Geometric analysis on Cantor sets and trees
type: Item
instance of: scholarly article
title: Geometric analysis on Cantor sets and trees
zbMATH Open document ID: 1376.46025
author: Anders Björn
author: Jana Björn
author: James T. Gill
author: Nageswari Shanmugalingam
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2017-04-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.0566
review text: Analysis on metric spaces is usually done on spaces that have some connectivity properties. The paper under review differs: the main metric spaces are Cantor sets. These sets can be seen as boundaries of rooted trees. A natural question is what the traces of first-order Sobolev spaces (defined on the trees) are on these Cantor sets. It turns out that these are Besov spaces. The authors investigate the quasisymmetric invariance of Besov spaces, which leads them to study quasiisometries between the boundaries of two rooted trees. The paper starts with introducing the trees that will be considered. Recall that a tree can be given a natural metric by saying that the distance between two vertices is the number of edges a geodesic connecting them has. There is also another common metric, the uniformizing metric, that gives weights to the edges. Namely, for \(\varepsilon>0\), \[ d_X(x,y)=\int_{[x,y]} e^{-\varepsilon |z|}\, d|z| \] is considered (\(d|z|\) stands for the measure that gives each edge Lebesgue measure \(1\)). Denoting by \(|x|\) the distance from the vertex~\(x\) to~\(0\), the authors verify that the weighted measure \[ d\mu(x)=e^{-\beta|x|}\, d|x| \] is doubling (where \(\beta>\log K\) and we consider a \(K\)-ary tree). In Section~4, the authors first recall the definition of the Newtonian Sobolev space. The purpose is to be able to state the Poincaré inequality on trees. After that, the authors verify that the trees satisfy a \(1\)-Poincaré inequality. Assuming that \(X\) is a regular \(K\)-ary tree and the uniformizing metric has parameter~\(\varepsilon\), the authors show that the boundary of \(X\) is an ultrametric space that is Ahlfors \(Q\)-regular, where \(Q=(\log K)/\varepsilon\). While the Newtonian Sobolev space ``lives'' on~\(X\), the considered Besov space ``lives'' on~\(\partial X\). The authors define the Besov space and provide different ways to calculate the norm. To show that Lipschitz continuous functions in the Besov space are dense is more…
reviewed by: Thomas Zürcher
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 30L99
zbMATH DE Number: 6704283
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: Newtonian spaces
zbMATH Keywords: embeddings
zbMATH Keywords: trace theorems
zbMATH Keywords: analysis on Cantor sets
Wikidata QID: Q109994362
MaRDI profile type: Publication
OpenAlex ID: W2109638356
arXiv ID: 1304.0566
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DOI: 10.1515/CRELLE-2014-0099
description: scientific article