The Sobolev extension problem on trees and in the plane

Abstract Let V be a finite tree with radially decaying weights. We show that there exists a set E ⊂ R 2 $E\subset {\mathbb{R}}^{2}$ 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 ) ${\Vert}{\Phi}{{\Vert}}_{{L}^{1,p}\left(V\right)}$ 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 ∈ L 2 , p ( R 2 ) $F\in {L}^{2,p}\left({\mathbb{R}}^{2}\right)$ so that ‖ F ‖ L 2 , p ( R 2 ) ${\Vert}F{{\Vert}}_{{L}^{2,p}\left({\mathbb{R}}^{2}\right)}$ has optimal order of magnitude.