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arXiv · 1310.0658

Uniform measures and uniform rectifiability

Xavier Tolsa
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Abstract

In this paper it is shown that if $μ$ is an n-dimensional Ahlfors-David regular measure in $R^d$ which satisfies the so-called weak constant density condition, then $μ$ is uniformly rectifiable. This had already been proved by David and Semmes in the cases n=1, 2 and d-1, and it was an open problem for other values of n. The proof of this result relies on the study of the n-uniform measures in $R^d$. In particular, it is shown here that they satisfy the "big pieces of Lipschitz graphs" property.

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Xavier Tolsa. 2014-12-16. Uniform measures and uniform rectifiability. https://doi.org/10.1112/jlms%2Fjdv013

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