Absolute continuity and $α$-numbers on the real line

Tuomas Orponen

Published 2017-03-08Version 1

Let $\mu,\nu$ be Radon measures on $\mathbb{R}$. 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 either one of the square functions $$\mathcal{S}^{2}_{\nu}(\mu)(x) = \sum_{x \in I \in \mathcal{D}} \alpha_{\mu,\nu}^{2}(I) \quad \text{or} \quad \mathcal{S}_{\nu}^{2}(\mu)(x) = \int_{0}^{1} \alpha_{\mu,\nu}^{2}(B(x,r)) \, \frac{dr}{r},$$ where $\mathcal{D}$ is the family of dyadic intervals of side-length at most one. Assuming that $\nu$ is doubling, and $\mu$ does not charge the boundaries of $\mathcal{D}$ (in the case of the first square function), I prove that $\mu|_{G} \ll \nu$, where $$G = \{x : \mathcal{S}_{\nu}(\mu)(x) < \infty\}.$$ If $\mu$ is also doubling, and the square functions satisfy suitable Carleson conditions, then absolute continuity can be improved to $\mu \in A_{\infty}(\nu)$. The results answer the simplest "$n = d = 1"$ case of a problem of J. Azzam, G. David and T. Toro.