A square function involving the center of mass and rectifiability

Abstract For a Radon measure $$\mu $$ μ on $$\mathbb {R}^d$$ R d , define $$C^n_\mu (x, t)= \left( \frac{1}{t^n} \left| \int _{B(x,t)} \frac{x-y}{t} \, d\mu (y)\right| \right) C n ( x , t ) = 1 ∫ B - y . This coefficient quantifies how symmetric the is by comparing center of mass at given scale and location to actual ball. We show that if n -rectifiable, then $$\begin{aligned} _0^\infty |C^n_\mu (x,t)|^2 \frac{dt}{t}< \infty \mu \text{-almost } \text{ everywhere }. \end{aligned}$$ 0 ∞ | 2 dt < -almost everywhere . Together with previous result Mayboroda Volberg, where they showed converse holds true, this gives new characterisation -rectifiability. To prove our main result, we also for an -uniformly rectifiable measure, $$|C_\mu ^n(x,t)|^2 \frac{dt}{t}d\mu Carleson $$\mathrm {spt}(\mu ) \times (0,\infty )$$ spt × that, whenever 1-rectifiable in plane, same Dini condition as above more general kernels. give uniform 1-rectifiability plane terms condition. uses classification $$\Omega Ω -symmetric measures from Villa (Rev Mat Iberoam, 2019).