Ahlfors type estimates for perimeter measures in Carnot-Carathéodory spaces (Q2434070)

The very interesting paper under review studies the interplay between the geometry of a hypersurface \(S=\partial\Omega\subset M^n\) in a Carnot-Carathéodory space and the \(s\)-Ahlfors regularity of its sub-Riemannian perimeter measure \(P_X(\Omega;\cdot).\) For this goal the authors establish comparison theorems for perimeter estimates between an hypersurface and its tangent space, and between a Carnot-Carathéodory geometry and its tangent Carnot group structure. In contrast to all results available up to now where vector fields are considered satisfying the finite rank condition at step \(r=2,\) the novel aspect of the paper is that no restrictions are imposed on the step of the Carnot-Carathéodory space.