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The Sobolev extension problem on trees and in the plane

Authors :
Carruth, Jacob
Israel, Arie
Publication Year :
2024

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.<br />Comment: 28 pages, 1 figure

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2406.12097
Document Type :
Working Paper