A square function involving the center of mass and rectifiability 

Author:  Villa, Michele^{1} 
Organizations: 
^{1}Research 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) 
Persistent link:  http://urn.fi/urn:nbn:fife2022091459045 
Language:  English 
Published: 
Springer Nature,
2022

Publish Date:  20220914 
Description: 
AbstractFor a Radon measure \(μ\) on \(\mathbb {R}^d\), define \(C^n_\mu (x, t)= \left( \frac{1}{t^n} \left \int _{B(x,t)} \frac{xy}{t} \, d\mu (y)\right \right)\). 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 1rectifiable in the plane, then the same Dini condition as above holds for more general kernels. We also give a characterisation of uniform 1rectifiability in the plane in terms of a Carleson measure condition. This uses a classification of Ωsymmetric measures from Villa (Rev Mat Iberoam, 2019). see all

Series: 
Mathematische Zeitschrift 
ISSN:  00255874 
ISSNE:  14321823 
ISSNL:  00255874 
Volume:  301 
Issue:  3 
Pages:  3207  3244 
DOI:  10.1007/s0020902203003w 
OADOI:  https://oadoi.org/10.1007/s0020902203003w 
Type of Publication: 
A1 Journal article – refereed 
Field of Science: 
111 Mathematics 
Subjects:  
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, HeriotWatt University and the University of Edinburgh. Open Access funding provided by University of Oulu including Oulu University Hospital. 
Copyright information: 
© The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 
https://creativecommons.org/licenses/by/4.0/ 