The Sobolev extension problem on trees and in the plane - Carruth Jacob; Israel Arie (Advanced Nonlinear Studies)
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) ${\Vert}{Φ}{{\Vert}}_{L^{1,p}\left(V\right)}$ has optimal order of magnitude. Here, L 1,p (V) is a weighted Sobolev space on V. (2) Given a function f:E→R $f:E→ \mathbb{R}$ , 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.