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

Abstract

Let $\mu ,\nu$ be Radon measures on $\mathbb{R}$, with $\mu$ non-atomic and $\nu$ doubling, and write $\mu ={\mu }_a+{\mu }_s$ for the Lebesgue decomposition of $\mu$ relative to $\nu$. For an interval $I\subset \mathbb{R}$, define ${\alpha }_{\mu ,\nu }(I):={\mathbb{W}}_1({\mu }_I,{\nu }_I)$, the Wasserstein distance of normalised blow-ups of $\mu$ and $\nu$ restricted to $I$. Let ${\mathcal{S}}_{\nu }$ be the square function $${\mathcal{S}}_{\nu }^2(\mu )=\sum _{I\in \mathcal{D}}{\alpha }_{\mu ,\nu }^2(I){\chi }_I,$$ where $\mathcal{D}$ is the family of dyadic intervals of side-length at most one. I prove that ${\mathcal{S}}_{\nu }(\mu )$ is finite ${\mu }_a$ almost everywhere, and infinite ${\mu }_s$ almost everywhere. I also prove a version of the result for a non-dyadic variant of the square function ${\mathcal{S}}_{\nu }(\mu )$. The results answer the simplest "$n=d=1"$ case of a problem of J. Azzam, G. David and T. Toro.

27 pages, 1 figure. v3: main results upgraded from sufficient conditions to characterisations

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