Absolute continuity and α-numbers on the real line

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

Tuomas Orponen

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors