Geometric analysis on Cantor sets and trees

TL;DR

This paper explores the geometric and functional analysis of Cantor sets viewed as boundaries of rooted trees, establishing precise relationships between Sobolev spaces, Besov spaces, and quasisymmetries, with implications for invariance properties.

Contribution

It provides explicit characterizations of Sobolev and Besov spaces on tree boundaries and establishes a correspondence between quasisymmetries and rough quasiisometries with sharp estimates.

Findings

Trace of Sobolev space is a Besov space with explicit smoothness.

Quasisymmetries extend to rough quasiisometries between trees.

Certain Besov spaces are invariant under quasisymmetries.

Abstract

Using uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets.