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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 Q − d, where 0 < d < Q, B α,p (E) = 0 holds if and only if αp ≤ d.

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).

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Guo Zhen Lu

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  1. Guo Zhen Lu
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Correspondence to Guo Zhen Lu.

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

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Lu, G.Z. Potential Analysis on Carnot Groups, Part II: Relationship between Hausdorff Measures and Capacities. Acta Math Sinica 20, 25–46 (2004). https://doi.org/10.1007/s10114-003-0297-8

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  • DOI: https://doi.org/10.1007/s10114-003-0297-8

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