SUFFICIENT CONDITIONS FORC1,α PARAMETRIZATION AND RECTIFIABILITY

摘要

We say a measure is C1, d-rectifiable if there is a countable union of C1, d-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in Rn to be C1, d-rectifiable, with (0,1]. The conditions involve a Bishop Jones type square function and all statements are quantitative in that the C1, constants depend on such a function. Along the way we also give sufficient conditions for C1, 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 Hlder and Lipschitz categories.