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An estimate for surface measure of small balls in Carnot groups. (English) Zbl 1440.43018
Summary: 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.
##### MSC:
 43A80 Analysis on other specific Lie groups 26D15 Inequalities for sums, series and integrals 58C35 Integration on manifolds; measures on manifolds
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