Vol. 12, No. 4, 2019

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Absolute continuity and $\alpha$-numbers on the real line

Tuomas Orponen

Vol. 12 (2019), No. 4, 969–996
DOI: 10.2140/apde.2019.12.969
Abstract

Let $\mu$, $\nu$ be Radon measures on $ℝ$, with $\mu$ nonatomic and $\nu$ doubling, and write $\mu ={\mu }_{a}+{\mu }_{s}$ for the Lebesgue decomposition of $\mu$ relative to $\nu$. For an interval $I\subset ℝ$, define ${\alpha }_{\mu ,\nu }\left(I\right):={\mathbb{W}}_{1}\left({\mu }_{I},{\nu }_{I}\right)$, the Wasserstein distance of normalised blow-ups of $\mu$ and $\nu$ restricted to $I‘$. Let ${\mathsc{S}}_{\nu }$ be the square function

${\mathsc{S}}_{\nu }^{2}\left(\mu \right)=\sum _{I\in \mathsc{D}}{\alpha }_{\mu ,\nu }^{2}\left(I\right){\chi }_{I},$

where $\mathsc{D}$ is the family of dyadic intervals of side-length at most 1. I prove that ${\mathsc{S}}_{\nu }\left(\mu \right)$ is finite ${\mu }_{a}$ almost everywhere and infinite ${\mu }_{s}$ almost everywhere. I also prove a version of the result for a nondyadic variant of the square function ${\mathsc{S}}_{\nu }\left(\mu \right)$. The results answer the simplest “$n=d=1$” case of a problem of J. Azzam, G. David and T. Toro.

Keywords
Wasserstein distance, $\alpha$-numbers, doubling measures
Primary: 42A99