Curvature dimension bounds and functional inequalities [Elektronische Ressource] : localization, tensorization and stability / vorgelegt von Kathrin Bacher

Curvature-Dimension Bounds andFunctional Inequalities: Localization,Tensorization and StabilityDissertationzurErlangung des Doktorgrades (Dr. rer. nat.)derMathematisch-Naturwissenschaftlichen FakultätderRheinischen Friedrich-Wilhelms-Universität Bonnvorgelegt vonKathrin BacherausBad OldesloeBonn 2009Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultätder Rheinischen Friedrich-Wilhelms-Universität Bonn1. Gutachter: Prof. Dr. Karl-Theodor Sturm2. Gutachter: Prof. Dr. Anton ThalmaierTag der Promotion: 5. März 2010Erscheinungsjahr: 2010AbstractThisworkisdevotedtotheanalysisofabstractmetricmeasurespaces (M;d;m)satisfying the curvature-dimension condition CD(K;N) presented by Sturm [Stu06a,Stu06b] and in a similar form by Lott and Villani [LV07, LV09].In the first part, we introduce the notion of a Borell-Brascamp-Lieb inequalityin the setting of metric measure spaces denoted by BBL(K;N). This inequalityholds true on metric measure spaces fulfilling the curvature-dimension conditionCD(K;N) and is stable under convergence of metric measure spaces with respect tothe L -transportation distance.2In the second part, we prove that the local version of CD(K;N) is equiv-alent to a global condition CD (K;N), slightly weaker than the usual global one.This so-called reduced curvature-dimension condition CD (K;N) has the localizationproperty.