Abstract:

Like Euclidean spaces, an (infinite) tree is equipped with a "Laplace operator", obtained by normalizing its adjacency operator (and subtracting the identity operator). Its kernel consist of harmonic functions, that have been widely studied in this environment too. The treee has a natural boundary, consisting of all "points at infinity", that is infinite geodesic rays starting at a fixed reference node. Harmonic functions can be represented as Poisson integrals of their "boundary values". This talk outlines a proof of the analogue on trees of a classical theorem (the local Fatou theorem) that links non-tangential convergence almost everywhere of harmonic functions to the finiteness of the so called "Lusin area function". The emphasis is on a wide class of non-homogeneous trees, where the Laplace operator induces a transient random walk, instead of the simpler case of homogeneous trees with isotropic Laplacian (that will be briefly mentioned too). This higher level generalization raises the question whether the same result would also hold for functions on the Euclidean disc that are in the kernel of elliptic differential operators other than the usual Laplacian.