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Characterization of n-rectifiability in terms of Jones’ square function: Part II

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Abstract

We show that a Radon measure \({\mu}\) in \({\mathbb{R}^d}\) which is absolutely continuous with respect to the n-dimensional Hausdorff measure \({\mathcal{H}^n}\) is n-rectifiable if the so called Jones’ square function is finite \({\mu}\)-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all n-rectifiable measures which are absolutely continuous with respect to \({\mathcal{H}^{n}}\). Further, in this paper we also investigate the relationship between the Jones’ square function and the so called Menger curvature of a measure with linear growth, and we show an application to the study of analytic capacity.

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Correspondence to Jonas Azzam.

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

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Azzam, J., Tolsa, X. Characterization of n-rectifiability in terms of Jones’ square function: Part II. Geom. Funct. Anal. 25, 1371–1412 (2015). https://doi.org/10.1007/s00039-015-0334-7

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  • DOI: https://doi.org/10.1007/s00039-015-0334-7

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