Search results for "Measure space"

showing 4 items of 24 documents

Differential of metric valued Sobolev maps

2020

We introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove that our notion is consistent with Kirchheim's metric differential when the source is a Euclidean space, and with the abstract differential provided by the first author when the target is $\mathbb{R}$.

metric measure spacesPure mathematicsFunction spaces; Metric measure spaces; Sobolev spaces01 natural sciencesMetric measure spacesfunction spacesSettore MAT/05 - Analisi Matematica0103 physical sciencesFOS: MathematicsTrigonometric functions0101 mathematicsMathematicsEuclidean space010102 general mathematicsTangentmetriset avaruudetFunctional Analysis (math.FA)Mathematics - Functional AnalysisSobolev spaceMetric spaceSobolev spacesFunction spaces010307 mathematical physicsfunktionaalianalyysiMetric differentialAnalysisJournal of Functional Analysis
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Mappings of finite distortion between metric measure spaces

2015

We establish the basic analytic properties of mappings of finite distortion between proper Ahlfors regular metric measure spaces that support a ( 1 , 1 ) (1,1) -Poincaré inequality. As applications, we prove that under certain integrability assumption for the distortion function, the branch set of a mapping of finite distortion between generalized n n -manifolds of type A A has zero Hausdorff n n -measure.

metric measure spacesPure mathematicsInjective metric spaceta111Mathematical analysisMathematicsofComputing_GENERALProduct metricEquivalence of metricsConvex metric spaceIntrinsic metricDistortion (mathematics)mappings of finite distortionMetric (mathematics)Metric mapGeometry and TopologyMathematicsConformal Geometry and Dynamics of the American Mathematical Society
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Existence of optimal transport maps in very strict CD(K,∞) -spaces

2018

We introduce a more restrictive version of the strict CD(K,∞) -condition, the so-called very strict CD(K,∞) -condition, and show the existence of optimal maps in very strict CD(K,∞) -spaces despite the possible lack of uniqueness of optimal plans. peerReviewed

metric measure spacesdifferentiaaligeometriaRicci curvatureoptimal mass transportationvariaatiolaskentaexistence of optimal mapsmittateoriametriset avaruudetbranching geodesics
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Non-branching geodesics and optimal maps in strong CD(K,∞) -spaces

2014

We prove that in metric measure spaces where the entropy functional is Kconvex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one optimal transport plan between any two absolutely continuous measures with finite second moments and this plan is given by a map. The results are applicable in metric measure spaces having Riemannian Ricci curvature bounded below, and in particular they hold also for Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below by some constant. peerReview…

metric measure spacesoptimal mapssMathematics::Metric GeometryMathematics::Differential Geometrynon-branching geodesic
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