Vol. 12, No. 4, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 12
Issue 7, 1643–1890
Issue 6, 1397–1642
Issue 5, 1149–1396
Issue 4, 867–1148
Issue 3, 605–866
Issue 2, 259–604
Issue 1, 1–258

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Subscriptions
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
 
Other MSP Journals
This article is available for purchase or by subscription. See below.
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 , 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.

PDF Access Denied

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/apde

We have not been able to recognize your IP address 100.24.122.228 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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