Singular points of H\"older asymptotically optimally doubling measures

TL;DR

This paper investigates how the doubling property of measures influences the regularity of their support, specifically analyzing the geometry near non-flat points for codimension-1 H"older doubling measures in four-dimensional space.

Contribution

It provides a detailed analysis of the support geometry of H"older doubling measures in , addressing the open problem of support regularity near non-flat points.

Findings

Supports exhibit regularity properties near non-flat points.

Provides geometric characterization of measures in  near singularities.

Extends previous results to a higher-dimensional setting.

Abstract

We consider the question of how the doubling characteristic of a measure determines the regularity of its support. The question was considered by David, Kenig, and Toro for codimension-1 under a crucial assumption of flatness, and later by Preiss, Tolsa, and Toro in higher codimension. However, their studies leave open the geometry of the support of such measures in a neighborhood about a non-flat point of the support. We here answer the question (in an almost classical sense) for codimension-1 H\"older doubling measures in $\RR^4$.