Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 1065–1094
Institute for Advanced Study, School of Mathematics
Princeton, NJ 08540, U.S.A.; ghinassi 'at' math.ias.edu
Abstract. We say a measure is C1,α d-rectifiable if there is a countable union of C1,α d-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in Rn to be C1,α d-rectifiable, with α ∈ (0,1]. The conditions involve a Bishop–Jones type square function and all statements are quantitative in that the C1,α constants depend on such a function. Along the way we also give sufficient conditions for C1,α parametrizations for Reifenberg flat sets in terms of the same square function. Key tools for the proof come from David and Toro's Reifenberg parametrizations of sets with holes in the Hölder and Lipschitz categories.
2010 Mathematics Subject Classification: Primary 28A75 28A12; Secondary 26A16, 28A78, 42B99.
Key words: ölder, Lipschitz, parametrizations, C1,α rectifiable measures, C1,α rectifiable sets, Jones beta numbers, Jones square functions, Reifenberg sets.
Reference to this article: S. Ghinassi: Sufficient conditions for C1,α parametrization and rectifiability. Ann. Acad. Sci. Fenn. Math. 45 (2020), 1065–1094.
https://doi.org/10.5186/aasfm.2020.4557
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