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An elementary approach to the dimension of measures satisfying a first-order linear PDE constraint


Author: Adolfo Arroyo-Rabasa
Journal: Proc. Amer. Math. Soc. 148 (2020), 273-282
MSC (2010): Primary 28A78, 49Q15; Secondary 35F35
DOI: https://doi.org/10.1090/proc/14732
Published electronically: August 7, 2019
MathSciNet review: 4042849
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Abstract: We give a simple criterion on the set of probability tangent measures $ \operatorname {Tan}(\mu ,x)$ of a positive Radon measure $ \mu $, which yields lower bounds on the Hausdorff dimension of $ \mu $. As an application, we give an elementary and purely algebraic proof of the sharp Hausdorff dimension lower bounds for first-order linear PDE-constrained measures; bounds for closed (measure) differential forms and normal currents are further discussed. A weak structure theorem in the spirit of [Ann. Math. 184(3) (2016), pp. 1017-1039] is also discussed for such measures.


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Additional Information

Adolfo Arroyo-Rabasa
Affiliation: Mathematics Institute, The University of Warwick, Coventry CV4 7AL, United Kingdom
Email: adolfo.arroyo-rabasa@warwick.ac.uk; and adolforabasa@gmail.com

DOI: https://doi.org/10.1090/proc/14732
Keywords: Hausdorff dimension, $\mathcal{A}$-free measure, PDE constraint, tangent measure, structure theorem, normal current
Received by editor(s): December 18, 2018
Received by editor(s) in revised form: April 15, 2019
Published electronically: August 7, 2019
Additional Notes: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement No. 757254 (SINGULARITY).
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2019 American Mathematical Society