The weak lower density condition and uniform rectifiability

Jonas Azzam¹ ; Matthew Hyde²

¹ University of Edinburgh, School of Mathematics
² University of Warwick, Mathematics Institute

Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 791-819.

Résumé

  We show that an Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ for which there exists $y \in B(x,r)$ and $0<t<r$ satisfying $\mathbf{H}^{d}_{\infty}(E\cap B(y,t))<(2t)^{d}-\epsilon(2r)^d$ is a Carleson set for every $\epsilon>0$. To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $\epsilon,\rho\in (0,1)$, $A>1$, and $X_n$ is a sequence of maximal $2^{-n}$-separated sets in $X$, and $\mathbf{B}=\{B(x,2^{-n})\colon x\in X_{n},n\in \mathbb{N}\}$, then   $\sum \left\{r_{B}^s\colon B\in \mathbf{B}, \frac{\mathbf{H}^s_{\rho r_{B}}(X\cap AB)}{(2r_{AB})^s}>1+\epsilon\right\} \le_{C,A,\epsilon,\rho,s} \mathbf{H}^s(X)$. This is a quantitative version of the classical result that for a metric space $X$ of finite $s$-dimensional Hausdorff measure, the upper $s$-dimensional densities are at most 1 $\mathbf{H}^s$-almost everywhere.  

Détail

DOI : 10.54330/afm.119478

Keywords: Uniform rectifiability, uniform measures