analysis of monotonicity. In a similar fashion, we may combine (4.15) and (4.17) in (4.6) to conclude,…”

A new characterization of convexity in free Carnot groups

“…The problem of fine regularity for convex functions has especially been investigated. See Capogna and Maldonado [7]; Capogna, Pauls, and Tyson [8]; Danielli, Garofalo, Nhieu, and Tournier [11]; Garofalo [14]; Garofalo and Tournier [15]; Gutiérrez and Montanari [17,18]; Juutinen, Lu, Manfredi, and Stroffolini [21]; Monti and Rickly [24], Rickly [25]; Sun and Yang [28,29]. (For the notion of "r-convex" function in Carnot groups, see Dah-Yan [9] and Sun and Yang [26,27].…”

A note on the engulfing property and the $\Gamma ^{1+ \alpha }$-regularity of convex functions in Carnot groups

“…In particular, several interesting results concerning Monge-Ampère-type equations and notions of Hessian measures have been proved by Gutierrez and Montanari [17], [18], Garofalo and Tournier [14], Danielli et al [4], [6], Lu et al [22]. At the moment it seems very difficult to obtain sub-Riemannian analogues of Euclidean regularity results such as Caffarelli's [3].…”