Effective Reifenberg theorems in Hilbert and Banach spaces
A famous theorem by Reifenberg states that closed subsets of R[superscript n] that look sufficiently close to k-dimensional at all scales are actually C [superscript 0, γ] equivalent to k-dimensional subspaces. Since then a variety of generalizations have entered the literature. For a general measur...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer Science and Business Media LLC
2021
|
| Online Access: | https://hdl.handle.net/1721.1/129741 |
| _version_ | 1811074099599900672 |
|---|---|
| author | Edelen, Nicholas Naber, Aaron Valtorta, Daniele |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Edelen, Nicholas Naber, Aaron Valtorta, Daniele |
| author_sort | Edelen, Nicholas |
| collection | MIT |
| description | A famous theorem by Reifenberg states that closed subsets of R[superscript n] that look sufficiently close to k-dimensional at all scales are actually C [superscript 0, γ] equivalent to k-dimensional subspaces. Since then a variety of generalizations have entered the literature. For a general measure μ in R[superscript n], one may introduce the k-dimensional Jones’ β-numbers of the measure, where β [superscript k][subscript μ] (x, r) quantifies on a given ball B[subscript r](x) how closely in an integral sense the support of the measure is to living inside a k-dimensional subspace. Recently, it has been proven that if these β-numbers satisfy the uniform summability estimate ∫ [superscript 2][subscript] β[superscript k][subscript μ](x,r)[superscript 2] dr/r < M, then μ must be rectifiable with uniform measure bounds. Note that one only needs the square of the β-numbers to satisfy the summability estimate, this power gain has played an important role in the applications, for instance in the study of singular sets of geometric equations. One may also weaken these pointwise summability bounds to bounds which are more integral in nature. The aim of this article is to study these effective Reifenberg theorems for measures in a Hilbert or Banach space. For Hilbert spaces, we see all the results from R[superscript 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 ∫[superscript 2][subscript] β[superscript k][subscript μ] (x,r)[superscript 1] dr/r < M without power gain, then μ must again be rectifiable with measure estimates. Improving this estimate in order to obtain a power gain turns out to be a subtle issue. For k = 1 we will see for a uniformly smooth Banach space that if ∫ [superscript 2][subscript] β[superscript 1][subscript μ](x,r)[superscript α] dr/r < M[superscript α/2], where α is the smoothness power of the Banach space, then μ is again rectifiable with uniform measure estimates. |
| first_indexed | 2024-09-23T09:43:09Z |
| format | Article |
| id | mit-1721.1/129741 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:43:09Z |
| publishDate | 2021 |
| publisher | Springer Science and Business Media LLC |
| record_format | dspace |
| spelling | mit-1721.1/1297412022-09-30T16:26:27Z Effective Reifenberg theorems in Hilbert and Banach spaces Edelen, Nicholas Naber, Aaron Valtorta, Daniele Massachusetts Institute of Technology. Department of Mathematics A famous theorem by Reifenberg states that closed subsets of R[superscript n] that look sufficiently close to k-dimensional at all scales are actually C [superscript 0, γ] equivalent to k-dimensional subspaces. Since then a variety of generalizations have entered the literature. For a general measure μ in R[superscript n], one may introduce the k-dimensional Jones’ β-numbers of the measure, where β [superscript k][subscript μ] (x, r) quantifies on a given ball B[subscript r](x) how closely in an integral sense the support of the measure is to living inside a k-dimensional subspace. Recently, it has been proven that if these β-numbers satisfy the uniform summability estimate ∫ [superscript 2][subscript] β[superscript k][subscript μ](x,r)[superscript 2] dr/r < M, then μ must be rectifiable with uniform measure bounds. Note that one only needs the square of the β-numbers to satisfy the summability estimate, this power gain has played an important role in the applications, for instance in the study of singular sets of geometric equations. One may also weaken these pointwise summability bounds to bounds which are more integral in nature. The aim of this article is to study these effective Reifenberg theorems for measures in a Hilbert or Banach space. For Hilbert spaces, we see all the results from R[superscript 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 ∫[superscript 2][subscript] β[superscript k][subscript μ] (x,r)[superscript 1] dr/r < M without power gain, then μ must again be rectifiable with measure estimates. Improving this estimate in order to obtain a power gain turns out to be a subtle issue. For k = 1 we will see for a uniformly smooth Banach space that if ∫ [superscript 2][subscript] β[superscript 1][subscript μ](x,r)[superscript α] dr/r < M[superscript α/2], where α is the smoothness power of the Banach space, then μ is again rectifiable with uniform measure estimates. NSF (Grant DMS-1606492) 2021-02-10T19:38:46Z 2021-02-10T19:38:46Z 2018-11 2018-06 2020-09-24T20:46:49Z Article http://purl.org/eprint/type/JournalArticle 0025-5831 1432-1807 https://hdl.handle.net/1721.1/129741 Edelen, Nick et al. "Effective Reifenberg theorems in Hilbert and Banach spaces." Mathematische Annalen 374, 3-4 (November 2018): 1139–1218 © 2018 Springer-Verlag GmbH Germany, part of Springer Nature en https://doi.org/10.1007/s00208-018-1770-0 Mathematische Annalen Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ Springer-Verlag GmbH Germany, part of Springer Nature application/pdf Springer Science and Business Media LLC Springer Berlin Heidelberg |
| spellingShingle | Edelen, Nicholas Naber, Aaron Valtorta, Daniele Effective Reifenberg theorems in Hilbert and Banach spaces |
| title | Effective Reifenberg theorems in Hilbert and Banach spaces |
| title_full | Effective Reifenberg theorems in Hilbert and Banach spaces |
| title_fullStr | Effective Reifenberg theorems in Hilbert and Banach spaces |
| title_full_unstemmed | Effective Reifenberg theorems in Hilbert and Banach spaces |
| title_short | Effective Reifenberg theorems in Hilbert and Banach spaces |
| title_sort | effective reifenberg theorems in hilbert and banach spaces |
| url | https://hdl.handle.net/1721.1/129741 |
| work_keys_str_mv | AT edelennicholas effectivereifenbergtheoremsinhilbertandbanachspaces AT naberaaron effectivereifenbergtheoremsinhilbertandbanachspaces AT valtortadaniele effectivereifenbergtheoremsinhilbertandbanachspaces |