Derivations and Alberti representations

By Andrea Schioppa. 2013

50 citations recorded by the source corpus.

Abstract We relate generalized Lebesgue decompositions of measures in terms of curve fragments (“Alberti representations”) and Weaver derivations. This correspondence leads to a geometric characterization of the local norm on the Weaver cotangent bundle of a metric measure space ( X , μ ) : the local norm of a form df “sees” how fast f grows on curve fragments “seen” by μ. This implies a new characterization of differentiability spaces in terms of the μ-a.e. equality of the local norm of df and the local Lipschitz constant of f. As a consequence, the “Lip–lip” inequality of Keith must be an equality. We also provide dimensional bounds for the module of derivations in terms of the Assouad dimension of X.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0f4c4d7e66542e2f5d34c87bd2abd78b869987d0

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On quotients of spaces with Ricci curvature bounded below (2017, background): “Note that it was shown in [67, 25] that if the Lip-lip-condition holds for some C > 0 then it holds for C = 1.”
Well posedness of Lagrangian flows and continuity equations in metric measure spaces (2014, background): “2, the (extended) distance is hidden in the choice of the family A, we call the next result “superposition in metric measure spaces”, because in most cases A consists precisely of distance functions from a countable dense set (see also the recent papers [Bate 2013; Schioppa 2014] for related results”