Tangential limits of potentials on homogeneous trees (Q1868132)

Let \(\mathbf{T}\) be a homogeneous tree of homogeneity \(q+1\), and \(\Delta\) be the boundary of \(\mathbf{T}\), and \(G\) the Green function on \(\mathbf{T}\times\mathbf{T}\). The main result is: If we assume that \(f\) is nonnegative and \(Gf(s)=\sum G(s,t)f(t)\) is finite and that \(f\) satisfies a growth condition, \(\sum f(t)^{p}q^{-\gamma|t|}\), for some \(\gamma\). Then the limit of \(Gf(s)\) as \(s\) tends to \(b\in\Delta\) within a tangential region \(\Omega_{\tau,a}\), where the tangency \(\tau\) satisfies \(1\leq\tau\leq 1/\gamma\), is zero for all \(b\in\Delta\) except possibly for a set \(E\subset\Delta\) of zero \(\tau\gamma\)-dimensional Hausdorff measure.