On the Uniqueness of the Norton-Sullivan Quasiconformal extension
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910005108670464 |
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| author | Barrionuevo, José Afonso Gonçalves, Felipe Medeiros, José Victor Oliveira, Lucas |
| author_facet | Barrionuevo, José Afonso Gonçalves, Felipe Medeiros, José Victor Oliveira, Lucas |
| contents | We show that the extension map \[ \mathcal{E}_{NS}(f)(z)=\frac{f(x+y)+f(x-y)}{2}+i\frac{f(x+y)-f(x-y)}{2}\mbox{ for all }z=x+iy\in\mathbb{H}\,, \] defined by Norton and Sullivan in '96, is the only locally linear extension map taking bi-Lipschitz functions on $\mathbb{R}$ to quasiconformal functions on $\mathbb{H}$, modulo the action of a group isomorphic to the linear group. In fact, we discovered many other extension like this one (lying in the orbit of such group action), such as: $f(x)\mapsto f(x)+i(f(x)-f(x-y))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19805 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Uniqueness of the Norton-Sullivan Quasiconformal extension Barrionuevo, José Afonso Gonçalves, Felipe Medeiros, José Victor Oliveira, Lucas Classical Analysis and ODEs Complex Variables Functional Analysis 30C62, 30C70, 30C62 We show that the extension map \[ \mathcal{E}_{NS}(f)(z)=\frac{f(x+y)+f(x-y)}{2}+i\frac{f(x+y)-f(x-y)}{2}\mbox{ for all }z=x+iy\in\mathbb{H}\,, \] defined by Norton and Sullivan in '96, is the only locally linear extension map taking bi-Lipschitz functions on $\mathbb{R}$ to quasiconformal functions on $\mathbb{H}$, modulo the action of a group isomorphic to the linear group. In fact, we discovered many other extension like this one (lying in the orbit of such group action), such as: $f(x)\mapsto f(x)+i(f(x)-f(x-y))$. |
| title | On the Uniqueness of the Norton-Sullivan Quasiconformal extension |
| topic | Classical Analysis and ODEs Complex Variables Functional Analysis 30C62, 30C70, 30C62 |
| url | https://arxiv.org/abs/2409.19805 |