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Transactions of the American Mathematical Society

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Square functions and uniform rectifiability


Authors: Vasileios Chousionis, John Garnett, Triet Le and Xavier Tolsa
Journal: Trans. Amer. Math. Soc. 368 (2016), 6063-6102
MSC (2010): Primary 42B20, 42B25; Secondary 28A75
DOI: https://doi.org/10.1090/tran/6557
Published electronically: November 12, 2015
MathSciNet review: 3461027
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Abstract: In this paper it is shown that an Ahlfors-David $ n$-dimensional measure $ \mu $ on $ \mathbb{R}^d$ is uniformly $ n$-rectifiable if and only if for any ball $ B(x_0,R)$ centered at $ \operatorname {supp}(\mu )$,

$\displaystyle \int _0^R \int _{x\in B(x_0,R)} \left \vert\frac {\mu (B(x,r))}{r... ...rac {\mu (B(x,2r))}{(2r)^n} \right \vert^2\,d\mu (x)\,\frac {dr}r \leq c\, R^n.$

Other characterizations of uniform $ n$-rectifiability in terms of smoother square functions are also obtained.

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Additional Information

Vasileios Chousionis
Affiliation: Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland
Address at time of publication: Department of Mathematics, University of Connecticut, 196 Auditorium Road U-3009, Storrs, Connecticut 06269-3009
Email: vasileios.chousionis@helsinki.fi, vasileios.chousionis@uconn.edu

John Garnett
Affiliation: Department of Mathematics, 6363 Math Sciences Building, University of California at Los Angeles, Los Angeles, California 90095-1555
Email: jbg@math.ucla.edu

Triet Le
Affiliation: Department of Mathematics, David Rittenhouse Laboratory, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104
Email: trietle@math.upenn.edu

Xavier Tolsa
Affiliation: Institució Catalana de Recerca i Estudis Avançats (ICREA) and Departament de Matemàtiques, Universitat Autònoma de Barcelona, Catalonia
Email: xtolsa@mat.uab.cat

DOI: https://doi.org/10.1090/tran/6557
Received by editor(s): July 25, 2014
Published electronically: November 12, 2015
Additional Notes: The first author was funded by the Academy of Finland Grant SA 267047, and also partially supported by the ERC Advanced Grant 320501, while visiting Universitat Autònoma de Barcelona
The second author was partially supported by NSF DMS 1217239 and the IPAM long program Interactions Between Analysis and Geometry, Spring 2013.
The third author was partially supported by NSF DMS 1053675 and the IPAM long program Interactions Between Analysis and Geometry, Spring 2013.
The fourth author was funded by an Advanced Grant of the European Research Council (programme FP7/2007-2013), by agreement 320501, and also partially supported by grants 2009SGR-000420 (Generalitat de Catalunya) and MTM-2010-16232 (Spain).
Article copyright: © Copyright 2015 American Mathematical Society

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