The Sobolev extension problem on trees and in the plane

TL;DR

This paper establishes an equivalence between Sobolev extension problems on finite weighted trees and in the plane, linking discrete and continuous Sobolev space extensions with optimal norm control.

Contribution

It demonstrates the existence of a set in the plane where Sobolev extension problems on trees and in the plane are equivalent, connecting discrete and continuous Sobolev space theories.

Findings

Existence of a set in the plane for Sobolev extension equivalence.

Equivalence of extension problems on trees and in the plane.

Optimal order of magnitude for Sobolev norms in extensions.

Abstract

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 $\phi$ on the leaves of $V$, extend it to a function $\Phi$ on all of $V$ so that $||\Phi||_{L^{1,p}(V)}$ 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.