A square function involving the center of mass and rectifiability

Abstract

For a Radon measure $\mu$ on $\mathbb{R}^d$, define $C^n_\mu(x, t)= \ (\frac{1}{t^n} \ |\int_{B(x,t)} \frac{x-y}{t} \, d\mu(y)\ | \ )$. This coefficient quantifies how symmetric the measure $\mu$ is by comparing the center 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{-almost everywhere}. $ Together with a previous result of Mayboroda and Volberg, where they showed that 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 Carleson measure on $\mathrm{spt}(\mu) \times (0,\infty)$. We also show that, whenever a measure $\mu$ is $1$-rectifiable in the plane, then the same Dini condition as above holds for more general kernels. Moreover, we give a characterisation of uniform 1-rectifiability in the plane in terms of a Carleson measure condition.