Abstract
Let , be Radon measures on , with nonatomic and doubling, and write for the Lebesgue decomposition of relative to . For an interval , define , the Wasserstein distance of normalised blow-ups of and restricted to . Let be the square function
where is the family of dyadic intervals of side-length at most 1. I prove that is finite almost everywhere and infinite almost everywhere. I also prove a version of the result for a nondyadic variant of the square function . The results answer the simplest “” case of a problem of J. Azzam, G. David and T. Toro.
Citation
Tuomas Orponen. "Absolute continuity and $\alpha$-numbers on the real line." Anal. PDE 12 (4) 969 - 996, 2019. https://doi.org/10.2140/apde.2019.12.969
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