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2019 Absolute continuity and $\alpha$-numbers on the real line
Tuomas Orponen
Anal. PDE 12(4): 969-996 (2019). DOI: 10.2140/apde.2019.12.969

Abstract

Let μ , ν be Radon measures on , 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 , 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 ) = I 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.

Citation

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Tuomas Orponen. "Absolute continuity and $\alpha$-numbers on the real line." Anal. PDE 12 (4) 969 - 996, 2019. https://doi.org/10.2140/apde.2019.12.969

Information

Received: 15 March 2017; Revised: 12 June 2018; Accepted: 14 July 2018; Published: 2019
First available in Project Euclid: 30 October 2018

zbMATH: 06991223
MathSciNet: MR3869382
Digital Object Identifier: 10.2140/apde.2019.12.969

Subjects:
Primary: 42A99

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 4 • 2019
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