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
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Proc. Amer. Math. Soc. 148 (2020), 273-282 Request permission

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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