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

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A Correction to this article was published on 10 January 2019


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

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


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

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.

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Correspondence to Xavier Tolsa.

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Communicated by L. Ambrosio.

The author was supported by the ERC Grant 320501 of the European Research Council (FP7/2007-2013) and also partially supported by the Grants 2014-SGR-75 (Catalonia), MTM2013-44304-P (Spain), and by the Marie Curie ITN MAnET (FP7-607647).

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Tolsa, X. Characterization of n-rectifiability in terms of Jones’ square function: part I. Calc. Var. 54, 3643–3665 (2015).

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