Planar $W^{1,\,1}$-extension domains
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913177305874432 |
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| author | Koskela, Pekka Rajala, Tapio Zhang, Yi Ru-Ya |
| author_facet | Koskela, Pekka Rajala, Tapio Zhang, Yi Ru-Ya |
| contents | We show that a bounded planar simply connected domain $Ω$ is a $W^{1,\,1}$-extension domain if and only if for every pair $x,y$ of points in $Ω^c$ there exists a curve $γ\subset Ω^c$ connecting $x$ and $y$ with $$ \int_γ\frac{1}{χ_{\mathbb R^2\setminus \partialΩ}(z)}\,ds(z) \le C|x-y|.$$ Consequently, a planar Jordan domain $Ω$ is a $W^{1,\,1}$-extension domain if and only if it is a $BV$-extension domain, and if and only if its complementary domain $\tilde Ω$ is a $W^{1,\,\infty}$-extension domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09167 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Planar $W^{1,\,1}$-extension domains Koskela, Pekka Rajala, Tapio Zhang, Yi Ru-Ya Functional Analysis Complex Variables 46E35, 30C20 We show that a bounded planar simply connected domain $Ω$ is a $W^{1,\,1}$-extension domain if and only if for every pair $x,y$ of points in $Ω^c$ there exists a curve $γ\subset Ω^c$ connecting $x$ and $y$ with $$ \int_γ\frac{1}{χ_{\mathbb R^2\setminus \partialΩ}(z)}\,ds(z) \le C|x-y|.$$ Consequently, a planar Jordan domain $Ω$ is a $W^{1,\,1}$-extension domain if and only if it is a $BV$-extension domain, and if and only if its complementary domain $\tilde Ω$ is a $W^{1,\,\infty}$-extension domain. |
| title | Planar $W^{1,\,1}$-extension domains |
| topic | Functional Analysis Complex Variables 46E35, 30C20 |
| url | https://arxiv.org/abs/2512.09167 |