On the differentiability of Lipschitz functions with respect to measures in the Euclidean space View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2016-02

AUTHORS

Giovanni Alberti, Andrea Marchese

ABSTRACT

For every finite measure μ on Rn we define a decomposability bundle V(μ,·) related to the decompositions of μ in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on Rn is differentiable at μ-a.e. x with respect to the subspace V(μ,x), and prove that this differentiability result is optimal, in the sense that, following (Alberti et al., Structure of null sets, differentiability of Lipschitz functions, and other problems, 2016), we can construct Lipschitz functions which are not differentiable at μ-a.e. x in any direction which is not in V(μ,x). As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) k-dimensional normal currents, which we use to extend certain basic formulas involving normal currents and maps of class C1 to Lipschitz maps. More... »

PAGES

1-66

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00039-016-0354-y

DOI

http://dx.doi.org/10.1007/s00039-016-0354-y

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1035724589


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