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arXiv · 2211.04222

On the density problem in the parabolic space

Andrea Merlo·Mihalis Mourgoglou·Carmelo Puliatti
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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.

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Andrea Merlo, Mihalis Mourgoglou, Carmelo Puliatti. 2022-11-08. On the density problem in the parabolic space. https://arxiv.org/abs/2211.04222

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