Uniform measures and uniform rectifiability

Xavier Tolsa

doi:10.1112/jlms/jdv013

Uniform measures and uniform rectifiability

Xavier Tolsa

Research output: Contribution to journal › Article › Research › peer-review

8 Citations (Scopus)

Abstract

© 2015 London Mathematical Society. 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.

Original language	English
Pages (from-to)	1-18
Journal	Journal of the London Mathematical Society
Volume	92
Issue number	1
DOIs	https://doi.org/10.1112/jlms/jdv013
Publication status	Published - 1 Jan 2014

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abstract = "{\textcopyright} 2015 London Mathematical Society. In this paper, it is shown that if μ is an n-dimensional Ahlfors-David regular measure in ℝd 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 ℝd. In particular, it is shown here that they satisfy the 'big pieces of Lipschitz graphs' property.",

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