2013/04/02 by Anders Björn, Jana Björn, James T. Gill +1
Mathematics · #Analytic and geometric function theory #Besov space #Boundary (topology) #Cantor set #Combinatorics #Computer science #Functional analysis #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Interpolation space #Mathematical analysis #Mathematics #Pure mathematics #Smoothness #Sobolev space #Space (punctuation) #TRACE (psycholinguistics) #Tree (set theory) #Type (biology) #Uniformization (probability theory) #math.FA #msc:30L05 #msc:30L10 #msc:31E05 #msc:46E35 #msc:51M10
paper · pdf · doi:10.1515/crelle-2014-0099
published as J. Reine Angew. Math. 725 (2017), 63-114
arxiv created 2013/04/02 · openalex publication_date 2014/11/28 · arxiv updated 2017/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.