Radon measures and Lipschitz graphs

TL;DR

This paper characterizes Radon measures supported on Lipschitz graphs in Euclidean space by analyzing their behavior on dyadic cubes and conical annuli, extending classical rectifiability criteria to more general measures.

Contribution

It introduces a new characterization of Lipschitz graph rectifiability for arbitrary Radon measures using coarse doubling ratios, expanding previous results limited to pointwise doubling measures.

Findings

Provides a measure-theoretic criterion for Lipschitz graph rectifiability.

Extends classical rectifiability characterizations to general Radon measures.

Uses dyadic cube analysis and conical annuli to detect measure support on Lipschitz graphs.

Abstract

For all $1\leq m\leq n-1$, we investigate the interaction of locally finite measures in $\mathbb{R}^n$ with the family of $m$-dimensional Lipschitz graphs. For instance, we characterize Radon measures $\mu$, which are carried by Lipschitz graphs in the sense that there exist graphs $\Gamma_1,\Gamma_2,\dots$ such that $\mu(\mathbb{R}^n\setminus\bigcup_1^\infty\Gamma_i)=0$, using only countably many evaluations of the measure. This problem in geometric measure theory was classically studied within smaller classes of measures, e.g.~for the restrictions of $m$-dimensional Hausdorff measure $\mathcal{H}^m$ to $E\subseteq \mathbb{R}^n$ with $0<\mathcal{H}^m(E)<\infty$. However, an example of Cs\"{o}rnyei, K\"{a}enm\"{a}ki, Rajala, and Suomala shows that classical methods are insufficient to detect when a general measure charges a Lipschitz graph. To develop a characterization of Lipschitz graph rectifiability for arbitrary Radon measures, we look at the behavior of coarse doubling ratios of the measure on dyadic cubes that intersect conical annuli. This extends a characterization of graph rectifiability for pointwise doubling measures by Naples by mimicking the approach used in the characterization of Radon measures carried by rectifiable curves by Badger and Schul.