On the density problem in the parabolic space

TL;DR

This paper extends classical geometric measure theory results to parabolic spaces, establishing rectifiability criteria, characterizations via densities, and analyzing uniform measures, with applications to quantitative parabolic rectifiability.

Contribution

It introduces rectifiability criteria and density characterizations for measures in parabolic spaces, expanding the understanding of geometric measure theory in this setting.

Findings

Proved a Marstrand-Mattila rectifiability criterion for measures of general dimension.

Characterized intrinsic rectifiable measures through densities.

Showed that the weak constant density condition implies the bilateral weak geometric lemma.

Abstract

In this work we extend many classical results concerning the relationship between densities, tangents and rectifiability to the parabolic spaces, namely $\mathbb{R}^{n+1}$ equipped with parabolic dilations. In particular we prove a Marstrand-Mattila rectifiability criterion for measures of general dimension, we provide a characterisation through densities of intrinsic rectifiable measures, and we study the structure of $1$-codimensional uniform measures. Finally, we apply some of our results to the study of a quantitative version of parabolic rectifiability: we prove that the weak constant density condition for a $1$-codimensional Ahlfors-regular measure implies the bilateral weak geometric lemma.