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

## TL;DR

This paper studies the relationship between absolute continuity and $ ext{alpha}$-numbers on the real line, showing how a square function built from Wasserstein distances characterizes the Lebesgue decomposition of measures.

## Contribution

It establishes that the square function based on $ ext{alpha}$-numbers is finite almost everywhere on the absolutely continuous part and infinite on the singular part of a measure's Lebesgue decomposition on $ ext{R}$.

## Key findings

- The square function $ ext{S}_
u( extmu)$ is finite $ ext{ extmu}_a$-almost everywhere.
- The square function $ ext{S}_
u( extmu)$ is infinite $ ext{ extmu}_s$-almost everywhere.
- Results provide a solution to the $n=d=1$ case of a problem posed by Azzam, David, and Toro.

## 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 $$\mathcal{S}^{2}_{\nu}(\mu) = \sum_{I \in \mathcal{D}} \alpha_{\mu,\nu}^{2}(I)\chi_{I},$$ 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.