Square functions and uniform rectifiability
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- by Vasileios Chousionis, John Garnett, Triet Le and Xavier Tolsa PDF
- Trans. Amer. Math. Soc. 368 (2016), 6063-6102 Request permission
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 $\operatorname {supp}(\mu )$, \[ \int _0^R \int _{x\in B(x_0,R)} \left |\frac {\mu (B(x,r))}{r^n} - \frac {\mu (B(x,2r))}{(2r)^n} \right |^2 d\mu (x) \frac {dr}r \leq c R^n.\] Other characterizations of uniform $n$-rectifiability in terms of smoother square functions are also obtained.References
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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
- MR Author ID: 858546
- 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
- MR Author ID: 639506
- ORCID: 0000-0001-7976-5433
- Email: xtolsa@mat.uab.cat
- 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). - © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6063-6102
- MSC (2010): Primary 42B20, 42B25; Secondary 28A75
- DOI: https://doi.org/10.1090/tran/6557
- MathSciNet review: 3461027