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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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