Derivations on Metric Measure Spaces.

[摘要] In this thesis we study derivations on metric spaces with a prescribed measure.Such objects share similar properties as vector fields on smooth manifolds, such as a locality property and a pushforward construction.Our approach follows Weaver;;s theory of metric derivations.As in his framework, our derivations form a module over the ring of essentially bounded functions with respect to the given measure.From this linear algebraic structure, it is then reasonable to consider emph{linearly independent} sets of derivations.Our main result is as follows.For $k = 1, 2$, we characterize measures on $R^k$ which admit linearly independent sets of $k$ derivations.They are precisely the measures that are absolutely continuous to Lebesgue $k$-measure.The proof in $R^2$ uses new results about the structure of null sets due to Alberti, Cs;;ornyei, and Preiss.We also consider derivations on metric spaces that admit a doubling measure and a weak $(1,p)$-Poincar;;e inequality.Using properties of pushforward derivations, our main result then implies a special case of Cheeger;;s conjecture, which concerns the non-degeneracy of Lipschitz images of such spaces.

[发布机构] University of Michigan

[效力级别] Metric

[关键词] Derivation;Metric;Measure;Lipschitz;Mathematics;Science;Mathematics