On the density problem in the parabolic space

2022/11/08 by Merlo, Andrea, Mourgoglou, Mihalis, Puliatti, Carmelo · 1 citation
#28A75 #28A78 #30L99 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2211.04222

Abstract

In this work we extend many classical results concerning the relationship between densities, tangents and rectifiability to the parabolic spaces, namely ℝⁿ⁺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.

Cited by

• Marstrand's density theorem for arbitrary norms in the plane

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