Derivations and Alberti representations

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)(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.