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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2410.18422 |
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| _version_ | 1866913590962814976 |
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| author | Casey, Emily |
| author_facet | Casey, Emily |
| contents | Carleson's $\varepsilon^2$-conjecture states that for Jordan domains in $\mathbb{R}^2$, points on the boundary where tangents exist can be characterized in terms of the behavior of the $\varepsilon$-function. This conjecture, which was fully resolved by Jaye, Tolsa, and Villa in 2021, established that qualitative control on the rate of decay of the Carleson $\varepsilon$-function implies the existence of tangents, up to a set of measure zero. We prove that quantitative control on the rate of decay of this function gives quantitative information on the regularity of the boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18422 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative control on the Carleson $\varepsilon$-function determines regularity Casey, Emily Classical Analysis and ODEs Carleson's $\varepsilon^2$-conjecture states that for Jordan domains in $\mathbb{R}^2$, points on the boundary where tangents exist can be characterized in terms of the behavior of the $\varepsilon$-function. This conjecture, which was fully resolved by Jaye, Tolsa, and Villa in 2021, established that qualitative control on the rate of decay of the Carleson $\varepsilon$-function implies the existence of tangents, up to a set of measure zero. We prove that quantitative control on the rate of decay of this function gives quantitative information on the regularity of the boundary. |
| title | Quantitative control on the Carleson $\varepsilon$-function determines regularity |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2410.18422 |