Absolute continuity and α-numbers on the real line

Orponen, Tuomas

doi:10.2140/apde.2019.12.969

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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

Abstract

Let $μ$ , $ν$ be Radon measures on $ℝ$ , with $μ$ nonatomic and $ν$ doubling, and write $μ = μ_{a} + μ_{s}$ for the Lebesgue decomposition of $μ$ relative to $ν$ . For an interval $I \subset ℝ$ , define $α_{μ, ν} (I) : = W_{1} (μ_{I}, ν_{I})$ , the Wasserstein distance of normalised blow-ups of $μ$ and $ν$ restricted to $I ‘$ . Let $S_{ν}$ be the square function

S_{ν}^{2} (μ) = \sum_{I \in D} α_{μ, ν}^{2} (I) χ_{I},

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

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Keywords

Wasserstein distance, $\alpha$-numbers, doubling measures

Mathematical Subject Classification 2010

Primary: 42A99

Milestones

Received: 15 March 2017

Revised: 12 June 2018

Accepted: 14 July 2018

Published: 20 October 2018

Authors

Tuomas Orponen
Department of Mathematics and Statistics University of Helsinki Helsinki Finland

Vol. 12, No. 4, 2019