The Sobolev extension problem on trees and in the plane
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910491826192384 |
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| author | Carruth, Jacob Israel, Arie |
| author_facet | Carruth, Jacob Israel, Arie |
| contents | Let $V$ be a finite tree with radially decaying weights. We show that there exists a set $E \subset \mathbb{R}^2$ for which the following two problems are equivalent: (1) Given a (real-valued) function $ϕ$ on the leaves of $V$, extend it to a function $Φ$ on all of $V$ so that $||Φ||_{L^{1,p}(V)}$ has optimal order of magnitude. Here, $L^{1,p}(V)$ is a weighted Sobolev space on $V$. (2) Given a function $f:E \rightarrow \mathbb{R}$, extend it to a function $F \in L^{2,p}(\mathbb{R}^2)$ so that $||F||_{L^{2,p}(\mathbb{R}^2)}$ has optimal order of magnitude. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12097 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Sobolev extension problem on trees and in the plane Carruth, Jacob Israel, Arie Functional Analysis Classical Analysis and ODEs 46E35 Let $V$ be a finite tree with radially decaying weights. We show that there exists a set $E \subset \mathbb{R}^2$ for which the following two problems are equivalent: (1) Given a (real-valued) function $ϕ$ on the leaves of $V$, extend it to a function $Φ$ on all of $V$ so that $||Φ||_{L^{1,p}(V)}$ has optimal order of magnitude. Here, $L^{1,p}(V)$ is a weighted Sobolev space on $V$. (2) Given a function $f:E \rightarrow \mathbb{R}$, extend it to a function $F \in L^{2,p}(\mathbb{R}^2)$ so that $||F||_{L^{2,p}(\mathbb{R}^2)}$ has optimal order of magnitude. |
| title | The Sobolev extension problem on trees and in the plane |
| topic | Functional Analysis Classical Analysis and ODEs 46E35 |
| url | https://arxiv.org/abs/2406.12097 |