Geometric analysis on Cantor sets and trees

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.

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• Bounded geometry and $p$-harmonic functions under uniformization and hyperbolization
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• Dirichlet-to-Neumann maps on Trees
• Systems involving mean value formulas on trees
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• Characterization of trace spaces on regular trees via dyadic norms