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

## TL;DR

This paper introduces a simple algebraic criterion for estimating the Hausdorff dimension of measures constrained by first-order linear PDEs, providing elementary proofs and discussing related measure structures.

## Contribution

It presents a new, elementary algebraic approach to lower bounds on the Hausdorff dimension of PDE-constrained measures, simplifying previous complex proofs.

## Key findings

- Established a criterion for tangent measures and dimension bounds.
- Provided elementary proofs for sharp Hausdorff dimension bounds.
- Discussed measure structures and differential forms related to PDE constraints.

## Abstract

We give a simple criterion on the set of probability tangent measures $\mathrm{Tan}(\mu,x)$ 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.