Riesz transforms and the BAUPP and BWGL criteria for uniform rectifiability
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914068433993728 |
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| author | Tolsa, Xavier |
| author_facet | Tolsa, Xavier |
| contents | In this note it is shown that if $μ$ is an $n$-Ahlfors regular measure in $\mathbb R^{n+1}$ such that the $n$-dimensional Riesz transform is bounded in $L^2(μ)$ and the so-called BAUPP (bilateral approximation by unions of parallel planes) condition holds for $μ$, then $μ$ satisfies the BWGL (bilateral weak geometric lemma), and so $μ$ is uniformly $n$-rectifiable. In this way, one can solve the David-Semmes problem in codimension one without relying on the BAUP (bilateral approximation by unions of planes) criterion of David and Semmes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26547 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Riesz transforms and the BAUPP and BWGL criteria for uniform rectifiability Tolsa, Xavier Classical Analysis and ODEs 42B20, 28A75 In this note it is shown that if $μ$ is an $n$-Ahlfors regular measure in $\mathbb R^{n+1}$ such that the $n$-dimensional Riesz transform is bounded in $L^2(μ)$ and the so-called BAUPP (bilateral approximation by unions of parallel planes) condition holds for $μ$, then $μ$ satisfies the BWGL (bilateral weak geometric lemma), and so $μ$ is uniformly $n$-rectifiable. In this way, one can solve the David-Semmes problem in codimension one without relying on the BAUP (bilateral approximation by unions of planes) criterion of David and Semmes. |
| title | Riesz transforms and the BAUPP and BWGL criteria for uniform rectifiability |
| topic | Classical Analysis and ODEs 42B20, 28A75 |
| url | https://arxiv.org/abs/2509.26547 |