Characterization of rectifiable measures in terms of 𝛼-numbers

Azzam, Jonas; Tolsa, Xavier; Toro, Tatiana

doi:10.1090/tran/8170

Characterization of rectifiable measures in terms of $\alpha$-numbers
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by Jonas Azzam, Xavier Tolsa and Tatiana Toro PDF

Trans. Amer. Math. Soc. 373 (2020), 7991-8037 Request permission

Abstract:

We characterize Radon measures $\mu$ in $\mathbb {R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $\mu$-measure zero by countably many $d$-dimensional Lipschitz images and $\mu \ll \mathcal {H}^{d}$. The characterization is in terms of a Jones function involving the so-called $\alpha$-numbers. This answers a question left open in a former work by Azzam, David, and Toro.

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