In this paper,we establish the relationship between Hausdorff measures and Bessel capac- ities on any nilpotent stratified Lie group G (i.e.,Carnot group).In particular,as a special corollary of our much more general ...In this paper,we establish the relationship between Hausdorff measures and Bessel capac- ities 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_(α,p)(E)=0 implies (?)^(Q-αp+(?))(E)=0 for all (?)>0.On the other hand,(?)^(Q-αp)(E)<∞ implies B_(α,p)(E)=0.Conse- quently,given any set E(?)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).展开更多
基金Research supportde partly by the U.S.National Science Foundation Grant No.DMS99-70352
文摘In this paper,we establish the relationship between Hausdorff measures and Bessel capac- ities 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_(α,p)(E)=0 implies (?)^(Q-αp+(?))(E)=0 for all (?)>0.On the other hand,(?)^(Q-αp)(E)<∞ implies B_(α,p)(E)=0.Conse- quently,given any set E(?)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).
