Derivations and Alberti representations

Abstract

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.MSC: 53C23, 46J15, 58C20

The Lip-lip equality is stable under blow-up

“…Later we showed [Sch14b,Sch13] that the Lip-lip inequality always self-improves to an equality (see also [CKS15] for another argument); this might be interpreted as saying that the Lip-lip equality provides an asymptotically quantitative characterization of differentiability spaces; however, there is a more precise result in terms of the quantitative characterization of the local norm for Weaver forms (Theorem 2.29) which will be used in this paper.…”

“…In many applications in analysis on metric spaces the assumption that X(µ) has finite index (and is hence finitely generated) is not restrictive: for example it holds if either µ or X are doubling (see Corollary 5.136 in [Sch13]). …”

“…Note that locality allows to extend the action of derivations on Lipschitz functions so that if f ∈ Lip(X) and D ∈ X(µ), Df is well-defined (see Remark 2.115 in [Sch13]). We now pass to consider some algebraic properties of X(µ).…”

“…Unfortunately, f x will be bad only at some locations near x and at some scales (which can be small but are bounded away from 0); thus it is then necessary to glue several f x 's together. It is worth to point out that lifting depends on the structure theory for Weaver derivations developed in [Sch14b,Sch13]. In fact, (Y x , ν x ) is not a differentiability space, but there is still a decent theory (in particular finite-dimensionality) for the L ∞ -modules of Weaver derivations and forms.…”

“…This can be done from two equivalent points of view: [Bat12, Sec. 4] choosing a candidate chart and producing a non-differentiable function for that chart, [Sch13,Sec. 5.3] showing that the space of "germs" of Lipschitz functions is infinite-dimensional.…”