The Sobolev extension problem on trees and in the plane
Let V be a finite tree with radially decaying weights. We show that there exists a set E⊂R2 $E\subset {\mathbb{R}}^{2}$ 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‖L1,p(V) ${\Vert}{\Ph...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2025-01-01
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| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2023-0158 |
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| Summary: | Let V be a finite tree with radially decaying weights. We show that there exists a set E⊂R2
$E\subset {\mathbb{R}}^{2}$
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‖L1,p(V)
${\Vert}{\Phi}{{\Vert}}_{{L}^{1,p}\left(V\right)}$
has optimal order of magnitude. Here, L
1,p
(V) is a weighted Sobolev space on V. (2) Given a function f:E→R
$f:E\to \mathbb{R}$
, extend it to a function F∈L2,p(R2)
$F\in {L}^{2,p}\left({\mathbb{R}}^{2}\right)$
so that ‖F‖L2,p(R2)
${\Vert}F{{\Vert}}_{{L}^{2,p}\left({\mathbb{R}}^{2}\right)}$
has optimal order of magnitude. |
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| ISSN: | 2169-0375 |