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

Arroyo-Rabasa, Adolfo

doi:10.1090/proc/14732

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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