Geometric second derivative estimates in Carnot groups and convexity

Abstract

Abstract. We prove some new a priori estimates for H2-convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group G. Such estimates are global and are geometric in nature as they involve the horizontal mean curvature H of ∂Ω. As a consequence of our bounds we show that if G has step two, then for any smooth H2-convex function in Ω ⊂ G vanishing on ∂Ω one has

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].…”

Subelliptic geometric Hardy type inequalities on half-spaces and convex domains

“…. , r, is the Riemannian outer unit normal to ∂G + (see [11]) and d ∈ R. The Euclidean distance to the boundary ∂G + is denoted by dist(x, ∂G + ) and defined as follows…”

Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups

“…. , n) are left-invariant vector fields on the Heisenberg group, ν is the Riemannian outer unit normal (see [9]) to the boundary. Also, there is the L p -generalisation of the above inequality…”

“…. , r, is the Riemannian outer unit normal to ∂G + (see [9]) and d ∈ R. Let us define the so-called angle function…”