Villa, M. A square function involving the center of mass and rectifiability. Math. Z. 301, 3207–3244 (2022). https://doi.org/10.1007/s00209-022-03003-w

### 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)
Language: English
Published: Springer Nature, 2022
Publish Date: 2022-09-14
Description:

# Abstract

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