English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

For a Radon measure $\mu$ on $\mathbb{R}^d$, define $C^n_\mu(x, t)= \(\frac{1}{t^n} \ |\int_{B(x,t)} \frac{x-y}{t} \, d\mu(y)\ | \ )$. Thiscoefficient quantifies how symmetric the measure $\mu$ is by comparing thecenter of mass at a given scale and location to the actual center of the ball.We show that if $\mu$ is $n$-rectifiable, then  $ \int_0^\infty |C^n_\mu(x,t)|^2 \frac{dt}{t} < \infty \, \, \mu\mbox{-almosteverywhere}. $  Together with a previous result of Mayboroda and Volberg, where they showedthat the converse holds true, this gives a characterisation of$n$-rectifiability. To prove our main result, we also show that for an$n$-uniformly rectifiable measure, $|C_\mu^n(x,t)|^2 dt/t d\mu$ is a Carlesonmeasure on $\mathrm{spt}(\mu) \times (0,\infty)$. We also show that, whenever ameasure $\mu$ is $1$-rectifiable in the plane, then the same Dini condition asabove holds for more general kernels. Moreover, we give a characterisation ofuniform 1-rectifiability in the plane in terms of a Carleson measure condition.

A square function involving the center of mass and rectifiability