Geometric analysis on Cantor sets and trees
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 complicated than in the classical situation. However, Cantor sets have an ultrametric structure, which leads to the density result. Section~6 deals with traces of Newtonian functions. Some results follow from \textit{A. Gogatishvili} et al. [Math. Nachr. 283, No. 2, 215--231 (2010; Zbl 1195.46033)]. However, the paper under review extends these results in the setting of the paper. The upshot of the section is to find the trace for Newtonian Sobolev functions. The authors identify which Besov space does the job. Furthermore, the obtained result is proved to be sharp. One goal of the article is to study compositions of Besov mappings and quasisymmetric mappings. Section~7 talks about quasisymmetric mappings and their Hölder and bi-Hölder regularity. Under certain restrictions it is shown that if \(f\) is bi-Hölder and \(u\) in a certain Besov space, then \(u\circ f\) is in a Besov space as well. The next section strengthens the result of the previous section by using that Besov functions are traces of Sobolev functions. This also makes it necessary to look at extensions of quasisymmetries from the boundary to the tree. The last section proves a result about extending a quasiisometry between rooted trees to the boundary such that the boundary mapping is quasisymmetric. Furthermore, it contains a rigidity result.
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