Potential Analysis on Carnot Groups, Part II: Relationship between Hausdorff Measures and Capacities

In this paper, we establish the relationship between Hausdorff measures and Bessel capacities on any nilpotent stratified Lie group G (i. e., Carnot group). In particular, as a special corollary of our much more general results, we have the following theorem (see Theorems A and E in Section 1): Let Q be the homogeneous dimension of G. Given any set E G, B ,p (E) = 0 implies Qp* (E) = 0 for all > 0. On the other hand, Qp(E) > implies B ,p (E) = 0. Consequently, given any set E G of Hausdorff dimension Q d, where 0 > d > Q, B ,p (E) = 0 holds if and only if p d. A version of O. Frostmans theorem concerning Hausdorff measures on any homogeneous space is also established using the dyadic decomposition on such a space (see Theorem 4.4 in Section 4).