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 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) has optimal order of magnitude. Here, L1,p(V) is a weighted Sobolev space on V. (2) Given a function f:E → R, extend it to a function F ∈ L2,p(R²) so that ||F||_L2,p(R²) has optimal order of magnitude.