University of Oulu

Villa, M. A square function involving the center of mass and rectifiability. Math. Z. 301, 3207–3244 (2022).

A square function involving the center of mass and rectifiability

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Author: Villa, Michele1
Organizations: 1Research Unit of Mathematical Sciences, University of Oulu, P.O. Box 8000, 90014, Oulu, Finland
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 0.5 MB)
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Language: English
Published: Springer Nature, 2022
Publish Date: 2022-09-14


For a Radon measure \(μ\) on \(\mathbb {R}^d\), define \(C^n_\mu (x, t)= \left( \frac{1}{t^n} \left| \int _{B(x,t)} \frac{x-y}{t} \, d\mu (y)\right| \right)\). This coefficient quantifies how symmetric the measure \(μ\) is by comparing the center of mass at a given scale and location to the actual center of the ball. We show that if \(μ\) is \(n\)-rectifiable, then

\[\begin{aligned} \int _0^\infty |C^n_\mu (x,t)|^2 \frac{dt}{t}< \infty~~ \mu \text{-almost } \text{ everywhere }. \end{aligned}\]

Together with a previous result of Mayboroda and Volberg, where they showed that the converse holds true, this gives a new characterisation of \(n\)-rectifiability. To prove our main result, we also show that for an \(n\)-uniformly rectifiable measure, \(|C_\mu ^n(x,t)|^2 \frac{dt}{t}d\mu\) is a Carleson measure on \(\mathrm {spt}(\mu ) \times (0,\infty )\). We also show that, whenever a measure \(μ\) is 1-rectifiable in the plane, then the same Dini condition as above holds for more general kernels. We also give a characterisation of uniform 1-rectifiability in the plane in terms of a Carleson measure condition. This uses a classification of Ω-symmetric measures from Villa (Rev Mat Iberoam, 2019).

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Series: Mathematische Zeitschrift
ISSN: 0025-5874
ISSN-E: 1432-1823
ISSN-L: 0025-5874
Volume: 301
Issue: 3
Pages: 3207 - 3244
DOI: 10.1007/s00209-022-03003-w
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: M. Villa was supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. Open Access funding provided by University of Oulu including Oulu University Hospital.
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