A square function involving the center of mass and rectifiability

Abstract

For a Radon measure $\mu$ on ${\mathbb{R}}^d$, define $C_{\mu }^n(x,t)=(\frac{1}{t^n}|{\int }_{B(x,t)}\frac{x-y}{t}d\mu (y)|)$. 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_μ(x,t)|^2 \frac{dt}{t} < \infty \, \, μ\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, $|C_{\mu }^n(x,t){|}^{2}dt/td\mu$ 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.

arXiv comment36 pages. Updated references, added a remark, slight change of format. To appear in Math. Zeitschrift