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
where
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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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). https://doi.org/10.1007/s00526-015-0917-z
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DOI: https://doi.org/10.1007/s00526-015-0917-z