Boundedness of the density normalised Jones' square function does not imply $1$-rectifiability

Abstract

Recently, M. Badger and R. Schul proved that for a $1$-rectifiable Radon measure $\mu$, the density weighted Jones' square function $$J_1(x)={\sum _{Q\in \mathcal{D}}}_{\ell (Q)\le 1}{\beta }_{2,\mu }^2(3Q)\frac{\ell (Q)}{\mu (Q)}{1}_Q(x)$$ is finite for $\mu$-a.e. $x$. Answering a question of Badger-Schul, we show that the converse is not true. Given $\epsilon >0$, we construct a Radon probability measure on $[0,1{]}^2\subset {\mathbb{R}}^2$ with the properties that $J_1(x)\le \epsilon$ for all $x\in \mathrm{spt}\mu$, but nevertheless the $1$-dimensional lower density of $\mu$ vanishes almost everywhere. In particular, $\mu$ is purely $1$-unrectifiable.

23 pages, 4 figures

References in corpus (2)

• Multiscale analysis of 1-rectifiable measures: necessary conditions
• A characterization of $1$-rectifiable doubling measures with connected supports

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• Necessary condition for rectifiability involving Wasserstein distance $W_2$
• Radon measures and Lipschitz graphs
• Sufficient condition for rectifiability involving Wasserstein distance $W_2$