Characterization of rectifiable measures in terms of -numbers

We characterize Radon measures μ \mu in R n \mathbb {R}^{n} that are d d -rectifiable in the sense that their supports are covered up to μ \mu -measure zero by countably many d d -dimensional Lipschitz images and μ ≪ H d \mu \ll \mathcal {H}^{d} . The characterization is in terms of a Jones function...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2020-11, Vol.373 (11), p.7991-8037
Main Authors: Azzam, Jonas, Tolsa, Xavier, Toro, Tatiana
Format: Article
Language:English
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Summary:We characterize Radon measures μ \mu in R n \mathbb {R}^{n} that are d d -rectifiable in the sense that their supports are covered up to μ \mu -measure zero by countably many d d -dimensional Lipschitz images and μ ≪ H d \mu \ll \mathcal {H}^{d} . The characterization is in terms of a Jones function involving the so-called α \alpha -numbers. This answers a question left open in a former work by Azzam, David, and Toro.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/8170