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Absolute continuity and $${\alpha}$$-numbers on the real line. (English) Zbl 1403.28003
Summary: Let $$\mu,\nu$$ be Radon measures on $$\mathbb{R}$$, 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 $$I\subset\mathbb{R}$$, 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 $\mathcal{S}^{2}_{\nu}(\mu)=\sum\limits_{I\in\mathcal{D}}\alpha_{\mu,\nu}^{2}(I)\chi_{I},$ where $$\mathcal{D}$$ is the family of dyadic intervals of side-length at most 1. 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 nondyadic 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.

##### MSC:
 28A33 Spaces of measures, convergence of measures 28A75 Length, area, volume, other geometric measure theory
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##### References:
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