A square function involving the center of mass and rectifiability
Michele Villa · Mathematische Zeitschrift · 2022
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 μ n ( x , t ) = 1 t n ∫ B ( x , t ) x - y t d μ ( y ) . This coefficient quantifies how symmetric the measure $$\mu $$ 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 $$\mu $$ μ is n-rectifiable, then $$\begin{aligned} \int _0^\infty |C^n_\mu (x,t)|^2 \frac{dt}{t}< \infty \mu \text{-almost } \text{ everywhere }. \end{aligned}$$ ∫ 0 ∞ | C μ n ( x , t ) | 2 dt t < ∞ μ -almost everywhere . 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, $$|C_\mu ^n(x,t)|^2 \frac{dt}{t}d\mu $$ | C μ n ( x , t ) | 2 dt t d μ is a Carleson measure on $$\mathrm {spt}(\mu ) \times (0,\infty )$$ spt ( μ ) × ( 0 , ∞ ) . We also show that, whenever a measure $$\mu $$ 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 $$\Omega $$ Ω -symmetric measures from Villa (Rev Mat Iberoam, 2019).