Absolute continuity and α-Numbers on the real line

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http://hdl.handle.net/10138/307023

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Orponen , T 2019 , ' Absolute continuity and α-Numbers on the real line ' , Analysis & PDE , vol. 12 , no. 4 , pp. 969-996 . https://doi.org/10.2140/apde.2019.12.969

Title: Absolute continuity and α-Numbers on the real line
Author: Orponen, Tuomas
Contributor: University of Helsinki, Department of Mathematics and Statistics
Date: 2019
Language: eng
Number of pages: 28
Belongs to series: Analysis & PDE
ISSN: 1948-206X
URI: http://hdl.handle.net/10138/307023
Abstract: Let mu be Radon measures on 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 of R, define alpha(mu,nu) (I) := W-1 (mu(I), nu(I)), the Wasserstein distance of normalised blow-ups of mu and nu restricted to I. Let S nu be the square function S-nu(2) (mu) = Sigma alpha(2)(mu,nu)(I) chi(1), where D is the family of dyadic intervals of side-length at most 1. I prove that 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 S-nu(mu). The results answer the simplest "n = d = 1" case of a problem of J. Azzam, G. David and T. Toro.
Subject: Wasserstein distance
alpha-numbers
doubling measures
RECTIFIABILITY
OPERATORS
WEIGHTS
111 Mathematics
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