Vol. 12, No. 4, 2019

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

Tuomas Orponen

Vol. 12 (2019), No. 4, 969–996
DOI: 10.2140/apde.2019.12.969

Let μ, ν be Radon measures on , with μ nonatomic and ν doubling, and write μ = μa + μs for the Lebesgue decomposition of μ relative to ν. For an interval I , define αμ,ν(I) := W1(μI,νI), the Wasserstein distance of normalised blow-ups of μ and ν restricted to I. Let Sν be the square function

Sν2(μ) = IDαμ,ν2(I)χ I,

where D is the family of dyadic intervals of side-length at most 1. I prove that Sν(μ) is finite μa almost everywhere and infinite μs almost everywhere. I also prove a version of the result for a nondyadic variant of the square function Sν(μ). The results answer the simplest “n = d = 1” case of a problem of J. Azzam, G. David and T. Toro.

Wasserstein distance, $\alpha$-numbers, doubling measures
Mathematical Subject Classification 2010
Primary: 42A99
Received: 15 March 2017
Revised: 12 June 2018
Accepted: 14 July 2018
Published: 20 October 2018
Tuomas Orponen
Department of Mathematics and Statistics
University of Helsinki