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

Abstract

We give a simple criterion on the set of probability tangent measures Tan(μ, x) of a positive Radon measure μ, which yields lower bounds on the Hausdorff dimension of μ. 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.