, Volume 20, Issue 1, pp 25-46

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

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Abstract

In this paper, we establish the relationship between Hausdorff measures and Bessel capacities on any nilpotent stratified Lie group $\Bbb G$ (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 $\Bbb G$ . Given any set E $\Bbb G$ , B α,p (E) = 0 implies ℋ Q−αp+ ε(E) = 0 for all ε > 0. On the other hand, ℋ Q−αp (E) < ∞ implies B α,p (E) = 0. Consequently, given any set E $\Bbb G$ of Hausdorff dimension Qd, where 0 < d < Q, B α,p (E) = 0 holds if and only if αpd.

A version of O. Frostman’s 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).

Research supported partly by the U. S. National Science Foundation Grant No. DMS99–70352