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arXiv · 1806.01250

Effective Reifenberg theorems in Hilbert and Banach spaces

Nicholas Edelen·Aaron Naber·Daniele Valtorta
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Abstract

The aim of this article is to study effective Reifenberg theorems for measures in a Hilbert or Banach space. For Hilbert spaces, we see all the results from $\mathbb{R}^n$ continue to hold with no additional restrictions. For a general Banach spaces we will see that the classical Reifenberg theorem holds, and that a weak version of the effective Reifenberg theorem holds in that if one assumes a summability estimate $\int_0^2 \beta^k(x,r)^1 \frac{dr}{r} 1$ any power gain at all may fail, even for uniformly smooth Banach spaces.

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Nicholas Edelen, Aaron Naber, Daniele Valtorta. 2018-06-04. Effective Reifenberg theorems in Hilbert and Banach spaces. https://arxiv.org/abs/1806.01250

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