Boundary convergence of harmonic functions on homogeneous trees and (possibly) buildings
Massimo A. Picardello · 2009
We prove admissible convergence to the boundary of functions that are harmonic on subsets of an infinite homogeneous tree with respect to the adjacency operator. The approach is based on a discrete Green formula, suitable estimates for the Green and Poisson kernel and an analogue of the Lusin area function. This is joint work with L. Atanasi (TAMS 2008). A similar approach might work for harmonic functions on affine buildings. We give an outline of such an approach for the building of PGL3. Massimo Picardello (Mathematics Department, University of Roma “Tor Vergata”) Bou dary convergence of harmonic functions on homogeneous trees and (possibly) building Graz, June 29 , 2009 2 / 66 Area function in the half-plane To start, let us review the continuous case: the Lusin area theorem in the half-plane (or equivalently the disc). Let f be harmonic in the half-plane H+, for instance. ω ∈ R generic point in the real axis. Γα(ω) cone of width α in H+ with vertex in ω (note: in the hyperbolic distance of H+ this is a tube with axis {ω + iτ, τ > 0}) Area function: Aα(ω) := ∫