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).