Derivations on Metric Measure Spaces.

Derivation;Metric;Measure;Lipschitz;Mathematics;Science;Mathematics

Gong, JasunTappenden, Jamie ;

University of Michigan

关键词: Derivation;    Metric;    Measure;    Lipschitz;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/60807/jgong_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 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.