Acta Mathematica Sinica

, Volume 20, Issue 1, pp 25–46

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

ORIGINAL ARTICLES

DOI: 10.1007/s10114-003-0297-8

Cite this article as:
Lu, G.Z. Acta Math Sinica (2004) 20: 25. doi:10.1007/s10114-003-0297-8

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\(\Bbb G\), Bα,p(E) = 0 implies ℋ Q−αp+ ε(E) = 0 for all ε > 0. On the other hand, ℋ Q−αp(E) < ∞ implies Bα,p(E) = 0. Consequently, given any set E\(\Bbb G\) of Hausdorff dimension Qd, where 0 < d < Q, Bα,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).

Keywords

Sobolev spacesStratified groupsBessel capacitiesHausdorff measuresRadon measures

MR (2000) Subject Classification

46E3541A1022E25

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA