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Transactions of the American Mathematical Society

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Characterization of rectifiable measures in terms of $\alpha$-numbers


Authors: Jonas Azzam, Xavier Tolsa and Tatiana Toro
Journal: Trans. Amer. Math. Soc. 373 (2020), 7991-8037
MSC (2010): Primary 28A12, 28A75, 28A78
DOI: https://doi.org/10.1090/tran/8170
Published electronically: September 9, 2020
MathSciNet review: 4169680
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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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Additional Information

Jonas Azzam
Affiliation: School of Mathematics, University of Edinburgh, JCMB, Kings Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland
MR Author ID: 828969
ORCID: 0000-0002-9057-634X
Email: j.azzam@ed.ac.uk

Xavier Tolsa
Affiliation: ICREA, Passeig Lluís Companys 23 08010 Barcelona, Catalonia; and Departament de Matemàtiques and BGSMath, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia
MR Author ID: 639506
ORCID: 0000-0001-7976-5433
Email: xtolsa@mat.uab.cat

Tatiana Toro
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350
MR Author ID: 363909
Email: toro@uw.edu

Received by editor(s): March 20, 2019
Received by editor(s) in revised form: March 16, 2020
Published electronically: September 9, 2020
Additional Notes: The second author was supported by the ERC grant 320501 of the European Research Council, and also partially supported by the grants 2017-SGR-395 (Catalonia), MTM-2016-77635-P and MDM-2014-044 (MICINN, Spain).
The third author was partially supported by the Craig McKibben & Sarah Merner Professor in Mathematics and by NSF grant number DMS-1664867
Article copyright: © Copyright 2020 American Mathematical Society