An estimate for surface measure of small balls in Carnot groups

We introduce a family of quasidistances in ${\mathbb R}^d$, such that some of them are equivalent to natural distances on Carnot groups. We find the sufficient conditions for the balls w.r.t. a quasidistance from our family to be comparable to ellipsoids. Using comparability to ellipsoids we find asymptotics of surface measure of intersections of small balls with linear submanifolds and the conditions for finiteness of the integral w.r.t. the surface measure of negative power of the distance. We provide several examples of Carnot groups, where comparability to ellipsoids can be shown for natural distances, and therefore we can study the asymptotics and finitness of the integrals explicitly. We also show an example of a Carnot group, where the comparability to ellipsoids does not hold.