Characterization of n-rectifiability in terms of Jones’ square function: part I

Abstract

In this paper it is shown that if $\mu$ is a finite Radon measure in ${\mathbb R}^d$ which is n-rectifiable and $1\le p\le 2$, then

$$\begin{aligned} \displaystyle \int _0^\infty \beta _{\mu ,p}^n(x,r)^2\,\frac{dr}{r}<\infty \quad \mathrm{for}\,\, \mu {\text {-}}\mathrm{a.e.}\,\, x\in {\mathbb R}^d, \end{aligned}$$

where

$$\begin{aligned} \displaystyle \beta _{\mu ,p}^n(x,r) = \inf _L \left( \frac{1}{r^n} \int _{\bar{B}(x,r)} \left( \frac{\mathrm{dist}(y,L)}{r}\right) ^p\,d\mu (y)\right) ^{1/p}, \end{aligned}$$

with the infimum taken over all the n-planes $L\subset {\mathbb R}^d$. The $\beta _{\mu ,p}^n$ coefficients are the same as the ones considered by David and Semmes in the setting of the so called uniform n-rectifiability. An analogous necessary condition for n-rectifiability in terms of other coefficients involving some variant of the Wasserstein distance $W_1$ is also proved.