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

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 ${\mathcal{S}}_{\nu }$ be the square function

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

where $\mathcal{D}$ is the family of dyadic intervals of side-length at most 1. I prove that ${\mathcal{S}}_{\nu }\left(\mu \right)$ 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 ${\mathcal{S}}_{\nu }\left(\mu \right)$. The results answer the simplest “$n=d=1$” case of a problem of J. Azzam, G. David and T. Toro.